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More than one nature for natural units

2011-11-30T09:31:00.001+01:00

Hey blog, long time no see!Bee has put together a nice video on natural units. There are one or two aspects that I would put slightly differently and rather than writing a comment I thought it might better be to write a post myself.

The first thing is that strictly speaking, there is not the natural unit system, it depends on the problem you are interested in. For example, if you are interested in atoms, the typical mass is that of the electron, so you will likely be interested in masses as multiples of $m_e$. Then, interactions are Coulomb and you will want to express charges as multiples of the electron charge $e$. Finally, quantum mechanics is your relevant framework, so it is natural to express actions in multiples of $\hbar$. Then a quick calculation shows that this unit system of setting $m_e=e=\hbar=1$ implies that distances are dimensionless and the distance $r=1$ happens to be the Bohr radius that sets the natural scale for the size of atoms. Naturalness here lets you guess the size of an atom from just identifying the electron mass, the electric charge and quantum mechanics to be the relevant ingredients.

When you are doing high energy particle physics quantum physics and special relativity are relevant and thus it is convenient to use units in which $\hbar=c=1$ which is Bee's example. In this unit system, masses and energy have inverse units of length.

If you are a classical relativist contemplating solutions of Einstein's equations, then quantum mechanics (and thus $\hbar$) does not concern you but Newton's constant $G$ does. These people thus use units with $c=G=1$. Confusingly, in this unit system, masses have units of length (and not inverse length as above). In particular, the length scale of a black hole with mass M, the Schwarzschild radius is $R=2M$ (the 2 being there to spice up life a bit). So you have to be a bit careful when you convert energies to lengths, you have to identify if you are in a quantum field theory or in a classical gravity situation.

My other remark is that it is conventional how many independent units you have. Many people think, that in mechanics you need three (e.g. length, mass and time, meters, kilograms and seconds in the SI system) and a fourth if you include thermodynamics (like temperature measured in Kelvins) and a fifth if there is electromagnetism (like charge or alternatively current, Amperes in SI). But these numbers are just what we are used to. This number can change when we change our understanding of a relation from "physical law" to "conversion factor". The price is a dimensionful constant: In the SI system, it is a law that in equipartition of energy $E=\frac 12k_bT$ and Coulombs law equates a mechanical force to an electrostatic expression via $F=\frac{qQ} 1{4\pi\epsilon_0r}$ and it is a law that light moves at a speed $c=s/t$.

But alternatively, we could use these laws to define what we actually mean by Temperature (then measured in units of energy), charge (effectively setting $4\pi\epsilon_0$ to unity and thereby expressing charge in mechanical units) and length (expressing a distance by the time light need to traverse it). This eliminates a law and a unit. What remains of the law is only the fact that one can do that without reference to circumstances, that a distance from here to Paris does not depend for example on the time of the year (and thus on the direction of the velocity of the earth on its orbit around the sun and thus potentially relative to the ether). If the speed of light would not be constant and we would try to measure distances by the time it takes light to traverse them then distances would suddenly vary when we would say that the speed of light varies.

There is even an example that you can increase the number of units to more than what we are used to (although a bit artificial): It is not god given what kinds of things we consider 'of the same type' and thus possible to be measured in the same units. We are used to measuring all distances in the same unit (like for example meters) or derived units like kilometers or feet (with a fixed numerical conversion factor). But in nautical situations it is common to treat horizontal distance to be entirely different from vertical distances. Horizontal distances like the way to the next island you would measure in nautical miles while vertical distances (like the depth of water) you measure in fathoms. It is then a natural law that the ratio between a given depth and a given horizontal distance is constant over time and there is dimensionful constant (fathoms per mile) of nature that allows to compute a horizontal distance from a depth.

Bitcoin explained

2011-06-03T15:53:00.000+02:00

As me, you might have recently heared about "Bitcoin", the internet currency that tries to be safe without a central authority like a bank or a credit card company that say which transactions are legitimate. So far, all mentions in blogs, podcasts or the press that I have seen had in common that they did not say how it works, what are the mechanisms that make sure Bitcoins operate like money. So I looked it up and this is what I found:

Bitcoin uses to cryptographic primitives: hashes and public key encryption. I case you don't know what these are: A hash is a function that reads in a string (or file or number, those are technically all the same) and produces some sort of checksum. The important properties are that everybody can do this computation (with some small amount of effort) and produce the same checksum. On the other hand, it is "random" in the sense that you cannot work backwards, i.e. if you only know the checksum you effectively have no idea about the original string. It is computationally hard to find a string for a given checksum (more or less the best you can do is guess random strings, compute their checksums until you succeed). A related hard problem is to find such a string with prescribed first $N$ characters.

This can be used as a proof of effort: You can pose the problem to find a string (possibly with prescribed first characters) such that the first $M$ digits of the checksum have a prescribed value. In binary notation you could for example you could ask for $M$ zeros. Then on the average you have to make $2^M$ guesses for the string until you succeed. Presenting such a string then proves you have invested an effort of $O(2^M)$. The nice thing is that this effort is additive: You can start your string with the characters "The message '....' has checksum 000000xxxxxxxxxxx" and continue it such that the checksum of the total string starts with many zeros. That proves that in addition to the zeros your new string has, somebody has already spent some work on the string I wrote as dots. Common hash functions are SHA-1 (and older and not as reliable: MD5).

The second cryptographic primitive is public key encryption. Here you have two keys $A$, the public key which you tell everybody about and $B$ your secret key (you tell nobody about). These have the properties that you can use one of the keys to "encrypt" a string and then the other key can be used to recover the original string. In particular, you need to know the private key to produce a message that can be decrypted with the public key. This is called a "signature": You have a message $M$ and encrypt it using $B$. Let us call the result $B(M)$. Then you can show $A$ and $M$ and $B(M)$ to somebody to prove that you are in possession of $B$ without revealing $B$ since that person can verify that $B(M)$ can be decrypted using $A$. Here, an example is the RSA algorithm.

Now to Bitcoin. Let's go through the list of features that you want your money to have. The first is that you want to be able to prove that your coins belong to you. This is done by making coins files that contain the public key $A$ of their owner. Then, as explained in the previous paragraph you can prove that you are the legitimate owner of the private key belonging to that coin and thus you are its owner. Note that you can have as many public-private key pairs as you like possibly one for every coin. It is just there to equate knowing of a secret (key) to owning the coin.

Second you want to be able to transfer ownership of the coin. Let us assume that the recipient has the public key $A'$. Then you transfer the coin (which already contains your public key $A$) by appending the string "This coin is transfered to the owner of the secrete key to the public key $A'$". Then you sign the whole thing with your private key $B$. The recipient can now prove that the coin was transferred to him as the coin contains both your public key (from before) and your statement of the transfer (which only you, knowing $B$ can have authorized. This can be checked by everybody by checking the signature). So the recipient can prove you owned the coin and agreed to transfer it to him.

The last property is that once you transfered the coin to somebody else you cannot give it to a third person as you do not own it anymore. Or put differently: If you try to transfer a coin a second time that should not work and the recipient should not accept it or at least it should be illegitimate.

But what happens if two people claim they own the same coin, how can we resolve this conflict? This is done via a public time-line that is kept collaboratively between all participants. Once you receive a coin you want to be able to prove later that you already owned it at a specific time (in particular at the time when somebody else claims he received it).

This is done as follows: You compute the hash function of the transfer (or the coin after transfer, see a,bove including the signature of the previous owner of the coin that he has given it to you) and add it to the time line. This means you take the hash value of the time line so far, at the hash of the transfer and compute new hash. This whole package you then send to your network peers and ask them to also include your transfer in their version of the time line.

So the time line is a record of all the transfers that have happened in the past and each participant in the network keeps his own copy of it.

There could still be a conflict when two incompatible time lines are around. Which is the correct one that should be trusted? One could have a majority vote amongst the participants but (as everybody knows from internet discussions) nothing is easier than to come up with a large number of sock puppets that swing any poll. Here comes the proof of work that I mentioned above in relation to hash functions: There is a field in the time line that can be filled with anything in the attempt to construct something that has a hash with as many zeros as possible. Remember, producing $N$ leading zeros amounts to $O(2^N)$ work. Having a time line with many zeros demonstrates that were willing to put a lot of effort into this time line. But as explained above, this proof of effort is additive and all the participants in the network continuously try to add zeros to their time line hashes. But if they share and combine their time lines often enough such that they stay coherent they are (due to additivity) all working on fining zeros on the same time line. So rather than everybody working for themselves everybody works together as long as their time lines stay coherent. And going back through a time line it is easy to see how much zero finding work has been but in. Thus in the case of conflicting time lines one simply takes that that contains more zero finding work. If you wanted to establish an alternative time line (possibly one where at some point in time you did not transfer a coin but rather kept it to yourself so you could give it to somebody else later) to establish it you would have to outperform all other computers in the network that are all busy working on computing zeros for the other, correct, time line.

Of course, if you want to receive a bitcoin you should make sure that in the generally accepted time line that same coin has not already been given to somebody else. This is why the transfers take some time: You want to wait for a bit that the information that the coin has been transferred to you has been significantly spread on the network and included in the collective time line that it cannot be reversed anymore.

There are some finer points like how subdividing coins (currently worth about 13 dollars) is done and how new coins can be created (again with a lot CPU work) but I think they are not as essential in case you want to understand the technical basis of bitcoin before you but real money in.

BTW, if you liked this exposition (or some other here) feel free to transfer me some bitcoins (or fractions of it). My receiving address is

19cFYVExc2ZS4p7ZARGyENFijnV43y6ts1

Mixed superrationality does not beat pure in prisoner's dilemma

2011-03-24T16:23:00.000+01:00

The prisoner's dilemma is probably one of the most famous toy games of game theorists. It amounts to two criminals that being caught by the police are interrogated individually are offered the following deal: If both remain silent ("cooperate" with each other) both go to prison for $S$ ('short') years for small crimes that the police can prove. But if one prisoner admits the big crime ("defects") he goes free and the other spends $L$ ('long') years in prison. But if both admit the crime they both face a $M$ ('middle') year sentence. To be a dilemma the sentences should obey $0<S<M<L$ and by picking an appropriate normalisation of the unit of time, we can set $S=1$.

The standard (economist) analysis of the game goes as follows: I assume that the other prisoner has already made his decision. Then, no matter what he decided I am better off by defecting: If he cooperates, my choice is between going free and $S$ years while if he is defecting I can choose between $M$ and $L$. So I defect and he comes to the same conclusion, so we end up spending $M$ years in prison. Both defecting is in fact a Nash equilibrium.

That's not too exciting, as we could do better by both cooperating and serving only $S$ years, which is Pareto optimal but unstable because there is the temptation for each player to defect and then go free. So much for the classic analysis of this game (not iterated) which is a model for many decision problems where one has to decide between a personal advantage or the global optimum.

I first learned about this game many many years ago when still attending high school from a Douglas Hofstaedter column in the Scientific American. He makes the following observation: When defecting, I am counting on the fact that the other prisoner is not as clever as me. It only pays if the situation is asymmetric. But since the other prisoner is faced with the same problem, he will come up with the same solution so the asymmetric case of one player cooperating and the other defecting will not occur. Thus the only real possibilities are both cooperating (yielding $S$ years) and both defecting (yielding $M$ years) of which the obvious better choice is to cooperate. Hofstadter calls this argument "superrational". It is the realization that in the analysis of the Nash equilibrium the idea that my decision is independent of the other prisoner's decision might be wrong.

Then Hofstadter points out another version of this game: You receive a letter from a very rich person stating that she is studying human intelligence and she figured that you are one of the top ten intelligent people in the world. She offers you (and also the other nine top-brainers) the following game: On the bottom of the letter is a coupon. You can either ignore the letter (in which case nothing more will happen) or you write your name on the coupon and send it back. If out of the ten possible coupons she receives exactly one she gives the person who returned the coupon 100 Million dollars. If any other number of coupons arrive until the end of this year nobody will receive any money. And as a warning: You are watched over by a number of private investigators. If they notice you trying to find out who the other nine people are the whole thing is called off and again nobody will get any money. So don't even think about it.

This does not look very promising: Obviously, if you don't send in the coupon you won't get any money. So you have to send the coupon but so will the other nine and again you will receive nil. Too bad.

Well, unless you widen your strategy space and besides 'pure', deterministic strategies you also allow for 'mixed', i.e. probabilistic strategies. You could for example come up with the following strategy: You roll dice and then send the coupon only with probability $p$. Let's see which $p$ optimizes your expectation assuming the other nine player follow the same strategy: You only get the money if you send the letter (probability $p$) and all nine other don't (probablity $(1-p)^9$) so the expectation is $E=p(1-p)^9$. Setting to zero the $p$ derivative of $E$ gives $0=(1-p)^9-9p(1-p)^8=(1-p)^8(1-p-9p)$ thus $p=1/10$. So you could prepare ten envelopes but only one with the coupon and mail a random one of these to optimize your expectation.

But with this idea of taking into account also mixed strategies we can go back to the prisoner's dilemma and see what happens when both players defect with probability $p$ (this is the new part of the story I came up with this morning under the shower. Of course, I do not claim any originality here). Then the expected number of years I spend in prison is $p^2M+Lp(1-p)+(1-p)^2$. Quick check for $p$ being 0 or 1 I get back the two deterministic values. So can I do better? Obviously, this is a quadratic function of $p$ going through $(0,1)$ and $(1,M)$. So it has is minimum in the interior of the range $p\in[0,1]$ if the slope at $p=0$ is negative (remember $M>S=1$). But the slope is $2(M-L+1)p+L-2$ which is positive as long as $L>2$. But this is really the interesting parameter range for the game since for $L<2$ it is better for both players to always switch between cooperate-defect and defect-cooperate since the average sentence in the asymmetric case is shorter than the one year sentence of both cooperating. So, unless that is the case, always cooperating is still the better symmetric strategy of superrational players than the probabilistic ones.

Formulas in Blogger

2011-03-15T16:34:00.001+01:00

To include formulas in blogger.com I have so far used mimetex which uses an external server running a cgi-script to convert TeX-style formulas to picutres.

This did its job most of the time except that mathphys, the old machine in Bremen that hosted my mimetex service died a couple of months ago and that the formulas have that stupid box around them which is particularly annoying for single symbols (this could probably be fixed by investing some time staring at the stylesheet for this blog). This is very much 8bit pixel style and does not scale nicely but I never touched it since it allowed you to read what I wrote.

Now, some reader suggested MathJax which I try out here:

Let's start witha wave function $\psi$, we define the velocity field $\vec v= \frac1{2m}\Im(\frac{\nabla \psi}{\psi})$. This leads to a conserved current:
$$\frac{\partial\rho}{\partial t}= -\vec\nabla\cdot (\bar\psi\psi\vec v).$$
At first, I thought it does not work but it just takes some time to reload.

Bohmian mechanics threatend by Occam's razor

2011-03-02T17:46:00.001+01:00

Last semester, I have been running a seminar on "Foundation of Quantum Mechanics" (wiki page) for TMP students that had been disappointed that "Mathematical Quantum Mechanics" was not on foundations.

Overall, I am quite satisfied with the outcome. We had covered several approaches to foundational issues, in particular the relation of quantum to classical physics and here specifically the "measurement problem" (which I am convinced is not a problem but is explained withing quantum theory by decoherence). We will produce a reader with all the contributions and I myself will write some introduction (which I will post here as well once it is finished).

But today, want to discuss Bohmian mechanics which was one of the topics and which has strong support by some local experts. I never really cared about this approach (being one of the Gallic villages where a small group of people know they are doing it better than the rest of the ignorant world, much like algebraic QFT or loop quantum gravity) being satisfied with quantum physics without any extras.

But now was the time to find out what Bohmian mechanics is really about and in this post I would like to share my findings. The big question everybody asks really is "do they make any predictions that differ from usual quantum mechanics i.e. can it be distinguished by some sort of experiment or is it just an alternative interpretation?" but unfortunately I do not have a final answer. But more below.

Before I start, let me put it a bit in perspective: Inequalities of Bell type (an I would include the Kochen-Specker theorem and GHZ type experiments) show in effect that the world cannot be both "realistic" and "local". Realistic means here that all properties have values at any instant of time irrespective of whether they are measured or not while local means that any decision I take here and now (for example whether I measure the or component of the spin of my half of an EPR singlet state) cannot influence measurements that are so far away that they cannot be reached even at the speed of light.

Thus one has to give up either realism or locality. The common interpretation of quantum mechanics gives up realism, the component of the spin does not have a value when I measure the component but is local. In some of the popular literature you will find statements to the contrary but they are mistaken: It is true, there can be non-local correlations. But this is no different from classical physics: Most of the time the color of the sock on my right foot is correlated with the color of the sock on the left foot, even at the same instant of time (when they are space-like to each other). But the question of locality is not about states (which are always global) it is about operators or measurements. And measuring the color (as compared for example to the size) of one of the socks does not influence the other sock, the local operators do commute.

Bohmian mechanics insists on realism and the price it has to pay is to give up is locality. It does not violate causality in an obviously measurable way but doing the - or -measurement here influences what happens far far away. But enough of these philosophical remarks, let's look at some formulas.

In its pure form, Bohmian mechanics is about non-relativistic systems of particles with Hamiltonian of the form . Everybody knows that the norm-squared wave function in position representation gives the probability distribution of finding particle 1 at , particle 2 at etc. and there is a conserved current for this density. That is if you start with some distribution at an initial time then wait a bit while you flow according to the current you end up with the new at a later time.

The new thing for the Bohmians is to interpret this current as an actual current of particles with velocities . According to the Bohmians, these particles with joint coordinates are dots that for example show up on the screen of a double slit experiment. Obviously, if you start with a probability distribution of particle positions given by at an initial time and follow the deterministic flow equation for above, then at any later time the particles will be distributed according to . The Bohmians claim, that there are really particles and at any instant of time their position is and the velocity is no matter whether they are measured or not. That's it.

A few trivial remarks: This theory is non-local as the velocity of the -th particle does depend via the wave function on the positions of all the other particles. Bohmians say that this is not to worry about since their theory is non-relativistic and this is like for example the Coulomb interaction in non-relativistic quantum mechanics where the force on one electron depends on the instantaneous positions of the other charged particles.

The next remark is that in Bohm's theory there is also the wave function that follows the same Schroedinger equation as in usual quantum mechanics. Thus any question involving only the wave function trivially gives the same answer as in quantum mechanics. The equation of motion for the particle positions which are the new ingredient in the Bohm theory depend on the wave function but not the other way around. There is no feed-back and the wave function does not know about the . Any quantum mechanical measurement that in the end measures position (like for example Stern-Gerlach) gives the same result as the follow the wave function that determines the outcome in the usual interpretation.

All observables that are functions of the coordinates at one instant of time do commute with each other and one can thus give them all sharp values at that instant of time. Thus there is no problem with claiming those positions are even in the usual interpretation.

Position measurements at different times in general do not commute and thus they have no common meaning. Thus the only hope to find disagreement is in experiments that in the Bohmian interpretation require sharp positions at different instants of time.

When it comes to spin I have the impression that the Bohmians cheat a bit: They declare that "spin is no a property of a point-like particle" meaning that realism does not apply to the different components and like in the usual interpretation, the components do not have a meaning unless measured. One can read this as a manifestation of the preferred role the Bohmians give to observables that are a function of the position operators over all other operators. In effect they claim only those position observables deserve realism.

Of course, one can reformulate the Bell type experiments mentioned above in terms of positions (e.g. by translating spins into positions via Stern-Gerlach set-ups) but then the non-local flow equation seems to prevent any obvious contradictions with quantum mechanics.

There are more formal problems: For time-reversal invariant Hamiltonians, one can always choose the eigenfunctions of the Hamiltonian to be real. Thus for the wave-function to be such an eigenfunction , the particles don't move, even in i.e. the Coulomb field of a hydrogen atom. You may say that this is not the classical world but the quantum world and there are other equations of motion but I must say I find particles standing still even in the presence of forces a bit strange.

That that brings us to my main criticism: It is not clear to me how to observe the particle at . Do experiments measure the wave function (via ) or do they measure ? And if so, can I prepare (and later measure) without significantly disturbing the wave function? If that is the case I can of course check whether I put an electron in a hydrogen atom in an energy eigenstate at some and later check whether I find it at some other place (which quantum mechanics would predict).

There are if course ways to wiggle out: You could argue that this experiment is impossible since I would always disturb the wave function significantly by placing a particle at and thus everything get screwed up.

But this excuse is pretty much equivalent to "you cannot observe (directly)". But then we are adding something (the particles at positions ) to our theory which is not observable. And that sounds to me to be directly threatened by Occam's razor.

Anyway. Unless somebody explains to me how to measure , I maintain that adding to the theory is as good as adding invisible angels.

Update: The promised write-up is here.

Picking the bigger number of two even if one is unknown

2010-11-16T16:14:00.001+01:00

Here is a nice problem from the xkcd blog: Two real numbers, A and B, are drawn using some unknown, possibly probabilistic process and written on papers that go into two envelopes. You randomly pick one and open it to find some number on it. You now have to decide whether you want to receive that number as an amount in dollars or rather the number that is in the other envelope (which is still sealed). Can you come up with a process that with probability >50% picks the larger amount?

Think about it.

SPOILER ALERT

You can. You will need some function f that maps the real numbers to the open interval (0,1) in a strictly monotonic way. You could for example take f(x) = (1+tanh(x))/2. Assume the number that you found in the first envelope was X. Then throw an unfair coin such that with probability f(X) you keep X and otherwise take the other envelope. Obviously (?), if you started with the envelope with the smaller number you are more likely to switch than if you had started with the larger number.

This sounds a bit counter intuitive. How can you increase your expected payoff if you know nothing about the number in the second envelope?

You might have the idea that something fishy is going on. What comes to mind is for example that you can produce paradoxes when assuming that there is a uniform probability distribution on the reals (or integers). But I believe, that this is not what is going on here since I did not say how the numbers were picked. They could have been picked with any perfectly fine probability measure on the reals, nobody said all numbers were equally likely. Below I will compute the expected outcome for any probability distribution that might have been used and it always works, not just in average.

More precisely, I think this is unrelated to a similar puzzle: In that second puzzle, there are also two envelopes that contain numbers representing payout but none is opened but instead it is known that the number in one envelope is twice the number in the other. You just don't know if you have the half oder double. There, assuming your envelope contains X then you could be tempted to argue that with probability 50% the other contains 2X and with 50% it contains X/2 and thus the expectation value is 50% x 2 X + 50% x X/2= 5/4 X and thus you could increase your expectation by 25% by switching. But then you could increase it by another 25% by using the same argument again and switching back.

In the second puzzle, it is really the implied uniform distribution of X's that is the origin of the paradox: You can see this by giving the additional information that both numbers are definitely smaller than 100 trillion dollars. That sounds like a trivial information but note that the calculation of the expectation value changes: If X is greater than 50 trillion dollars, you know with certainty that the other number cannot be 2X and thus the expectation of taking the other envelope is nor 125%X but X/2. If you now carefully go through the expectation value calculation you will find that averaged over all values of X the expectation for switching is the same as for keeping the first envelope.

Some of my readers will notice that the second puzzle is related to recent arguments that were made in the Landscape scenario about the imminent end of the world.

Back to the first game. Let's do some calculation to compute the expectation of the outcome. We will assume that the numbers were picked according to some probability measure rho(x)dx and that has a finite expectation value, i.e. the integral E=E(X)=int x rho(x)dx converges.

Then the expected outcome of the strategy above is X with probability f(X) and E with probability (1-f(X)) (as in that case we take the number in the second envelope).

We can now compute the expectation E(f(X) X + (1-f(X))E)= E(f(X)X) + E - E E(f(X)). For simplicity assume that E=0. Otherwise we could pay out E immediately and then subtract E from all number in the envelopes. Thus the expected payout of our strategy is E(f(X)X) but it is easy to see that this is positive (and thus we make more than the average E=0): In computing

E(f(X)X) = int f(x) x rho(x) dx

we can for x<0 overestimate f(x) by f(0) and for x>0 underestimate f(x) by f(0) and then conclude (unless rho(x) = delta(x) and we always spit out 0s)

E(f(X)X) > f(0)E(X) = 0

Thus we do better on the average than by deciding for one envelope or the other not taking into account the contents of the first one.

Not that the difference to the first puzzle is that this works for any rho(x), we did not have to assume some (non existent) uniform rho(x) and the effect does not go away as soon as a cut-off is introduced contrary to the other puzzle.

Is there a prisoner's dilemma in vaccination?

2010-10-15T13:51:00.000+02:00

Recently, I have been thinking about vaccination strategies as I was confronted with opinions which I consider at least very risk inviting. Not to spill oil in the fire I will thus anonymize illnesses and use made up probabilities. But let me assure you that for the illness I have in mind the probabilities are such that the story is similar.

Let's consider illness X. For simplicity assume that if you meet somebody with that illness you'll have it yourself a bit later with 100% probability. If you have X then in 1 in 2000 cases you will develop complication C which is lethal. But C itself is not contagious.

Luckily, there exists a vaccination against X that is 100% effective, i.e. if vaccinated you are immune to X. But unfortunately, the veccination itself causes the deadly C in 1 in a million cases.

So, the question is: Should you get vaccinated?

Unfortunately, the answer is not clear: It depends on the probability that if not vaccinated you will run into somebody spreading X. If X is essentially eradicated there is no point in taking the vaccination risk but if X is common it is much safer to vaccinate.

The break even is obviously when that probability is 1 in 500. If it's less likely to meet somebody then it would be to your advantage not to vaccinate.

Unfortunately, the probability of meeting an X infected person depends on how well people are vaxinated: As X is so contagious, if the vaccination rate drops the probability of meeting somebody with X dramatically increases. That is, not vaccinating yourself might be profitable for you but if everybody follows the same strategy the vaccination rate might drop and the society as a whole will see many more cases of C.

If you assume in addition that your information is not perfect and you might be wrong in estimating the probabilities involved it is not clear to me to which fixed point this system evolves.

But it seems likely to me that there are situations where for the society as a whole it is much better if you get vaccinated even if this increases you personal risk of encountering C.

Opinions?

Tensor factor subalgebras question

2010-09-14T17:24:00.000+02:00

After some serious arm-twisting performed by some TMP students I accepted to run a seminar on Foundations of Quantum Mechanics under the condition that it would be a no-nonsense class. This is in addition to the String Theory Lectures I have to teach as well.

After coming back from our vacation and workshop I found myself in the situation that I had much more fun doing some reading for the seminar (reviews on decoherence etc) than preparing for the string class (given that I have already twice taught Intro to Strings and that there are David Tong's wonderful lecture notes which give you the impression that you could take a few pages of those and be well prepared for class).

In the preparation, I came across a wonderful video of a lecture by a well known physicist (I will link this later as I might take part of this as a quiz for the first session. Let me just mention that it contains the clearest version of Bell's inequality that I a am aware of).

I am convinced that a lot of possible confusion about quantum physics together with locality (let me only mention the three letters E, P and R) comes from the fact that people confuse the roles of observables and states: Observables can be local and causality is built in by asking operators localised at space like distances to commute while states are always global objects. There is nothing like "the wave function of electron 1" or only in the approximation where you ignore all the other particles. You cannot use it when talking about correlations etc. But this is not bad, even in classical (statistical) physics, there are non-local correlations, like the colors of the socks on my two feet. The fact that in addition to correlations, there can be entanglement in the quantum theory does not change that.

Furthermore, I find it helpful to think (of course I did not come up with this approach) of the Hilbert space (and its wave functions) as a secondary object and take the observables as a starting point (and not derived as the operators acting on the wave functions). Those then are the elements of a (C*)-algebra and the Hilbert space only arises as a representation of that algebra. Stone and von Neumann for example then tell you that there is essentially a unique representation if the algebra is that of canonical commutation relations.

States are then functionals w that map each observable A to a complex number w(A) (interpreted as the expectation value). This linear function has to be normalised, w(1)=1 and positive meaning that for all A one has w(A^* A)>=0 (did I tell you that formuals are broken?). Then the GNS construction is similar to a highest weight representation: Using w and the algebra, one can construct a Hilbert space: As a vector space you can take the algebra. It is a representation after defining the action to be simply left multiplication. The scalar product of the elements A and B can be given by w(A^* B). Positivity of w tells you this is at least positive semi-definite. One can quotient out the zero-space to obtain something potitive definite and then employ some C*-magic to show that the action by left multiplication can be lifted to the quotient. I have suppressed some topological fine-print here like taking completions etc.

The states correspond in general to density matrices (or reducible representations) and as always can be convex combined as x w1 + (1-x) w2, the extremal states corresponding to irreducible representations.

In quantum information applications (as well as EPR and decoherence), one often starts with a Hilbert space that is a tensor product H = H1 x H2. Restricting attention to the first factor only corresponds to taking the partial trace over H2 and in general turns pure states on H into mixed states on H2. This has the taste of "averaging over all possible states of H2" but in the algebraic formulation if becomes clear that one is only restricting a state w to the subalgebra of operators of the form A1 x id where id is the identitiy on H2.

What I do not understand yet and where I am asking your help is the following: How does the splitting into tensor factors really work on the algbraic side? In particular, assume I have a C*-algebra C and a pure state w. Now I take some subalgebra C1 of C and obtain a new state w1 on C1 by restricting w to this subalgebra. What is the relation of the two Hilbert spaces H and H1 I obtain from the GNS construction on w and and w1 respectively? What is a sufficient condition on C1 that I can regard H1 as a tensor factor of H as above?

A necessary condition are obviously dimensions in the finite dimensional case: Here, the C*-algebras are just the complex matrix algebras of size n x n and the irreducible representation is on C^n. This is only a non-trivial tensor product if n is not prime. But nothing stops me for example to start with the big algebra being the 17x17 matrices and the subalgebra being those matrices that have the last row and column filled with zeros. But C^16 is definitely not a tensor-factor of C^17.

Back from silence

2010-09-14T16:22:00.000+02:00

It has been very silent here recently (or not so recently) but there is no particular reason for this except that I have been busy with other things (including an update on my facebook relationship status) and small things have been posted to Twitter rather than the blog.

And if one is not constantly taking care of things they tend to degrade. So is this blog. What happened is that mathphys.jacobs-university.de, the computer that I have been using to host (background and other) images and the mimetex installation that was serving formulas for this blog as well as a number of other CGI scripts has died or at least is being turned off. Anyway, I have to relocate these things and I am still looking for a good solution. It should be a computer with a static, routed IP address on which I can install programs and in particular cgi-scripts of my liking. Here at LMU, this is probably not going to happen for reasons of security paranoia on the sysadmin side. In addition, mathphys was handling my email traffic, meaning that currently spam reaches my inbox and messages are threatened to be deleted by well meaning service providers. But this just means that the suffering is strong enough that I will be looking for a solution in the very near future. The solution will most likely be renting some virtual linux server. Suggestions in this direction would be more than welcome.

Not long ago, I have been attending the 40th incarnation of the Ahrenshoop Symposium once more organised by the Humboldt Uni crowd. This get together had a particularly interesting selection of talks many of which I really enjoyed. In particular I learned a lot and updated my options on F-Theory GUTs and AdS-Condensed Matter. Many thanks to the organisers! As you would expect, PDFs are online except for Sean's who gave a flip-chart talk (on four flip charts).

At that meeting I was asked what had happened to this blog and this post is supposed to be the answer to this question. I hope of course that more content will be here, soon. I was also asked to mention that it was Martin Rocek who got all the soap.

How to obtain a polymer Hilbert space

2010-02-10T14:37:00.003+01:00

On Monday, I will be at HU Berlin to give a seminar on my loop cosmology paper (at 2pm in case you are interested and around). Preparing for that I came up with an even more elementary derivation of the polymer Hilbert space (without need to mention C*-algebras, the GNS-construction etc). Here it goes:

Let us do quantum mechanics on the line. That is, the operators we care about are and . But as you probably know, those (more precisely, operators with the commutation relation ) cannot be both bounded. Thus there problems of domains of definition and limits. One of the (well accepted) ways to get around this is to instead work with Weyl operators and . As those will be unitary, they have norm 1 and the canonical commutation relations read (with the help of B, C and H) . If you later want, you can go back to and similar for .

Our goal is to come up with a Hilbert space where these operators act. In addition, we want to define a scalar product on that space such that and act as unitary operators preserving this scalar product. We will deal with the position representation, that is wave functions . and then act in the usual way, by translation and by multiplication . Obviously, these fulfil the commutation relation. You can think of and as the group elements of the Heisenberg group while and are in the Lie algebra.

Here now comes the only deviation from the usual path (all the rest then follows): We argue (motivated by similar arguments in the loopy context) that since motion on the real line is invariant under translation (at least until we specify a Hamiltonian) is invariant under translations, we should have a state in the Hilbert space which has this symmetry. Thus we declare the constant wave function to be an element of the Hilbert space and we can assume that it is normalised, i.e. .

Acting now with , we find that linear combinations of plane waves are then as well in the Hilbert space. By unitarity of , it follows that , too. It remains to determine the scalar product of two different plane waves . This is found using the unitarity of and sesquilinearity of the scalar product: . This has to hold for all and thus if it follows that the scalar product vanishes.

Thus we have found our (polymer) Hilbert space: It is the space of (square summable) linear combinatios of plane waves with a scalar product such that the are an orthonormal basis.

Now, what about and ? It is easy to see that when defined by a derivative as above acts in the usual way, that is on a basis element which is unbounded as can be arbitrarily large. The price for having plane waves as normalisable wave functions is, however, that is not defined: It would be . But for the two exponentials in the denominator are always orthogonal and thus not "close" as measured by the norm. The denominator always has norm 2 and thus the limit is divergent. Another way to see this is to notice that would of course act as multiplication by the coordinate , but times a plane wave is no longer a linear combination of plane waves.

To make contact with loop cosmology one just has to rename the variables: What I called for a simplicity of presentaion is the volume element in loop cosmology while the role of is played be the conjugate momentum .

If you want you can find my notes for the blackboard talk at HU here (pdf or djvu

Entropic Everything

2010-01-27T13:38:00.002+01:00

The latest paper by Eric Verlinde on gravity as an entropic force makes me wonder whether I am getting old: Let me admit it: I just don't get it. Is this because I am conservative or lack imagination or too narrow minded? If it were not for the author, I would have rated it as pure crackpottery. But maybe I am missing something. Today, there were three follow-up papers dealing with cosmological consequences (the idea being roughly that Verlinde uses the equipartition of energy between degrees of freedom each getting a share of 1/2 kT which is not true quantum mechanically at low temperatures as there the system is in the ground state with the ground state energy. As in this business temperature equals acceleration a la Unruh this means the argument is modified for small accelerations which is a modification of MOND type).

Maybe later I try once more to get into the details and might have some more sensible comments then but right now the way different equations from all kinds of different settings (Unruh temperature was already mentioned, E=mc^2, one bit per Planck area, etc) are assembled reminds me of this:

Instability of the QED vacuum at large fine structure constant

2010-01-19T15:16:00.000+01:00

Today, in the "Mathematical Quantum Mechnics" lecture, I learned that the QED vacuum (or at least the quantum mechanical sector of it) is unstable when the fine structure constant gets too big.

To explain this, let's go back to a much simpler problem: Why is the hydrogen-like atom stable? Well, a simple answer is that you just solve it and find the spectrum to be bounded above . But this answer does not extend to other problems that cannot be diagonalised analytically.

First of all, what is the problem we are considering? It's the potential energy of the electron which in natural (for atomic physics) units is . And this goes to negative infinity when goes to 0. But quantum mechanics saves you. Roughly speaking (this argument can be made mathematically sound in terms of Hardy's inequality), if you essentially localise the electron in a ball of radius and thus have the potential energy , Heisenberg's uncertainty implies the momentun is at least of the order and thus the kinetic energy is at least of the order . Thus, when becomes small and you seem to approach the throat of the potential the positive kinetic energy wins and thus the Hamiltonian of the hydrogen atom is bounded from below. This is the non-relativistic story.

Close to the nucleus however, the momentum can be so big that you have to think relativistically. But then trouble starts as at large momenta the energy grows only linearly with momentum and thus the kinetic energy only scales like which is the same as the potential energy. Thus a more careful calculation is needed. The result of it is that it depends on which term eventually wins. Above a critical value (which happens to be of order one) the atom is unstable and one can gain an infinite amount of energy by lowering the electron into the nucleus and quantum mechanics is not going to help.

Luckily, nuclei with large enough do not exist in nature. Well, with the exception of neutron stars which are effectively large nuclei. And there it happens. All the electrons are sucked into the nuceus and fuse with the protons to neutrons. In fact, the finite size of the nucleon is what regulates this process as the nature of the Coulomb potential is smeared out in the nucleus. But such a highly charged atom would be only of the size of the nucleus (about some femto meters) rather than the size of typical atoms.

But now comes QED with the possibility of forming electron-positron pairs out of the vacuum. The danger I am talking about is the fact that they can form a relativistic, hydrogen like bound state. And both are (as far as we know) point like and thus there is no smearing out of the charge. It is only that in this case which luckily is less than one. If it would be bigger you could create this infinte amount of energy from the vacuum by pair creation and bringing them on-shell in their relative Coulomb throat. What a scary thought. Especially, since is probably only the vev of some scalar field which can take other values in other parts of the multiverse which would then disappear with a loud bang.

Some things come to my mind which in principle could help but which turn out to make things worse: is not a constant but it's running and QED has asymptotic slavery which means at short distances (which we are talking about) it gets bigger and makes things worse. Further, we are treating the electromagnetic field classically which of course is not correct. But my mathematical friends tell me that quantizing it also worsens things.

We know, QED has other problems like the Landau pole (a finite scale where goes to infinity due to quantum effects). But it seems to me that this is a different problem since it already appears at .

Any ideas or comments?

Download compete twitter timeline

2009-10-09T14:41:00.002+02:00

Upon popular request, I wrote a small script to download all tweets of a given twitter id. Have fun!


#!/usr/bin/perl

use Net::Twitter;
$|=1;

unless(@ARGV){
    print "Usage: $0 twitter_id [sleep_seconds]\n";
    exit 0;
}

my ($follow,$sleeper)  = @ARGV;

# No account needed for this.
my $twit = Net::Twitter->new(username => 'MYNAME', password => 'XXX');

$p=1;
while(1){
    my $result = $twit->user_timeline({id => $follow, page => $p});
    
    foreach my $tweet (@{$result}){
 print "At ", $tweet->{'created_at'},"\n";
 print $tweet->{'text'},"\n\n";
    }
    ++$p;
    sleep $sleeper if $sleeper;
}

You might have to install the Net::Twitter module. This is most easily done as


sudo perl -MCPAN -e shell

and then (possibly after answering a few questions)


install Net::Twitter

Not so canonical momentum

2009-10-05T22:52:00.003+02:00

Two weeks ago, I was on Corfu where I attended the conference/school/workshop on particles, astroparticles, strings and cosmology. This was a three week event, the first being on more conventional particle physics, the second on strings and the last on loops and non-commutative geometry and the like. I was mainly there for the second week but stayed a few days longer into the loopy week.

I think it was a clever move by the organisers of the last week to give five hours to the morning lecturers rather than one or two as in the string week. So they had the time to really develop their subjects rather than just mention a few highlights. John Baez has already reported on some of the lectures.

I would like to mention something I learned about elementary classical mechanics and quantum mechanics which was just a footnote in Ashtekar's first lecture but which was new to me: One canonical variable can have several canonical conjugates! In the loopy context, this appears as both the old and the new connection variables have the same canonical momentum although they differ by the Imirzi parameter times the second fundamental form (don't worry if you don't know what this is in detail, what's important that the 'positions' are different in the two sets of variables although they have the same canonical momentum).

How can this be? I always thought that if is a canonical variable the conjugate variabel is determined by . What I had not realized is that you could for example take and obtain the same fundamental Poisson brackets (and consequently commuation relations after quantization). Similarly, you could add any function to the momentum without changing the commutation relations.

The origin of this abiguity can be found in the fact that also the Lagrangian is not unique: You can always add a total derivative without changing the action (at least locally, see below). For example, to obtain by the derivative formula, add to the action. The most general change would be to add .

What about the quantum theory? This is most easily seen by realising that upon a gauge transformatio , the action of a charge particle changes by . Thus our change in Lagrangian (with a corresponding change in the canonical momentum) can be viewed as a gauge transformation (even if no gauge field is around one could add a trivial one). Correspondingly, the wave function would have to be changed to as acting on by a canocially quantized is the same as acted on by .

So, it seems as if you would get exactly the same physics in the primed variables as in the unprimed ones. But we know that not all total derivatives have no influence on the qunatum theory the -angle being the most prominent example. How would that appear in our much simpler quantum mechanics example? Here, it is important to remember that one should only use gauge transformations that are trivial at infinity. Here, if you change the phase of the wave function too wildly at you might leave the good part of the Hilbert space: For example the kinetic energy being an unbounded operator is not defined on all of Hilbert space but only on a dense subspace (most often taken to be some Sobolev space). And that you might leave by adding a wild phase and end up in a different self adjoint extension of the kinetic energy.

I have no idea if all this is relevant in the loopy case and the old and new variables or the variables are related by a (generalized) gauge transformation but at least I found in amusing to learn that the canonical conjugate is not canonical.

Jazz makes you age faster

2009-08-08T11:19:00.003+02:00

Before I head off for the travel season (first vacation: St. Petersburg, Moscow, Transsiberian Railway, Irkutsk, Baikal Lake, Ulan Ude and Mongolian Border, then two weeks of workshop in Corfu, then meeting collaborators in Erlangen and finally lecuring in Nis, Yugoslawia) - you won't notice any change in posting frequency - I would like to leave you with the latest statistic I learned about in "Sueddeutsche Zeitung" today:

In 1982, the average age of the audience of jazz concerts (don't know if in Germany or worldwide or whatsoever) was 29, today it is 64. So, even assuming immortality of improvised music enthusiasts, in 27 years, they got older by 35 years!

Note well that for an average of 64, if I attend a jazz concert being 36, we need 78ers to get back to the average.

Thermodynamic (in)stability of hydrogen

2009-07-14T14:12:00.004+02:00

The interview season for the "Theoretical and Mathematical Physics" master programme at LMU is approaching quickly. We have to come up with new questions and problems that help us judge our applicants.

It turns out to be easy to find questions in quantum mechanics and those easily lead over to mathematics questions. However, we were always short on good stat mech problems. One possibility is to have an easy start with the harmonic oscillator and then couple that to a heat bath and compute the partition function (with geometric series featuring).

But this time, we thought we could vary this a bit and came to a surprising realization: Hydrogen is unstable! This was news to me but google finds a number of pages where this is discussed. Often wrongly, but the good explanation is in a 2001 paper by Miranda.

The idea is the following: Everybody knows that the energy of the -th level of the hydrogen atom has energy proportional to . This level has degeneracy since runs from 1 to and then runs from to . So the partition function is . First, we thought that this might be a function named after some 19th century mathematician but mathematica told us its name is actually since the exponent quickly approaches 1 for every positive .

The conclusion seems to be that there is something wrong with the hydrogen atom. And we have not even started to consider the positive energy scattering states. Obviously, this problem has an IR divergence and it is probably better to embed it in some cavity of finite radius. But still, you would think that then for a large cavity, most of the statistical weight would be in the highly excited states and the probability to be in the ground state would go to zero as the cavity gets larger.

The conclusion would be that a hydrogen atom at any temperature would almost never be in its ground state but always highly excited or even ionized. And all this only because the density of states diverges at 0. This looks like a situations worse than the Hagedorn transition that strings experience due to the exponentially growing density of states.

The solution in the above mentioned paper is quite simple: Rather than these scaling arguments one should put in some numbers! Let us start with the Bohr radius, which is m and the radius grows like . This means in ameter sized cavity we can only fit states up to roughly . However, at room temperature, Boltzmann exponent and . Thus, to balance the Boltzmann suppression of the higher levels compared to the ground state one has to take into account at the order of states and not just the first . Or put differently, one should use and exponentially large cavity. Otherwise the partition function is essentially cut off at and the probablility to find the ground state is very very very close to 1.

Wrocław summary

2009-07-06T18:37:00.001+02:00

So, I did not get around to live blog from the XXVth Max Born meeting "The Planck Scale". The main reason was, that there were no hot news or controversial things presented, rather people from the different camps talked about findings that a daily reader of hep-th had in some form or the other already noticed. I don't want to create the impression that it was boring, by no means. There were many interesting talks, there were just no breathtaking revelations. I myself am not an exception: I took the opportunity of having several loop-people in the audience to talk once more on the loop string, this time focussing on spontaneous breaking of diffeomorphism invariance.

By now, the PDFs are online and in a few days you will also find video footage. To get an idea what people discussed, the organizers had the idea to assemble tag clouds from the slides, some are above.

Let me mention a few presentations and speakers nevertheless. Steve Carlip talked about the notion of space-time being two dimensional at very short distances in several unrelated approaches. Related was a nice presentation of Silke Weinfurter on her papers with Visser on the scalar mode not decoupling in Horava gravity. That talk was probably on the most recent and hottest results and I had the impression that many other approaches still have to digest the lesson that it is non-trivial to modify gravity and still not throw out the baby with the bath tub.

Hermann Nicolai presented his work (together with Meissner) on a classically scale invariant version of the standard model in which the only dimensionful coupling (the Higgs squared term) arises from an anomaly. They claim that their model is compatible with the current data and would imply that LHC sees the Higgs and only the Higgs. Daniel Litim gave a nice overview over the asymptotic safety scenario for gravity. Bergshoeff and Skenderis talked about models related to 3d topologically massive gravity and Jose Figueroa-O-Farrill presented a summary of algebraic structures relevant for M2 theories.

Mavromatos discussed possible observations of time delays in gamma ray bursts and implications for bounding modifications of dispersion relations in quantum gravity. Steve Giddings talked about locality and unitarity in connection with black hole information loss and Catherine Meuseburger explained how in 3d gravity observers can make geometrical measurements with light rays to find the gauge invariant information on in which Ricci-flat world they are living.

I was surprised how many people still work on non-commutative geometry (in the various forms). The Moyal-plane, however, seems to be out of fashion (not so much because of UV-IR-mixing which I think is the main reason to be careful but many people seem to think they can work around that but are worried about unitarity on the other hand). Kappa-Minkowski is a space many people care about and Dopplicher explained why we live in quantum space-time. The general attitude seemed to be (surprisingly) that Lorentz-breaking in those theories is not an issue. However, Piacitelli, showed a calculation that should have been done quite a while ago: People say that although Lorentz invariance is broken that is not a problem since there is a twisted co-product version that preserves at least some related quantum symmetry. Piacitelly now spelled out what that means in everyday's terms: When you do a boost or rotation, twisting the co-product is equivalent to treating theta as a tensor and rotating that as well. Great, that explains why the formalism shows that rotational symmetry is preserved while the physics clearly says that a tensor background field singles out preferred directions. I had for a long time the suspicion that this is what is behind this Hopf-algebra approach but could never motivate myself enough to understand that in detail so I could confirm it.

In addition, there were many talks from loop-related people (also on spin foams, BF-type theories etc) about which I would like to mention just one: Modesto applied the reasoning found in the loop approach to cosmology (I would like to say more about this in a future posting) to a spherically symmetric space-time (i.e. what is Schwarzschild in the classical theory). What he finds is indeed Schwarzschild at large distances but the discretization inherent in that approach produces a solution that has a T-duality like R <--> l_p^2/R symmetry.

A great opportunity for meetings of this style with people coming from different approaches are always extended discussion sessions. Once more, those were a great plus (although not as controversial as a few years back in Bad Honnef), there were two, one on quantum gravity and one on non-commutative geometry.

There, once more, people complained that it is hard to do this kind of physics without new experimental input. Of course to a large degree, this is true. But to me it seems that also misses an important point: By no means, everything goes! At least you should be able to make sure you are really talking about gravity in the sense that in not so extreme regimes you recover well known physics (Newton's law for example). Above, I mentioned Horava gravity apparently failing that criterion and it seems many other approaches are not even there to be tested in that respect.

We often say, we work on strings because it is the only game in town. On that meeting you could have a rather different impression: It seemed more like everybody was playing more or less their on game and many didn't even know the name of their game. Another example of such a trivial non-trivial test is what your theory says about playing snooker: The kinetics of billard balls tests tensor products of Poincare representations of objects with trans-planckian momenta and energies. If your approach predicts weird stuff because it does not allow for trans-planckian energies my interpretation would be that you face hard times phenomenologically, even if your model agrees with CMB polarizations.

More conference blogging

2009-06-29T14:25:00.002+02:00

Instead of String '09 I decided to attend this year the Born Symposium on the Planck scale. There are a number of stringy speakers as well as quite a few people from the loop camp. Watch this space for some reports. The talks (including video) will be online as well (as opposed to Strings '09).

More quality journals by Elsevier

2009-05-06T11:25:00.002+02:00

After all the fun re the El Nashie fan board "Chaos, Solitons and Fractals" it seems Elsevier has put another nail in their coffin by allowing the pharma company Merck to run their pseudo scientific marketing journal under their flag: Merck Makes Phony Peer-Review Journal | blog.bioethics.net . Once more, John Baez has more details: The Foibles of Science Publishing | The n-Category Café

Great, I think nobody in the world can claim anymore that our libraries should throw big money at these commercial publishing houses because they provide the quality control that open access publication cannot provide.

Journal club cgi

2009-04-27T20:17:00.006+02:00

For our journal club, I wrote a small cgi script that provides a web page where people can dump arxiv.org identifies of papers they are interested in and then everybody can see title, abstract and authours as well as a link to the pdf.

You can copy the file to your cgi-bin directory, make it world executable and create a world writeable directory /opt/journal . You might need to install the XML::TreeBuilder module (as root run 'perl -MCPAN -e shell' and then do 'install XML::TreeBuilder'.


#!/usr/bin/perl
#
# Journal club cgi
# Written by Robert C. Helling (helling@atdotde.de)
# Published under the Gnu Public License (most recent version)
#

use XML::TreeBuilder;
use LWP::Simple;
use CGI;

my $g = CGI::new();
print "Content-type: text/html\n\n";
my @ids = ();

system "touch /opt/journal/numbers";
open (IN,'/opt/journal/numbers') ||die "Cannot open /opt/journal/numbers:$!!";;
while(){
chomp;
if(/(\d\d\d\d\.\d\d\d\d)/){
push @ids, $1;
}
elsif(/^\-\-\-/){
@ids = ();
}
}
close IN;

if($new_paper = $g->param('paper_id')){
my ($new_id) = $new_paper =~ /(\d\d\d\d\.\d\d\d\d)/;
push @ids,$new_id;
open (OUT, ">>/opt/journal/numbers") || die "Cannot write to /opt/journal/numbers:$!";
print OUT "$new_id\n";
print "\nNEW$new_id\n";
close OUT;
}

my %seen = ();
@ids = grep { ! $seen{$_} ++ } @ids;

dbmopen %papers, '/opt/journal/papers', 0666;

print join '<hr>', map {&show($_)} @ids;

print ($g->start_form(),
  $g->textfield('paper_id'),
  $g->submit(-name => "add paper"),
  $g->end_form());

sub show{
my $id = shift;
my $data;

unless($data = $papers{$id}){
$data = get("http://export.arxiv.org/oai2?verb=GetRecord\&identifier=oai:arXiv.org:$id\&metadataPrefix=arXivRaw");
$papers{$id} = $data;
}

my $all = XML::TreeBuilder->new;
$all->parse($data);

my $authors = $all->find('authors')->as_text;
my $title = $all->find('title')->as_text;
my $abstract = $all->find('abstract')->as_text;

return "$authors\n<a href=\"http://arxiv.org/pdf/$id\">$title</a>\n$abstract\n";
}

dbmclose %papers;

Posts disappearing

2009-03-01T23:27:00.002+01:00

As I have noted earlier, some posts by John Baez and others on certain crackpots having their own Elsevier journal to dump their E-infinity weirdness have disappeared from the net. Luckily, one brave soul keeps copies of this stuff.

Now, I had to learn, that they completely overtook the discussion section of the online version on a Scientific American article on dynamical triangulations and what is even more disturbing, made the weekly quality newspaper "Die Zeit" take down the online version of an article reporting these issues (in German). I would have thought it would take a lot to make a respectable paper with a law department to commit this kind of self censorship.

Initial data for higher order equations of motion

2009-02-17T20:59:00.000+01:00

Over lunch, today, we had a discussion on higher order quantum corrections in the effective action. You start out with a classical action that only contains terms with up to two derivatives. This corresponds to equations of motion that are second order in time. As such, for the physical degrees of freedom (but I want to ignore a possible gauege freedom here) you then have to specify the field and its time derivate on a Cauchy surface to uniquely determine the solution.

Loop corrections, however, tyically lead to terms with any number of derivatives in the effective action. Corresponding equations of motion allow then for more initial data to be specified. The question then is what to do with the unwanted solutions. If you want this is the classical version of unitarity.

Rather than discussing higher derivative gravity (where our lunch discussion took off) I would like to discuss a much simpler system. Say, we have a one dimensional mechanical system and the classical equation of motion is as simple as it can get, just . To simplify things, this is only first order in time and I would like to view a second order term already as "small" correction. The higher order equation would then be with small .

To find solutions, one uses the ansatz and finds . For small , the two exponents behave as which blows up and which approaches the solution of the "classical equation".

The general solution is a linear combination of the two exponential functions. We see that the solution blows up over a time-scale of unless the initial data satisfies the classical equation .

We can turn this around and say that if the classical equation is satisfied initially, we are close to the classical solution for long time (it's not exactly the same since differs from the "classical exponent" by order terms. For other initial data, the solution blows up exponentially on a "quantum time" inversely proportional to the small parameter .

This plot shows for . On the axis that goes into the picture there is a parameter for the initial conditions which is for data satisfying the classical equation initially. You can see that this parameter determines if goes to or over short time. Only the classical initial data stays small for much longer.

Unfortunately, this still leaves us with the question of why nature chooses to pick the "classical" initial data and seems not to use the other solutions. In the case of higher order gravity there is of course an anthropic argument that suggests itself but I would rather like to live without this. Any suggestions?

Better than refereeing fees

2009-02-17T16:30:00.003+01:00

A few days ago, I received an invitation to join a facebook group that demands that all journals should follow JHEP to pay their referees. So far, I did not sign up.

I am convinced the refereeing system has many flaws. But the pay of referees is not one of them. I don't know how much JHEP pays, I have heard the figure of 30$ per paper. What's that? Refereeing a string theory paper is a job that requires a specialist with a broad academic background. So, you would expect an hourly rate that is well above 100$ (judging for example the rate that lawyers demand). That means, by paying this specialist 30$ I expect that the refereeing takes him less than 20 minutes (including typing the report and uploading it to a web page). But that's exactly the problem with the refereeing system: The value you can add by refereeing a paper in 20 minutes negligible. You have to spend significantly more time with the paper to have a more significant opinion than you have after one minute of seeing the authors' names, reading the abstract and flipping through the pages.

On the other hand, referees are already paid for their refereeing: That's part of an academics job, and he/she already gets a salary from the university. That should already cover the refereeing as it is part of the job like it is to attend seminars and to discuss with other scientists.

The problem with the refereeing system really is that too often too little attention is given to the actual paper. Everybody knows first hand examples of excellent papers that were rejected for stupid reasons. On the other hand, there is a lot of very low quality stuff that gets printed, the Bogdanovs' papers and the El Nashie[no link so far] story being only the most prominent examples.

Of course, the refereeing process is most likely the only value that publishers add to a paper when it is promoted from a freely available preprint on arxiv.org to a published article. And we (that is our employers through their libraries) pay enormous sums for this more and more demanding justification. And giving this justification gets harder and harder with every b.s. paper that appears in print.

The flaw with the "you are already paid" argument is of course that refereeing is invisible and besides your obligation as a scientist there is little incentive to do a good job. Nobody (except maybe the editor) sees it and there is no reward, not even an idealistic one.

There is however one simple improvement that would be trivial to implement. I learned this from Vijay Balasubramanian a few years ago and I am convinced it should be introduced immediately: If a paper gets accepted, the identity of the referee should be published together with the paper while a referee that rejects a paper should stay anonymous.

This would give an incentive to do good work as a referee. If the paper's value is low and you still accepted it because you did not properly read it you will receive shame while if the paper is good people can see you put some effort into it. Keeping the identities of rejectors hidden of course prevents referees from accepting papers because of fear of any kind of "revenge" from the authors.

I am sure the quality of the refereeing process would increase significantly if this were implemented. Thus, I would urge you to support the publication of accepting referees names in the next discussion of the flaws of the refereeing system I am sure you will take part in over the next few weeks!

Update: Some brave soul has collected all the El Naschie stuff that seems to have disappeared from the web.

Reinstalling my Aspire One

2009-01-21T17:05:00.004+01:00

Did I mention before that the package selection tool (originally from Fedora) that comes with the Aspire One sucks big time? Yesterday, I came across zebra barcode reader I could use this together with the built in camera to keep track of the books I lend to various people. Zebra was of course not packaged and compiling it required C++ bindings for ImageMagick. Those I could not install due to dependency problems. I spent more or less the whole day installing this/compiling that/upgrading the other. In the end, I had zebra running (only to learn that ordinary bar codes are either too small to be resolved by the web cam or bringing them closer to the cam is too close for the camera to focus. Bummer. But enlarging the barcode on the copier made the computer recognize it but somehow defeats the purpose) but left the Aspire One in a state unable to boot.

Luckily, it only did not boot into X but I could still get a prompt and mount an external disk and backup my home dir. And now I install Ubuntu to it. I will continuously update this post as I keep going.

Step one was to get the installation disk onto a bootable USB stick. UNetbootin is a wonderful tool for this it only took me some time to find it. In the first go I ended up with a "Missing operating system" error when trying to boot from it. Fortunately, Bug #277903 in usb-creator (Ubuntu): “Missing Operating System [message at boot]” has the solution: Remove the partition using fdisk(!!!) and create new one using gparted. That worked. Now I install from that stick.

Stay with me and keep fingers crossed!

Update: I had expected to have to wait for long times fighting with various bits and pieces and having time to blog the solutions as I found them. But there were no solutions to find. Everything I tried so far just worked. Without any hassle. So nothing to say except I should have switched to Ubuntu already much earlier. OK, two things do not work so far: The mic (for skype for example) and the bluetooth USB stick. But I have not really tried.

A feature I would love

2009-01-18T15:51:00.003+01:00

I have a problem and it seems so far no one has come up with a good solution for it: Many web 2.0 services come in the form of a stream of items most of which I would like to see exactly once. Examples are RSS feeds, podcasts (here seeing could as well mean synching with the ipod), mail (in some respect), twitter, usenet, etc.

If I access them with a single program on a single device, this program keeps track if what I have seen so far and (at least in default mode) only presents me the new stuff. Excellent. Only that I use more than one device to access these services: I have my desktop, a laptop at home, a netbook on the road or in meetings and seminars, sometimes I even use other peoples devices.

What I want is a way that all these different devices have a possibility to share the information which items I have already seen. C'mon, this can't be that hard to implement. And I am sure I am not the only person with more than one computer.

The problem is most pronounced with RSS feeds. As mentioned some time ago, I used Liferea read blogs. This is where I first noted I would like to share the "read it flag" info between different computers. I thought that program might have a file like .newsrc in the old days where it records for each feed which posts have been seen before. Then it shouldn't be too hard to put this under subversion control or write a small perl script to merge those files. Except that information is not kept in a simple format. Instead, liferea keeps the downloaded content in some xml file and that file has flags. In order to merge the state of read items one would have to download them as does liferea and then sync the flags. Why do they have to mix the content and the meta information, why why why? I even looked at liferea's source but I couldn't see the possibility of an easy patch.

I "solved" this by giving up on liferea and moving to Google Reader instead. Since there, feeds are not read locally but on a single server, there is no problem with sharing state information. Only that I am not very comfortable with letting google know which feeds I like. And maybe at some point the GUI gets on my nerves or whatever. I don't think this is an ideal solution.

For more or less the same reason I use a similar approach to mail: All my different addresses' mail ends up in the inbox of my desktop computer (except for mailing lists etc which are sorted in appropriate alternative inboxes but on the same computer). In case I want to read mail on a different computer I ssh to the desktop. Except that it is not directly connected to the net. I first have to ssh to LMU's firewall. But that computer still cannot see my desktop. So from there I ssh to some PC in the Arnold Sommerfeld Center. From where eventually I can ssh to my computer (although only with a numeric ip since my computer is not important enough to get a DNS entry. And so far I was too lazy to hook it up to dyndns. But over recent months dhcp was kind enough to always give me the same ip and thus this chain of computers is supported by appropriate entires in .ssh/config . But I am digressing.

What I wanted to say about mail is not so much I don't want to use IMAP and one central mail server. It is also about saved mail in local folders. Those I have to many and too much volume to put them all in IMAP. At least on publicly accessible computers. And on my desktop (where I can install whatever I want) it would be of no use since this is always two hops away from the rest of the internet, at least for ingoing connections. OK, I could set up ssh tunnels. But those usually do not work reliably over longer times (we are talking at least weeks, I want to have reliable access to my mail even if I travel for longer time).

But again, the main problem seems to be sharing the 'read it' information.

One more incarnation of the same problem: ITunes does of course not exist for Linux. So I manage my ipod with Amarok. I would like to use my different computers to upload recent podcasts to the ipod but have not found a way to do this consistently.

Thermodynamics of gravitational systems

2009-01-02T15:43:00.002+01:00

In this last (first?) post of the year I would like to express some confusion I have with respect to applying thermodynamic reasoning to cosmology or in general situations governed by gravity. The main puzzle I would like to understand is the question regarding the entropy balance of the universe: According to the second law of thermodynamics, entropy is never decreasing (I hope this is the correct sign, I can never remember it. Let's see, S is minus trace rho log rho. If rho is proportional to the projector on an N dimensional subspace we have S = - N 1/N log(1/N) = log(N). Thus it is increasing if the probability spreads over a larger subspace. Good). So if it is increasing, it should have been minimal at the big bang which seems to be at conflict with the universe being a hot soup of all kinds of fluctuations right after the big bang.

With the popular science interpretation of entropy as a measure of disorder or negative information the early universe must have been highly ordered and should have contained maximal information, a notion which is highly counter intuitive. So this needs some clearing up.

The simplest resolution would be that it is compatible with observation to assume that the universe has infinite volume and if it has a finite entropy density the entropy is infinite and any discussion of increasing or decreasing entropy is meaningless as it will be infinite at any time and it does not make sense to talk about more or less infinite entropy.

We could however try to still make sense in a local, desitised version: We could make the usual cosmology assumption of the universe being pretty much homogeneous and talk solely of entropy densities (after all, we only observe a Hubble sized ball of it and should thus only make appropriate local statements). But since the universe is expanding should we use co-moving or constant volumes when computing the densities when applying a desitised second law? But I don't think this is the real problem.

I am much more worried about another point: I am not convinced it makes sense to apply thermodynamic reasoning to situations that involve gravity! Obviously, the universe as we see it is not in thermal equilibrium, all the interesting stuff we see are local fluctuations. So standard textbook equilibrium thermodynamics does not apply: Remember for example, temperature is a property of an equilibrium, the fact it is well defined is sometimes called the zeroth law and out of equilibrium situations do not have a temperature! Only if locally things are not too different from an equilibrium state one can assign something like a local temperature. But things are even worse: The usual systems that we are used to describe thermodynamically (steam engines, containers of gas etc) have the property that the equilibrium is an attractor of the dynamics: All kinds of small, local perturbations diffuse away exponentially fast. This is in line with our intuitive understanding of the second law: The homogeneous state is the one with the highest entropy and thus the diffusion is governed by the second law.

This is not the case anymore as soon as gravity is the dominating force: What is different here is that gravity is always attractive. Thus if you have a nearly homogeneous matter distribution with small local fluctuations, over-dense regions will gravitate even stronger and thus will be even denser while under-dense regions will gravitate less and will become even emptier. Thus the contrast is increasing over time (a feature which is of course essential to structure formation of galaxies, stars etc). But this means the equilibrium is unstable. This is at least in conflict with the naive understanding of the second law above.

Some deeper inspection reveals that when you axiomatise thermodynamics you usually make some assumption on convexity (or concavity, depending on whether you use intensive or extensive variables of state) of your favorite thermodynamic potential (free energy etc). IIRC this is something you impose. Your system has to fulfill this property in order to be described by thermodynamics. And it seems that gravity does not have this property (the stability) and it quite possible (if I am not mistaken, sitting here in a train without any books or internet access) thermodynamic arguments do not apply to gravity.

Note well that I am talking classically (actually even only about the weak field situation in which the fluctuations are well described by Newtonian gravity), I have not even mentioned black holes and their negative heat capacity due to Hawking radiation which should make you even more uneasy about thermodynamic stability.

There is however a related problem my classically relativistic friends told me about: When discussing cosmology, it is usually a good first approximation that the universe is homogeneous which supposedly it is at large scales. At small scales however, this is obviously not the case with voids, galaxies, stars, stones etc. But for the evolution at large scales you average all those local fluctuations and replace everything by the cosmological fluid.

The problem with the non-linear theory of gravity is however that it is by far not obvious that this averaging commutes with time evolution: That, starting from good initial conditions it does not matter if you first average and then compute the time evolution of the averaged matter density or if you first compute the time evolution and then to the spatial averaging. The first thing is of course what we always compute while the second thing is what really happens. An incarnation of this problem was an argument that was discussed a few years ago that what looks like the cosmological constant in our local patch of the universe is just a density fluctuation with a super-horizon wave length. At first you would reject such a suggestion since something that happens over regions that are causally disconnected from us should not influence our local observations. However, due to the non-linear nature of gravity this argument is too fast and needed a more thorough inspection. My impression is that eventually it was decided that this idea does not work. I would be happy to be informed by somebody follows these things more closely.

To wrap up, I feel that I would need to have to understand much more basic things about thermodynamics applied to gravity before I could make sensible statements about the entropy of the universe or Boltzmann brains and the similar.

Symmetry Breaking in Quantum Mechanics

2008-12-10T20:14:00.002+01:00

Today, in our Mathematical Quantum Mechanics lecture, I put my foot in my mouth. I claimed that under very general circumstances there cannot be spontaneous symmetry breaking in quantum mechanics. Unfortunately, there is an easy counter example:

Take a nucleus with charge Z and add five electrons. Assume for simplicity that there is no Coulomb interaction between the electrons, only between the electrons and the nucleus (this is not essential, you can instead take the large Z limit as explained in this paper by Frieseke. The only way the electrons see each other is via the Pauli exclusion principle. The Hamiltonian for this system has an obvious SO(3) rotational symmetry. The ground state, however is what chemists would call 1S^2 2S^2 2P^1. That is, there is one electron in a P-orbital and in fact this state is six-fold degenerate (including spin). Of course, there is a symmetric linear combination but in that six dimensional eigen-space of the Hamiltonian there are also linear combinations that are not rotationally invariant. Thus, the SO(3) symmetry here is in general spontaneously broken.

This is in stark contrast to the folk theorem that for spontaneous symmetry breaking you need at least 2+1 non-compact dimensions. This for example is discussed by Witten in lecture 1 of the IAS lectures or here and is even stated in the Wikipedia.

Witten argues using the Stone-von Neumann theorem on the uniqueness of the representation of the Weyl group (the argument is too short for me) and explicitly only discusses the particle on the real line with a potential (the famous double well potential where the statement is true). In 1+1 dimensions, there is the argument due to Coleman, that in the case of symmetry breaking you would have a Goldstone boson. But free bosons do not exist in 1+1d since the 2-point function would be a log which is in conflict with positivity.

I talked to a number of people and they all agreed that they thought that "in low dimensions (QM being 0+1d QFT), quantum fluctuations are so strong they destroy any symmetry breaking". Unfortunately, I could not get hold of Prof. Wagner (of the Mermin-Wagner theorem) but maybe you my dear reader have some insight what the true theorem is?

Spectral Action Part II

2008-12-08T23:26:00.001+01:00

After in part I, I have explained how to go back and forth between spaces with metric information and (C*)-algebras it's now time to add some physics: Let us now explore how to formulate an action principle that will lead to equations of motion when extremised.

Let me stress however, that although it may look differently, this is entirely classical physics, there is no quantum physics to be found anywhere here even though having possibly non-commutative algebras might remind you of quantum mechanics. This, however, is only a way of writing strange spaces and has nothing to do with the quantisation of a physical theory. We will even compute divergences of one loop Feynman diagrams, but still this only a technical trick to write the classical Einstein action (integral of the Ricci scalar over the manifold) and has nothing to do with quantum gravity! Our final result will be the classical equations of motion of gravity coupled to a gauge theory and a scalar field (the Higgs)!

The trick goes as follows: Last time, we saw that in order to encode metric information we had to introduce a differentiation operator so we could formulate the requirement that a function should have a gradient which has length less or equal to 1. One could have taken the gradient directly but there was a slight advantage to take the Dirac operator instead since that maps spinors to spinors instead of the gradient mapping scalars to vector fields.

Another advantage of the Dirac operator is that when it is squared it gives the Laplacian plus the Ricci scalar (which we want to integrate to obtain the Einstein Hilbert action) divided by some number which I vaguely remember to be 12 but which is not essential. In formulas, we have . Even better, when we are in d dimensions, taking the d-th power gives us the volume element. Thus, taking these two observations together and using the Clifford algebra, we find that taking powers of gives us . The second term is obviously what we need to integrate to obtain the EH action.

But how to get the integration? Here the important observation is that we can pretend that this operator is the kinetic term for some field. Then, we can compute the one loop divergence of this field. On one hand, we know that this is the functional trace of the log of this operator. On the other hand, we can compute the divergence of this expression either diagrammatically or with slightly advanced technology in terms of the heat-kernel.

I will explain the heat-kernel formalism at some other time. However, the result of that treatment is a series of "heat-kernel coefficients" which are scalars expressions of the curvature of mass dimension . That is is 1, is basically the Ricci scalar, is a linear combination of scalar contractions of the curvature squared and so on. All those can interpreted as the coefficients of a power series in a formal parameter , i.e. .

The important result now is that the effective action is the integral of over s from 0 to infinity and over all space-time points. Because of the negative powers of , this integral diverges at 0. It turns out, this diverge is nothing but the UV divergence of the one-loop diagrams. Now you can apply your favourite regularisation procedure (dimensinal regularisation, Pauli-Vilars, you name it). Here, for simplicity, we just use a cut-off and start our integration at instead of 0. Connes and Chamseddine do something very similar. Just instead of a sharp cut-off they use a function in the integral that decays exponentially when s approaches 0 (NB has mass-dimension -2 and thus acts as ).

For concreteness, take . Then the term leads to times 1 integrated over the space (i.e. a cosmological constant). The term from gives times the Einstein-Hilbert term and is times an integral of curvature squared. The remaining terms are finite when Lambda is removed. Thus, we find the Einstein action (including a huge cosmological constant) plus a curvature squared correction as the divergence of the one-loop effective action.

But the effective action can also be written as the functional trace of the log of the operator. For this, we don't need any field for which the operator is the kinetic operator of. Imagine we happen to know all the eigenvalues of . Then we have just found that the Einstein Hilbert action can be written as the divergent part of the sum of the log of all those eigenvalues. And this is the spectral action principle: .

As it happens, Connes and Chamseddine really know the eigenvalues of the Dirac operator on spheres. So, they can really do this sum (which turns out to be expressible in terms of zeta-functions). The spheres are of course compact and thus the eigenvalues are discrete. In our field theory argument above, however, we implicitly used the usual continuous momentum space arguments for the effective action. In the limit of large momentum (which is relevant for the divergence) corresponding to short distances this should not really matter, one can pretend that momentum space is actually continuous. However, with a cut-off, this is not precisely true and the discreteness comes in in sub-leading orders. The difference between the continuous computation and the discrete one is of course tiny for a large cut-off. And it is exactly this difference that lets the two authors find agreement to "astronomical precision" (p. 15).

OK, up to now, we have reformulated the gravitational action in terms of the spectrum of the Dirac operator. But what about gauge interactions. But every physicist should now now how to proceed: Do gravity in higher dimensions and perform a Kaluza-Klein reduction. If the compact space has a symmetry group G then besides some scalars you will find YM-theory with gauge group G.

In the non-commutative setting, one can as well take a non-commutative space for the compact directions. Here, Connes and Chamseddine argue for a minimal example (given some conditions of unclear origin, one might be suspicions that those are tuned to give the correct result). It is minimal in terms of irreducibility. However, the total space being the product (by definition reducible) of this compact space by a classical commutative 4d space time (again reducible) make the naturalness of this requirement a but questionable.

For the concrete specification of the compact space, some Clifford magic is employed (including Bott periodicity) but this is standard material. You end up with a non-commutative description of two points for the two spinor chiralities. The symmetry then determines the gauge group. Here I am not completely sure but it seems to me that they employ the usual representation theory arithmetic from GUT-theories to sort all standard model particles in nice representations.

The non-commutative formulation allows the KK-gauge boson A_mu to also have a leg in the compact direction between the two points. From the 4d perspective this is of course a scalar. This of course will be the Higgs. The two points have a finite distance (see the Landi notes for details) and give a mass term inversely proportional to the distance (that is opposite to a superficially similar D-brane construction as noted by Michael Douglas some years ago).

That's it. We have an algebraic formulation of the classical action for the standard model. Let's recap what went in: The NCG version of a space with metric information in terms of a Dirac operator. Some heat-kernel material to write the gravitational action in terms of eigenvalues, GUT-type representations theory and KK-theory. What kind of 'surprises' were found? GUT type relations are rediscovered, treating the discrete spectrum of on spheres can well be approximated by continuous momentum space for momenta large compared to the inverse radius. And there is a final observation detailed in appendix A of the paper: For there are some cancellations of unclear origin which make vanish. This however is a statement about a heat-kernel coefficient and does not have a priori any connection with the non-commutative approach. Furthermore, the physical implications are left in the dark.

KK description of Black Holes?

2008-12-05T07:54:00.002+01:00

Still no second part of the spectral action post. But instead a litte puzzle that came up over coffee yesterday: What is the Kaluza-Klein description of a black hole?

To be more explicit: Take pure gravity on R^4xK for compact K and imagine that it is large (some parsec in diameter say). Then you could imagine you have something that looks like a blackhole in this total space-time. What is its four dimenional description in KK theory?

With KK theory I mean the 4d theory with an infinite number of fields. I want to include the whole KK tower. This theory should be equivalent to the higher dimensional one since both are related by a (generalised) Fourier transform on K. One might be worried that a black hole is so singular that this Fourier transform has problems, does not converge or something. But if that is your worry, take a black hole that is not eternal but one that is formed by the collision of graviational waves say. In the past, those waves came in from infinity and if you siufficiently go back in time all fields are weak. This weak field configuration should have no problem being described in KK language and then the evolution is done in the 4d perspective. What would the 4d observer see when the black hole forms in higher dimensions?

The question I would be most interested in is if there is always a black hole in terms of the 4d gravity or if the 4d gravity can remain weak and the action can be entirely in the other fields.

One scenario I could imagine is as follows: The 4d theory has besides the metric some gauge fields and some dilatons. If the black hole is well localised in K then many higher Fourier modes of K will participate. From the 4d perspective, the KK momentum is the charge under the gauge fields and the unit is dependent on the dilaton. So could it be that there is a gauge theory black hole, i.e. a charged configuration that is confined to small region of space time where the coupling is strong with all the causality implications of black holes in gravity?

Spectral Actions Imprecisely

2008-12-02T14:14:00.003+01:00

A post from Lubos triggered me to write a post on non-commutative geometry models for gravity plus the standard model as for example promoted by Chamseddine and Connes in a paper today.

Before I start I would like to point out that I have not studied this paper in any detail but only read over it quickly. Therefore, there are probably a number of misunderstandings on my side and you should read this as a report of my thoughts when scanning over that paper rather than a fair representation of the work of Chamseddine and Connes.

Most of what I know about Connes' version of non-commutative geometry (rather than the *-product stuff which has only a small overlap with this) I know from the excellent lecture notes by Landi. If you want to know more about non-commutative geometry beyond the *-product this is a must read (except maybe for the parts on POSETs which are a hobby horse of the author and which can be safely ignored).

But enough of these preliminary remarks. Let's try to understand the spectral action principle!

Every child in kindergarden knows that if you have a compact space you get a commutative C*-algebra for free: You just have to take the continuous functions and add and multiply complex conjugate them point-wise. As norm you can take the supremum/maximum norm (here the compactness helps). This is what is presented in every introduction section of a talk on non-commutative geometry, but onfortunately, this is completely trivial.

The non-trivial part (due to Gelfand, Naimark and Segal) is that it works also the other way round: Given a (unital) commutative C*-algebra, one can construct a compact space such that this C*-algebra is the algebra of the functions on it. Furthermore, if one has got the algebra from the functions on a space, the new space is homeomorphic to (i.e. the same as) the original one.

How can this work? Of course, anybody with some knowledge in algebraic geometry (a similar endeavour but there one deals with polynomials rather than continuous functions) knows how: First, we have to find the space as a set of points. Let's assume that we started from a space and we know what the points are. Then for each point x we get a map from the functions on the manifold to the numbers, we simply map f to f(x). A short reflection reveals that this is in fact a representation of the algebra which is one-dimensional and thus irreducible. It turns out that all irreducible representations of the algebra are of this form. Thus, we can identify the points of the space with the irreducible representations of the algebra.

I could have told an equivalent story in therms of maximal ideals which arise as kernels of the above maps, i.e. the ideals of functions that vanish at x.

Next, we have to turn this set into a topological space. One way to do this is to come up with a collection of all open or all closed sets. In this case, however, it is simpler to define the topology in terms of a closure map, that is a map that maps a set of points to the closure of this set. Such a map has to obey a number of obvious properties (for example if I apply the closure a second time it doesn't do anything or a set is always contained in its closure). In order to find this map we have to make use of the fact that if a continuous functions vanishes on a set of points then by continuity it vanishes as well on the limit points of that set, that is on its closure. Therefore, I can define the closure of a set A of points as the vanishing points of all continious functions that vanish on A. This definition then has an obvious reformulation in terms of irreducible representations instead of points. Think about it, as a homework!

Now that we have a topological space, we want to endow it with a metric structure. But instead of giving a second rank symmetric tensor, we specify a measure for the distance between two points x and y as this is closer to our formalism so far. How can we do this in terms of functions?

The trick is now to introduce a derivative. This allows us to restrict our attention to functions which are differentiable and whose gradient is nowhere greater then 1, i.e. the supremum norm of the gradient is bounded by one (this is where the metric enters implicitly since we use it to compute the length of the gradient). Amongst all such functions f we maximize l=|f(x)-f(y)|. Take now the shortest path between x and y (a geodesic). Since the derivative of f along this paths is bounded by one l cannot be bigger than the distance between x and y. And taking the supremum over all possible f we find that l becomes the distance.

Again, I leave it as a homework to reformulate this construction in the algebraic setting in terms of irreducible representations and the distance between them. There is only a slight technical complication: We started with the algebra of scalar functions on the manifold but the gradient maps those to vector fields which is a different set of sections. This makes life a bit more complicated. Connes' solution is here to forget about scalar functions and take spinors (sections of a spinor bundle precisely) instead. If those exist, all the previous constructions work equally well. But now we can use the Dirac operator and this maps spinors to spinors (with another slight complication for Weyl spinors in even dimensions).

In the algebraic setting, the Dirac operator D is just some abstract linear (unbounded) operator D which fulfills a number of properties listed on p2 of the Chamseddine_Connes paper and once that is given by some supernatural being in addition to the algebra, you can actually reconstruct a Riemannian manifold from an abstract commutative C*-algebra and D.

Next, we have to write down actions. Unfortunately, I now have to run to get to a seminar on the status of LHC. This will be continued, so stay tuned!

What is (the) time?

2008-11-30T15:38:00.002+01:00

After seeing Sean revise his view on Templeton funded events and submitting an essay to the FQXi essay contest it seems that this is now officially PC.

So, nothing stands in my way to submit my own.

Picking winners

2008-11-14T18:12:00.003+01:00

I just came across a post at fontblog where it is described how they picked a winner from 36 contributors: They take a dice and throw it once. The number shown is the number of further throws of the dice that are added up to yield the winning number. Obviously, any number 1..36 can be picked, but the distribution is not even: Contributor 1 is picked if the first throw yields a one and the second one as well: probability 1/36. Contributor 2 will be picked by two sequences: 1 2 and 2 1 1 giving 1/6 x 1/6 + 1/6 x 1/6^2 and so on. Conributor 36 is only picked when 7 sixes are thrown in a row, i.e. with probability 1/6^7.

Of course I could not resist and write a perl program to compute the probability distribution, here it is:

Somewhat surprisingly, contributor 29 was eventually picked although he only had a probability of 0.28% a tenth of the average 1/36.

And here is the program:


#!/usr/bin/perl

%h = (0 => 1);

for $t(1..6){
    %h = nocheinwurf(%h);

    print "$t:\n";
    foreach $s(sort {$a <=> $b} keys %h){
        print "$s:".$h{$s}/6**$t." ";
        $total[$s] += $h{$s}/6**($t+1);
    }
    print "\n\n";
}

print "total:\n";
foreach $s(1..36){
    print "$s $total[$s]\n";
}

sub nocheinwurf{
    my %bisher = @_;
    
    my %dann;

    for $w(1..6){
        foreach $s(keys %bisher){
            $dann{$s+$w} += $bisher{$s};
        }
    }
    return(%dann);
}

Acer Aspire One and Fonic UMTS

2008-10-27T09:22:00.003+01:00

Sorry for these uninspired titles of computer related posts. I picked them for search engine optimization since I had not been able to find information on these topics easily in google and at least want others to have a simpler life.

So this post is about how I managed to hook up my shiny new (actually there are already the first few spots on the keyboard...) netbook to the inceadible Fonic UMTS flat rate. They offer a USB stick (a Huawai E160 internally) and a SIM card for 90 Euros. There is no further monthly cost and you only pay 2.50 Euro per calender day you use it. That includes the first GB at UMTS speed per day, if you need more it drops down to a GPRS connection.

The fact that you can read this post is proof that I got it to work as I type this in the departure lounge of Schönefeld airport waiting for my flight to Munich.

Basically, there are two pieces of software that are required: usb_modeswitch which turns the stick from a USB storage device to a serial device. I wanted to compile it myself which lead me to learn (I didn't expect that) that the GNU Linux of the Aspire One comes without the C compiler and I had to manually install gcc. Furthermore, this had to be selected from the list of packages since just selecting the "development environment" of the grouped package selection lead to a version conflict that the stupid package manager (I was complaining about before) was unable to resolve. But once gcc is there, compilation is a matter of seconds. I also had to move the config file to /etc and change a few semicolons which are the comment markers to activate the sections that refer to the E160.

So now, to make a connection I turn on the Aspire One and insert the Fonic stick. Then, as root, I run

usb_modeswitch -c /etc/usb_modeswitch.conf -W

The actual connection is made with umtsmon. I have to wait a few seconds after running the usb_modeswith as otherwise the stick will not be detected. In the config the APN has to be set to pinternet.interkom.de and the checkbox noauth as to be ticked while "replace default route" has to be unchecket since the version of pppd that comes with the Aspire One does not understand this option (which apparently was introduced by SuSE and adopted bey Debian). Then you click connect and it should work!

Update: I forgot to mention that because you had to disable the "replace default route" option, the default route (if existent) will not be disabled. Thus you either have to do it by hand (using 'route') or better just make sure you don't have already a network connection running when trying to connect eg. WiFi (and why would you want to connect UMTS if you already had a better newtowrk connection???)

Why gold?

2008-10-26T18:06:00.002+01:00

I have seen this being discussed in several newspaper articles: In view of the financial crisis people withdraw their money from the bank and buy gold. They do this to such an extend that Germany's main internet gold coin dealer has stopped accepting orders. Amazing. What do they think they are doing?

Ok, you don't trust your local bank XY anymore and, depending where you are, you either have so much money that you are beyond the limit or you even don't trust the banking guarantee system anymore. Therefore, you don't want to leave your money in your account. Fine, you are a pessimist.

But why on earth are you buying gold? Obviously you are so afraid that you are willing to renounce the interest you could be getting for your money. But why don't you keep your money in coins and bills and put it in a save place (safe, mattress, cookie jar).

But that's not good enough for you. You don't even trust your (or any other) currency anymore and want to avoid the risk of inflation. That is you don't trust that somebody is willing to give you enough real stuff for your bills. That is, you mistrust your complete local economy.

You really want to make sure. Therefore you buy gold. Because that keeps its value. That you know.

But why on earth do you believe that? Why do you think that the value of gold is in any way less symbolic or just based on common agreement than the agreement of many people that you will give you food if you provide them with enough pieces of paper on which somebody printed the portrait of some former president in green color?

I have bad news for you: You cannot eat the gold, at least it has zero nutrition value (see wiki article linked above). You can use it for electric contacts that will not corrode. And yes, you can make jewelery from it (as it is done with a third of the gold that is mined). But again, the value of the jewelery is just that "everbody knows gold is precious". And this is only the case as long as everybody believes that. But there ist not much you can actually do with gold that you cannot for example to with copper or palladium (which is an excellent catalyst for many reactions involving hydrogen by the way). The value of gold is only high because everybody believes that. But that's the same for dollar bills or any other currency (leaving out the Icelandic crone for the moment).

And it is not beyond historical example that people stopped believing that gold has high value: As pointed out buy my history teacher in high school this was the main strategic mistake of the Spanish crown: not realising that you can destroy the marked if you suddenly have a lot of it. The Spanish kept importing huge amounts of gold that they had found in the Americas without noticing early enough that the others lost their interest in gold as soon as the Spanish suddenly had so much. At the same time, others found that it's a much better to invest into the real economy.

It took some more years to understand that the value of a currency is not so much based on all the gold that the central bank holds but much more on the economy that backs it.

And of course the telephone desinfactants that stranded on the earth in our past as told by the hitchhiker's guide to the galaxy also had the great idea to use leaves as bills. Which lead them to burn down all their woods.

Aharonov Bohm phase from self-adjointness

2008-10-18T09:02:00.002+02:00

This week started the course in "Mathematical Quantum Mechanics" that I am co-teaching with Laszlo Erdös this term. Since I had to rush a bit in the end of yesterday's lecture, I composed some notes on how to extend the momentum operator in a self-adjoint way for the particle on the interval and how one can see an Aharonov-Bohm phase being the ambiguity of this procedure.

Seen it all before

2008-10-16T19:20:00.000+02:00

You thought you saw something new? Watch this:

Unbounded operators are not defined on all of H

2008-10-16T10:40:00.002+02:00

I am looking for an elementary proof of the fact that an unbounded operator cannot have the whole Hilbert space as its domain of definition. In the textbooks I had a look at this follows from the closed graph theorem which then is proved using somewhat heavy functional analysis machinery. What I am looking for is something that is accessible to physicists that have just learned about unbounded operators and that could be turned into a homework problem. If you know such a reference or could give me a hint (in the comments or to helling@atdotde.de) I would greatly appreciate it!

Anti crackpot blog and falsifyability

2008-09-17T09:49:00.003+02:00

There is a new blog dealing with crackpots in HEP. It probably grew out of the recent hype about LHC being the doomsday machine. I left a comment to the first post that I would like to pull out of the comment threat so I reproduce it here:

You would like to add that I think that criteria for good science are unfortunately slightly more complicated than just asking for "falsifyability".

This notion comes from the (pre war) neo-positivist school of thought with Karl Popper being one of the major figures. It is a tremendous progress after realising that strictly speaking you will never be able to prove empirical statements like "all ravens are black" (the classic example) by looking only at a finite subsample of all ravens.

This is how far the education of a typical physicist in philosophy of science goes.

Unfortunately, falsifyability is not a good criterium either from the standpoint of logic at least: The thing is that non-trivial scientific statemens are much more complex than a simple "all ravens are black". When you say such a thing (or write it in a paper), there are many qualifyers that go with it, at least implicitly. You have to know what exactly "raven" means, what exatly you call "black" and not "dark grey", you have to say how you measure the color and that might come with a theory of itself that makes sure your measuring aparatus acutally measures color (as for example percieved by the eye).

Speaking in terms of formal logic, all these presuppositions are connected to the "all ravens are black" by a long long chain of "and"s.

Now imagine you observe a white raven (after having seen a million black ravens). At least you would be tempted to further investigate whether that bird is really a raven or was painted white. It's typically not the first reaction to throw away the "black ravens" theory if it had a lot of support before. Often the reaction is that you rather fiddle with the presuppositions of your theory (for example by changing slightly the definition of what you call 'raven').

The "all ravens are black" part is somewhat protected after receiving some positive evidence for some time and you rather add bits and pieces to the silent background theory before giving up the big statements.

Thus we have to agree that it's not that easy to falsify even simple statments like "all ravens are black" that easily. Furthermore, this is not what happens in science (historically speaking)! You could probably still find an ether theory where the ether is dragged along by the earth in its orbit around the sun and that has many more very unusual properties that is not ruled out empirically. Still nobody in her right mind believes in it since relativity is some much more successful!

All this is much better explained in Thomas Kuhn's "Structure of Scientific Revolutions" and I urge everybody with a slight interst in these matters to read this classic. He argues much more for the consensus of the scientific community to be the relevant criterium (which might be a self referential criterium for science consiracy theorists).

I believe that naive following this wrong criterium of "falsifyability" has gotten beyond the standard model theory in a lot of trouble in the public perception. I would attribute a lot of this to trying to follow the wrong criterium.

Timing of arXiv.org submissions

2008-09-15T20:35:00.002+02:00

I have learned an interesting fact from Paul Ginsparg's talk at Bee's Science in the 21st Century conference: around the middle of his presentation he showed a histogram of at what time of the day people submit their preprints to the astro-ph archive. This histogram showed an extremely pronounced peak right after the deadline. After. I had suspected that preprints would pile up before the deadline because people still want to make it into tomorrow's listing because they fear to be scooped. But this is not the case. People want to be on the top of the listing!

My immediate reaction was that the number of nutters amongst astrophysicists is higher than expected. But the next news was at least as shocking: It seems to work! The papers submitted just after the deadline accumulate 90 citations on the average compared to 45 for generic astro-ph papers.

There seem to be two possible explanations of this fact (which is still there for other archives but not quite as pronounced): It could just work and the reader's attention span is so short (you might want to blame Sesame Street with its 1'30" spots for it) that they get distracted before they reach the end of the listing of new papers (this is the only place where submission time matters according to Ginsparg). Alternatively, people who care to submit right after the deadline also care about their citation count a lot and use all means to get it up.

Some further investigation seems to show that the second effect is much stronger than the first. Somebody from the audience suggested to study the correlation with self citations and small citation circles with submission time.

What I would want to look at would be if those are also the people that have a large percentage of their citations from revised versions of preprints (i.e. are those the people who care to write these emails begging for citations --- I have to admit I was one of them today)?

New Toy: Acer Aspire One 150L

2008-09-12T14:35:00.004+02:00

Since yesterday 11pm I have my shiny new toy: A Acer Aspire One 150L Netbook Computer. Intel Atom CPU, 1GB RAM, 1200x600 pixels and less than 1kg of weight should make it the ideal travel companion. The "L" in the name means that this is the first computer I ever bought that came with Linux preinstalled. What distinguishes it from its competitors (like the Asus Eec PC) is that it has a proper hard drive which has comfortable 120GB and not just a tiny solid state disk. The thing comes for 349 Euros from amazon.de (I have seen it for US$350 at Best Buy in San Francisco).

Of course, such a brand new device comes with a number of things to tweek. And not all solutions easily found by just googling. Therefore, I will keep updating this post to record what I have done so far:

The thing comes out of the box and boots in a few seconds into some desktop with application icons all over the place. What is missing is a terminal window! But you can use the file system browser to run /usr/bin/terminal which is a good start.
During set-up you have to come up with a password. It turns out, this is then set both as root password and as a password for the preconfigured user "User". I have not yet dared to set the root password to something else and to rename user "User" to "robert" since the build in applications might assume that I am User.
Once you have a terminal you should run (as root) xfce-setting-show as this allows you to turn on the pop-up menu when right clicking on the desktop.
In this pop-up menu you can find a package manager. I becomes obvious that this Linux is based on fedora and the conflict resolution just plain sucks. But it's better than nothing and you can eventually install openssh to be able to log in to uni. I hope that I will have a Debian running on this machine but currently the corresponding web sites still look a bit scary.
They have preinstalled a vpnc client. But they forgot to include the tunneling kernel module tun.ko . This gives nasty error messages about /dev/net/tun . Luckily, this module is available here . Just download it to /lib/modules/2.6.23.9lw/kernel/drivers/net/ and reboot (or insmod it).

Update: Included some links.

Cloud computing

2008-07-29T15:10:00.003+02:00

This morning I read an article in this week's Zeit "Die angekündigte Revolution". Its author claims that the fact that in the future we will not have computational power in our homes (or with us) but rather use computing centers that are centralised and accessible over the net. He equates that with the revolutions that came about as an centralised electricity supply was established (instead of factories having their own generators in the basement) and centralised water supply made people independent of a well in the backyard.

I am not so sure.

First of all we've already had that: In the past, there were mainframes connected to many terminals. The pain that came with this set-up was only relieved when computational power appeared on everybody's desks thanks to PCs. So why go back to those old days?

In addition, what I am observing is that the computational power I have local access to is exponentially growing for as long as I can think. And I see no end to that trend. Currently, my mobile phone has more memory than my PC a few years ago and has a processor that can do amazing things. The point is not that I need that but that it's so cheap I don't mind having it.

True enough, I only very rarely really need CPU cycles. But when I need them (computing something in mathematica or visiting a web page with some broken java or javascript that makes my browser busy) I usually want it right now. It's not that I am planning it and could as well outsource it to some centralised server.

It might be a bit different with storage. Having some centralised storage that can be accessed from everywhere (and where other people worry about backup so I don't have to) could be useful. Not only for backup and mobile access to all kinds of documents. All that assuming that data privacy has been taken care of. But also things like configuration files (like bookmarks), documents, media files. That already partly exists (at least for specific applications) but nothing unified, as of today (as far as I am aware of).

But I cannot see people giving up local computational power. Recently the part of PCs where performance has been growing most strongly are the video cards (by now massive multiprocessor rendering engines). That development was of course driven by the video game market. I don't see how that would be moved to a centralised computer.

As of today, I do not know anybody that uses Google Docs. It's more a proof of concept that an application for everyday use. If I really want to collaborate on documents I would rather use subversion or CVS. Again, that has a centralised storage but computation is still local.

Let me finish with two rants about related problems that I recently had: First, I use liferea as my RSS aggregator. That is a nice little program with intuitive user interface that allows me to quickly catch up with the feeds I am interested in. Unfortunately, it keeps its state (which posts it has already downloaded and which posts I have looked at) in a stupid format: Its actually a whole directory of xml and html files! So to continue reading blogs on my laptop from where I left of on my PC requires scp'ing that whole directory. Not to mention there is no way to somehow 'merge' two states...

The other thing is email. You might think this is trivial, but to me it seems it is not. My problem is first I am getting a lot and want my computer to do some of the processing for me. Thus I have the mail delivery program sort mail from mailing lists into appropriate folders (including a spam folder). Then, on a typical day I want to read it with an advanced reader (which in my case is alpine, the successor of pine). The killer function being to automatically save incoming mail in a folder matching my nickname for the author or the author's name and saving outgoing mail to a
folder according to the recipient. Not to mention I have about half a gig of old mail distributed over 470 folders, more than what one can easily deal with one of the GUI clients like thunderbird.

That is all nice and well. Once I am at my PC I just run alpine. Or if I am at home or travelling and connecting for a somewhat decent machine (i.e. one that has an ssh client or at least allows me to install one) I ssh to my PC and read mail there (actually, I ssh to the firewall of the physics department from there ssh to one of the computers of the theory group and from there eventually to my PC as it is well hidden from the outside world due to some other people's idea of computer security).

What if that other computer cannot do ssh but there is only a web browser? My current (and for my upcoming four week holiday in the south west of the USA) solution is to go to a page that has an ssh client as a java applet and then to step one above. But that is a but like the mathematician in the joke that sees the dust bin in his hotel room burning and takes the buring bin bin to the physicist's hotel room thereby reducing the problem to an already solved one (the physicist had extinguished a fire in the previous joke).

Why is it so hard these days to set up decent webmail? At least for the duration of a holiday? My problem is that there are three classes of computers: Type one are connected to the internet but I do not have sufficient privileges to install software. Type two I have the privileges but they don't give me an IP that is routed to the internet (even more: that accepts incoming connections). Type three: A computer to which I have root access and which as a routed IP but where the software installation is so out of date I cannot install software with one of the common tools (i.e. apt-get) without risiking to install/upgrade other packages that require at least a reboot. I should mention that that computer is physically located some hundred kilometers away from me and I am the only person who could reboot it. A major update is likely to effectively make me lose that computer.

These things used to be so much easier in the past: Since the days of my undergraduate studies I always had some of my own linux boxes hooked up to some university (or DESY in that case) network. On that I could have done the job. But with recent obsession with (percieved) security, you only get DHCP addresses (with which one can deal using dyndns etc) but also which are behind firewalls that do not allow for incoming connections. Stupid, stupid, stupid!!!

I am really thinking about renting one of those (virtual) servers at one of the hosters which you can now do for little money to solve all these problems. But that should not really be neccessary!

Observing low scale strings without landscape problems

2008-07-29T13:50:00.002+02:00

Tom Taylor is currently visiting Munich and a couple of days ago he had a paper with Dieter Lüst and Stephan Stieberger which discusses (besides many detailed tables) a simple observation: Assume that for some reason the string scale is so much lower than the observed 4d Planck scale that it can be reached by LHC (a possible but admittedly unlikely scenario) and in addition the string coupling is sufficiently small. Then they argue the 2 to 2 gluon amplitude is dominated by the first few Regge poles.

The important consequence of this observation is that this implies that the amplitudes are (up to the string scale, the only parameter) are independent of the details of the compactification and the way susy is broken: This amplitude is the same all over the landscape in all 10^500 vacua!

Observationally this would mean the following: At some energy there would be a resonance in the gg->gg scattering (or even better several). The angular distribution of the scattering products are characteristic for the spins of the corresponding Regge poles (i.e. 0 for the lowest, 1 for the next etc) and most importantly, the decay width can be computed from the energy of the resonance (which itself measures the free parameter, the string scale).

Of course, those resonances could still be attributed to some particles but the spin and decay width would be very characteristic for stings. As I said, all this is with the proviso that the string scale is so low it can be reached by LHC (or any accelerator looking for these resonances) and that the coupling is small (which is not so much a constraint as the QCD coupling is related to the string coupling and at those scales is already very small).

Formulas down

2008-07-08T16:39:00.003+02:00

The computer that serves formulas for this blog (via mimeTeX) is down. Since it's located under my old desk in Bremen I cannot just reboot it from here. Please be patient (or let me know another host for mimeTeX and a simple way to migrate all the old posts...).

Update:
Having said this mathphys.iu-bremen.de is up again thanks to Yingiang You!

Chaos: A Note To Philosophers

2008-07-02T15:43:00.004+02:00

For some reasons (not too hard to guess) I was recently exposed to a number of texts (both oral and written) on the relation between science (or physics) and religion. In those, a recurring misconception is a misunderstanding of the meaning of "chaos":

In the humanities it seems, an allowed mode of argument (often used to make generalisations or find connections between different subjects) is to consider the etymology of the term used to describe the phenomenon. In the case of "chaos", wikipedia is of help here. But at least in math (and often in physics) terms are thought to be more like random labels and yield no further information. Well, sometimes they do because the people which coined the terms wanted them to imply certain connections, but in case of doubt, they don't really mean something.

A position which I consider not much less naive than a literal interpretation of a religious text when it comes to questions of science (the world was created in six days and that happened 6000 years ago) is to allow a deity to intervene (only) in the form of the fundamental randomness of the quantum world. For example, this is quite restricting and most of the time, this randomness just averages out for macroscopic bodies like us making the randomness irrelevant.

But for people with a liking for a line of argument like this, popular texts about chaos theory come to a rescue: There, you can read that butterflies cause hurricanes and that this fact fundamentally restricts predictability even on a macroscopic scale --- room for non-deterministic interference!

Well, let me tell you, this argument is (not really surprisingly) wrong! As far as the maths goes, the nice property of the subject is, that it is possible to formalise vague notions (unpredictable) and see how far they really carry. So, what is meant here is that the late time behavior is a dis-continuous function of the initial conditions at t=0. That is, if you can prepare the initial conditions only up to a possible error of epsilon, you cannot predict the outcome (up to an error delta that might be given to you by somebody else) even by making epsilon as close to 0 as you want.

The crucial thing here if of course what is meant by "late time behavior": For any late but finite time t (say in ten thousand years), the dependence on initial conditions is still continuous, for any given delta, you can find an epsilon such that you can predict the outcome within the margin given by delta. Of course the epsilon will be a function of t, that is if you want to predict the farther future you have to know the current state better. But this epsilon(t) will always (as long as the dynamics is not singular) be strictly greater than 0 allowing for an uncertainty in the initial conditions. It's only in the limit of infinite t that it might become 0 and thus any error in the observation/preparation of the current state, no matter how small leads to an outcome significantly different from the exact prediction.

Ping

2008-06-24T20:22:00.002+02:00

I have been silent here for far too long, mainly because there were a number of other things that had higher priority on my list. So, just as some sort of sign of life here is a video of a cool gadget that I have been toying around with recently:

The Wiimote can be found for less than 30 Euros on Ebay and there are probably many fun things to do with the IR sensor and the accelerometer. Maybe combined with an Arduino.

The whiteboard functionality just cries for an application combined with a USB digital TV stick and gromit. This should be just in time for a Kloppomat for the semi-finals starting tomorrow (which will be watched chez the Christandls).

Currently, I am trying to do damage control on my failure to update my laptop from Etch to Lenny (this post is written while I wait for the backup of home directories to be finished).

There are many other items in my pipeline to to be written about. Let me list them so I have some additional motivation in the near future:

In april I attended a conference on different approaches to quantum gravity which was a lot better than it sounds. I would like to comment at least on the talks on exact renormalisation group and a non-gaussian fixedpoint of gravity and

the talks by the loop people (including Ashtekar's talk which he last week also presented as an ASC colloquium here at LMU). Just as a teaser: In the extended discussion sessions at the conference Thiemann claimed that by coupling to his version of gravity he can quantise any theory. We was then asked about anomalous theories and still claimed that that wouldn't matter. Ashtekar however seemed to smell that that was not the best answer and chimed in by mentioning that maybe later the anomalous theories do not have flat low energy limits. Nice move according to the classic "blame it on the part of the theory nobody has any idea about" strategy.

I have a little project going trying to join some archeological data with google earth/maps. What seemed like a trivial coordinate transformation in spherical coordinates turned out to be much more complicated due to the precision required (one has to take into account not only that the earth is an ellipsoid but also that there are different reference ellispoids used by different people etc).

So watch this space!

Standing on the shoulders of giants

2008-04-10T10:22:00.002+02:00

As mentioned before, since getting an IPod for christmas I am a huge fan of podcasts. I find it's like listening to the radio but you decide what they talk about. Currently, my favourite feeds are Chaos Radio Express (in German) and the In Our Time BBC programme.

Recently, I was listening to an episode about Newton's Principia that discussed the scientific trends of the time when Newton published his seminal book. In Cambridge, I had learned before that Newton's remark that he was standing on the shoulders of giants was not meant as modest as it might sound. In fact, it's meaning was rather sarcastic as it was referring to Robert Hooke whose bad posture mad it obvious that Newton was in fact saying that he did not learn anything from him.

To me, that had always sounded rather reasonable given that the law that carries Hooke's name does not sound particularly deep from a modern perspective: It states that to leading order the elastic deformation is linear in applied force, basically a statement saying that no surprise happens and the first order does not vanish. Formulated this way quite a minor contribution compared to Newton's axioms and his inverse square law for gravity.

From the BBC programe, however, I learned that the situation is not that simple: In fact, Hooke had already found an inverse square law for gravity experimentally and suggested to Newton in a letter that that might me responsible for the elliptic motions of the planets. Hooke himself did (could?) not prove that and was asking Newton for his opinion.

Later in their life, the two men both of difficult character had an ongoing dispute about scientific priorities and this is where the famous quotation is from. The wikipedia page contains more information about it and buts it in the context of (wave) optics rather than gravity.

Einstein book and Einstein thoughts

2008-04-04T12:19:00.003+02:00

Getting presents is not always easy especially if you you believe the presenter has put some thought into picking the present but failed due to lack of knowledge in the area of the present: Sometime in high school, for my birthday, a close friend gave me a cardboard circle of fifth so I could look up how many sharps there are in A major or G minor. That seemed like a good idea since I like to play music a lot. Except those kinds of things the cardboard display showed you are supposed to know and reproduce from memory (if not spinal cord) if you want to get into jazz improvisation. Thus, I could only produce some "uhm, thank you, how nice....".

Same thing happens with popular science physics books. I have not read any in many years since the density of information new to me is usually extremely low. Maybe I browse a bit in a book shop to see which topics are covered and read a page or two to see how a controversial topic is covered. I think this is the same for any worker in the field. Therefore, in the discussions of the controversial physics books of recent years, the authors' response to criticism of string theorists was often "you have not read the book" and indeed, this was true most of the time. But still, people haveing browsed through the book as above usually knew what was going on even without reading the book cover to cover.

That is the background to my reaction when my parents gave me a book for christmas which they had bought on their US trip in autumn: "My Einstein", a collection of essays edited by John Brockman. I assumed this would be just another Einstein book and one that was even a bit late for the Einstein year 2005. So the book sat on my shelf for a couple of months. But a few weeks ago I stated reading and was surprised: This was the most interesting book with a physics theme I have read in years! I can strongly recommend it!

The idea of the book is to ask 24 experts in fields related to Einstein's work or life to say what "Einstein" means to them. And the positive thing is that this is not 24 introductions to special relativity but 24 aspects of the physicist, the man, the pop star, the philosopher, the politician in 2006, more than fifty years after his death.

The authors include John Archibald Wheeler (the only one which actually interacted with Einstein), Lenny Susskind, Anton Zeilinger, Lee Smolin, George Smoot, Frank Tipler, George Dyson (the son of Freeman Dyson who was baby sit by Einstein's secretary), Maria Spirolulu, Lawrence Krauss and Paul Steinhardt. All of them find interesting and very diverse aspects of the Einstein topic and reading the book a number of physics questions came to my mind.

One essay was pointing out that what was peculiar about Einstein's way of thinking was that it was based on thought experiments and thus much more driven by elegance than by observation in the lab. This was illustrated by the fact that the reasoning that lead to special relativity was based on an analysis of Maxwell's equations (which are of course symmetric under Lorentz transformations, a fact which was known to Lorentz) rather than on an analysis of the Michelson Morley experiment.

One should note however that this argument is not logically tight: Of course it much more aesthetic to deduce from the fact that the speed of light comes out of Maxwell's equations that it should be universal and that if should be the same in all directions. However this does not follow directly as one can see by considering the acoustic analogue: From an analysis of kinetic gas theory one can deduce sound waves and the speed of sound can be expressed in terms of the molar weight of the gas etc. One finds a wave equation and that equation is invariant under boosts where the speed of light is replaced by the speed of sound. Under those acoustic Lorentz transformations, the speed of sound is the same in all frames. However, this is not a symmetry of the rest of nature and thus there is a preferred frame, the frame in which the air is at rest.

It could have been, that the world is invariant under Galilei transformations and Maxwell's equations hold only in a preferred frame (the rest frame of the ether say). This possibility cannot be ruled out by pure thought (like a Gedankenexperiment), one has to see which possiblity nature has chosen. And this is done in a Michelson Morley experiment for example.

But still: Buy that book and you will enjoy it!

You have to look hard to see quantum gravity

2008-03-06T17:59:00.003+01:00

Of course, we still don't know what the true theory of quantum gravity looks like. But many people have explained over and over again that there are already extremely tight constraints on what this theory will look like: That theory in the small should better look like the standard model for energies up to about 100GeV and it has to look like General Relativity for length scales starting from a few micrometers to tests at very high precision at scales of the size of the solar system (screwing up the laws of gravity at that scale just by a tiny amount will change the orbits of the moon --- known with millimeter precission --- an the planets on the long run and will very likely render the solar system unstable at time scales of the age of the solar system). At larger scales up to the Hubble scale we still have good evidence for GR, at least if you accept dark matter (and energy). These are the boundary conditions a reasonable candidate for quantum gravity has to live with.

Considerations like this suggest that most likely you will need observations close to the Planck scale to see anything new but not so radically new that you would have seen it already.

I know only of two possible exceptions: Large extra dimensions (which lower the Planck scale significantly) allowing for black hole production at colliders (which I think of as possible but quite unnatural and thus unlikely) and cosmological gravitational waves (aka tensor modes). At least string theory has not spoken the final verdict on those but there are indications that those are way beyond detection limits in most string inspired models (if you accept those as being well enough understood to yield reliable predictions).

Let me mention two ideas which I consider extremely hard to realise such they yield observable predictions and are now yet ruled out: The first example is a theory where you screw with some symmetry or principle by adding terms to your Lagrangian which have a dimensionful coefficient corresponding to high energies. Naively you would think that this should influence your theory only at those high energies and these effects are hidden for low energy observers like us unless we look really hard.

One popular example of such theories are theories that mess with the relativistic dispersion relation and for example introduce an energy dependent speed of light. Proponents of such theories suggest one should look at ultra high energy gamma rays which have traveled a significant fraction of the universe. Those often come in bursts of very short duration. If one assumes those gammas were all emitted at the same time but then one observes that the ones of higher energies within one burst arrive here either systematically earlier or later this would suggest that the speed at which they travel depends on the energy.

The problem with screwing with the dispersion relation is that you are very likely breaking Lorentz invariance. The proponents of such theories then say that this breaking is only by a tiny amount visible only at large energies. Such arguments might actually work in classical theories. But in a quantum theory, particles running in loops are transmitting such breaking effects to all scales.

Another way to put this: In the renormalisation procedure, you should allow for all counter terms allowed by you symmetries. For example in phi^4 theory, there are exactly three renormalisable, Lorentz invariant counter terms: phi^2, phi^4 and phi Box phi. Those correspond to mass, coupling constant and wave function renormalisation. But if your theory is not Lorentz invariant at some scale you have no right to exclude terms like for example phi d_x phi (where d_x is the derivative in the x-direction). Once those terms are there, they have no reason to have tiny coefficients (after renormalisation group flow). But we know with extremely high precission that those coefficients are extremely tiny if non-zero at all in the real world (in the standard model say).

So if your pet theory breaks Lorentz invariance you should better explain why (after taking into account loop effects) we don't see this breaking already today. So far, I have not seen any proposed theory that achieves this.

There is an argument of a similar spirit in the case of non-commutative geometry. If you start out with [x,y] = i theta then theta has dimension length^2 (and in more than 2D breaks Lorentz invariance but that's another story). If you make it small enough (Planck length squared, say) it's suggestive to think of it as a small perturbation which will go unnoticed at much larger scales (like the quantum nature of the world is not visible if you typical action is much larger than h-bar). Again, this might be true in the classical theory. But once you consider loop effects you have UV/IR mixing (the translation from large energy scales to low energy scales) and your tiny effect is seen at all scales. For example in our paper we worked it out in a simple example and demonstrated that a 1/r^2 Coulomb type force law is modified and the force dies out exponentially over distance scales of the order of sqrt(theta), the length scale you were going to identify with the Planck scale in the first place and, whoops, your macroscopic force is gone...

A different example are variable fundamental constants. At first, those look like an extremely attractive feature to a string theorist: We know that all those constants like the fine structure constant are for example determined by details of your compactification. Those in turn are parametrised by moduli and thus it's very natural to think of fundamental constants as scalar field dependent. Concretely, for example for the fine structure constant, in your Lagrangian you replace F^2 by sF^2 for some scalar s which you add and which by some potential is stabilised at the value that corresponds to alpha=1/137.

The problem is once again that either the fixing is so tight that you will not see s changing or s is effectively sourced by electric fields and shows up in violations of the equivalence principle as it transmits a fifth force. You might once more be able to avoid this problem by extreme fine tuning the parameters of this model just to avoid detection (you could make the coupling much weaker than the coupling to gravity). But this fine tuning is once more ruined by renormalisation and after including the first few loop orders, s will not anymore couple only to F^2 but to all standard model fields and have an even harder time to play hide and seek and up to today remain unobserved (an yes, tests of the equivalence principle have very high precission).

You might say that non-commutative geometry and varying constants are not really quantum gravity theories. But the idea should be clear: We already know a lot of things about physics and it's very hard to introduce radical new things without screwing up things we already understand. I'm not saying it's impossible. It's just that you have to be really clever and a naive approach is unlikely to get anywhere. So the New Einstein really has to be a new Einstein.

But in case you feel strong today, consider this job ad.

Zeit Sudoku Bookmarklet

2008-02-28T17:08:00.004+01:00

As you know, I am a Sudoku addict. For a nice ten minute break I often download the daily sudoku from Die Zeit. Until recently, from this bookmarked page, I had to change the level from "leicht" to "schwer" and then press the button for the PDF version to print out.

But now, they introduced even another page before that (where you have to choose between the Flash version and the puzzle from the printed (only weekly) newspaper or the oldstyle version that allows the download. You could no longer bookmark the next page as there the URL already contained some sort of session ID.

Of course, they want me to go through all these pages to make sure I do not miss any of the ads they want to present me. But I think that this new version requires a few mouse clicks too much and so I decided to have a look at the page's source code.

It turns out that the URL for the PDF no longer contains a session ID but instead contains today's date, slightly more than you can do with a static bookmark. But that got me thinking that one might be able to solve this problem with a bookmarklet, a bookmark that makes use of the fact you can have JavaScript in a URL and in a bookmark.

Last time I looked into JavaScript was roughly ten years ago and at that time it seemed like a very stupid idea to have some crippled language where you have to transmit all of the source code to the browser which then slowly interprets it if you can do much much more powerful things on the server side with CGI scripts.

Since then, a lot of time has passed and I have heard many interesting things (let me mention only AJAX) suggesting I should maybe reconsider my old dismissal of JavaScript. I had a look though a number of reference sheets and here it is: My first own JavaScript sniplet: A sudoku download bookmarklet. Clicking on this link (or bookmarking it and retrieving the bookmark) brings you directly to the PDF version on the latest "schweres" sudoku! Here is the source:


javascript:var d=new Date();
  var m=d.getMonth()+1;
  if(m<10){m="0"+m};
  var t=d.getDate();
  if(t<10){t="0"+t};
 open('http://sudoku.zeit.de/sudoku/kunden/die_zeit/pdf/sudoku_' +  
      d.getFullYear()+'-'+m+'-'+t+'_schwer.pdf')

Geometric Hamilton Jacobi

2008-02-07T16:17:00.000+01:00

Today, over lunch, togther with Christian Römmelsberger, we tried to understand Hamilton-Jacobi theory from a more geometric point of view.

The way this is usually presented (in the very end of a course on classical mechanics) is in terms of generating functions for canonical transformations such that in the new coordinates the Hamiltonian vanishes. Here I will rewrite this in the laguage of symplectic geometry.

As always, let us start with a 2N dimensional symplectic space M with symplectic form . In addition, pick N functions such that the submanifolds are Lagrangian, that is is a Lagrangian subspace of (meaning that the symplectic form of any two tangent vectors of vanishes). If this holds, the can be regarded as position coordinates.

Starting from these Lagrangian submanifolds, we can locally find a family of 1-forms in the normal bundle such that . You should think that for appropriate momentum coordinates on the Lagrangian leaves of constant . But here, these are just coefficient funtions to make a potential for .

Now we repeat this for another set of position coordinates which we assume to be "sufficiently independent" of the meaning that . This implies that locally are coordinates on M. With the comes another 1-form and since are locally related by a "gauge transformation". We have for a function F.

Let's look at a little bit closer. A general normal 1-form would look like . But since we started from Lagrangian leaves, there is no in and thus . But expressing this in coordinates yields

Comparing coefficients we find and . You will recognize the expressions for momenta in terms of a "generating function".

What we have done was to take two Lagrangian foliations given in terms of and and compute a function from them. The trick is now to turn this procedure around: Given only the and a function of these and some N other variables , one can compute the as functions on M: Take a point and define by inverting . For this remember that was defined implicitly above: It is the coefficient of in .

Up to here, we have only played symplectic games independent of any dynamics. Now specify this in addition in terms of a Hamilton function h. Then the Hamilton-Jacobi equations are nothing but the requirement to find a generating function such that the are constants of motion.

Even better, by making everything (that is h and Q and F) explicitly time dependent, by the requirement that the action 1-form is invariant: giving we get a transforming Hamilonian and we can require this to vanish:

If we think of the Hamiltonian h given in terms of the coordinates this is now a PDE for F which has to hold for all . That is, writing F as a function of and it has to hold for all (fixed) as a PDE in the and t.

Breaking it softly with this song

2008-02-05T17:45:00.000+01:00

Today, I would like to discuss a topic that came up in our lunch seminar. The issue however is not specific to this paper but applies to many model building constructions including KKLT. It is about the nature of supersymmetry breaking in many string theory constructions. At least for me the issue was not completely resolved during the seminar. So I dug a bit into the literature. This made it a bit clearer to me but still I would not say I completely understand it.

Maybe somebody more knowledgeable out there reads this and can give me some hints. Others might find interesting what I found out so far. This twofold motivation for writing this up should explain the nature of this post: I will start out with some well known background on susy breaking which everyone who for example has read Wess and Bagger should know. But then it will get more and more technical and end up right in the middle of my confusion.

As you know supersymmetry is a beautiful idea if you want to solve the hierarchy problem of why the electroweak scale of 100GeV given by the Higgs mass is relatively stable to quantum corrections. In plain vanilla QFT in 4d, the mass term of a scalar is quadratically divergent. If you happen to regulate your divergent integrals with a cut off you would expect your renormalised scalar masses generically to be given by the cut-off scale (which in case of the standard model would probably as high as the Planck scale).

In a supersymmetric theory, however, the leading divergeces of bosons and fermions running in the loop cancel. The running of the scalar mass is then only logarithmic, something you can live with.

The problem with supersymmetry is only that it is not a symmetry of nature. There is no mass degeneracy between bosons and fermions. In fact, the difference in mass is so big that we have so far seen only half of the multiplets. Thus at best, supersymmetry is broken in the real world.

This breaking of susy can be realised to different degrees: The simplest form is that it is only spontaneously broken: The theory (given in terms of a Lagrangian) is supersymmetric, applying a susy transformation to a solution again gives a solution. It just happens that the ground state is not invariant under such a transformation and thus susy is not a symmetry of this state.

For simplicity of exposition, let us restrict ourselves to rigid supersymmetry. Then the action is given in terms if two functions, the Kähler potential K and the superpotential W. Let us focus on the superpotential (the argument could be extended to a non-trivial Kähler potential as well). It is a function of the chiral superfields.

If the ground state of the theory were supersymmetric it would have to have zero energy. But the scalar potential is given by (the sum is over all chiral superfields). This can only be zero if there is a point in field space where is stationary w.r.t. all fields . If these equations do not have a simultaneous solution, the energy cannot be zero and susy is spontaneously broken. The loop calculations are not affected by this and thus the mass still runs logarithmically. There are good reasons to believe that this is not how susy breaking works in the real world.

The other extreme is that the theory is supersymmetric right form the start. The Lagrangian contains all sorts of terms and there is no hint of susy at all. The theory is a supersymmetric as generic rock is spherically symmetric. Of course in this situation there are no cancellations between divergences and down the drain is our solution of the hierarchy problem.

There is however a possibility in the middle between the two extremes: The theory is not supersymmetric, there are terms in the Lagrangian that break this symmetry right from the start. But these are not the most general terms but only those which do not destroy the logarithmic running of the mass. This situation dubbed "soft breaking" is what people think nature has chosen and for example it is realised in the MSSM.

Before you start calculating loop diagrams to check if in some non-susy theory the breaking happens to be soft let me tell you about an alternative characterisation: A theory is softly broken if it is possible to replace some coupling constants (the numerical coefficients of the susy breaking terms in the Lagrangian) by dynamical (super)-fields and add some other terms for these fields in such a way that you end up with a supersymmetric theory. You arrange the extra terms in such a way that in this theory the supersymmetry is spontaneously broken in such a way that the derivative in the direction of the new fields (called the "hidden sector") of the superpotential does not vanish in the ground state. Thus the theory is only softly broken if it can be augmented by a hidden sector such that the bigger theory is only spontaneously broken.

For example if is one of the (super) fields we started with and is a hidden sector superfield. If the ground state now has (a so called F-term) then a term in the augmented superpotential contains , a mass term only for the scalar part of but not for its fermionic part. This example thus shows that a scalar mass term (leading of a splitting of the susy mass degeneracy) is a soft term.

Another nice thing about a theory with soft breaking is that (at least after augmenting the theory) you can still express everything in the supersymmetric language including Kähler- and superpotential which is not the case for a general non-supersymmetric theory.

So much for background. Now to the issue that shoed up in the seminar: In many phenomenological string constructions you start from some compactification that you know and which happens to be supersymmetric. But then you want to make it more realistic (and for example have a positive cosmological constant). To this end you then for example you add some branes to the background that break the supersymmetry completely (for example by adding anti-branes if you already have branes). Now we would like to understand the the physical consequences of these additional branes (which are often put deep in some warped throat to make their effect small). What is now often done is to analyse this set-up still using the language of supersymmetry (for example by computing some super-potential of the Gukov-Vafa-Witten type). But this is at best only satisfied if the susy breaking is soft (and can be made spontaneous by augmenting the theory). If there are however hard (i.e. not soft) breaking terms you will never see them in this treatment even though they are most likely deadly for phenomenology.

Thus the question is: If I add these non-susy branes (or fluxes for that matter) by hand, will susy be broken only softly? I cannot imagine a reason why this should be true in general. But these people who write these papers are much more clever than I and most likely I am missing something. Please tell me what that could be!

What I found was arxiv:hep-th/0209206 where Matthias Klein studies a typical (but simple enough) example: He considers a theory on which is supersymmetric in the bulk. This theory is then rewritten in terms of four dimensional fields treating the coordinate on the interval as a continious parameter on the bulk fields. In five dimensions, the minimal number of supercharges is eight, thus from the four dimensional perspective this theory has . This bulk theory is then coupled to theories on the two boundaries. But this is done in a way such that the two bulk boundary couplings preserve different parts and thus the whole set-up of bulk with two boundaries is not supersymmetric. You can think of this as a toy model for a space-time with a brane and an anti-brane. Both these branes by themselves (including the bulk) preserve some susy but both of them at once do not.

Now Klein computes corrections to the mass of scalars on the boundary at say which are massless at tree level. The simplest diagram where we can expect that there are no susy cancellations has to involve fields also on the boundary at and is thus at two loop level. A typical diagram looks like this:

Here, the fields with horizontal lines are on the first brane, the vertical fields live in the bulk and the fields in the circle live on the second brane. Let us denote the momentum in the circle by q and the momentum in the lower loop by k. For this diagram we compute

Note the discrete component of the momentum for the bulk fields. As this component is not conserved on the branes (as the branes break translation in variance) there are two independent components for the two lines. The signs in the nominator are remnants of phases evaluated on the branes (depending on the exact boundary conditions of the bulk fields at the branes). The components should have appeared in the nominator as well but give vanishing contributions upon .

Gamma gymnastics brings the nominator to the form which is then rewritten via . The first and third term cancel then upon a shift in the q integration. We are left with

We find that the q integration diverges logarithmically. That's not too bad. Next, we do the sum over the 's. With a little help of mathematica, we find

This is the real surprise: The integrand is exponentially suppressed for large k. Thus, it is actually UV finite! Thus the only UV divergence of this diagram is the logarithmic divergence of the q integration. Of course, this is only one of many diagrams. But the claim is that this is generic: At least for two loops, one loop is running only on one brane including bulk-brane vertices and is thus protected by the bulk-single brane susy to diverge only like a log. The other loop involves a propagator across the bulk which makes this integration finite. Thus all contributions to the scalar (and other) masses on one brane are only log divergent. The claim seems to be that this is also true at higher loops although I do not quite see how to substantiate this. But if this is true, although there is no supersymmetry from the outset, the hierarchy is still protected.

From this point of view, this set-up in which susy is preserved locally but which breaks it only globally is an example of soft breaking. I must say, this is against my intuition as it seems to me that constructing some branes that in total break susy is as bad as it can get and I see no reason why this should be soft in any way.

But if this is true is there a possibility to augment this theory and find that the breaking is spontaneous in that bigger theory?

Even more important: Is the breaking soft enough that we can write down a low energy effective action in the supersymmetric language in terms of a K&aauml;hler and a super potential? Or are there other terms as well?

Resistance is real

2007-12-11T11:52:00.001+01:00

Following Clifford's hint I have had my nerdness tested. The version 2 result is here:

It seems a science/math post is long overdue. So here it is: Resistance of a wave guide/coax cable. Readers of the Feynman Lectures know that you can model a coax cable by an infinite sequence of capacitors and inductance likethis

To compute the asymptotic (complex) resistance of this infinite circuit Feynamn instructs you look at a single iteration and to summarise the rest in some black box with resistance R.

The new resistance between the terminals is then easily computed (when driven with frequency w):

Now you argue that an infinite chain should not change its resistance if one link is added and thus R=R'. This quadratic equation is solved as

The final thing is to remember that the chain of L's and C's is a discrete version of a continuous chain and thus one should take both L and C to zero while keeping their ratio fixed. We end up with

Note that in this limit the frequency w dropped out. So far the Feynman lectures.

But there is one curious thing: Although we have been adding only capacitors and inductances which have purely imaginary resistances and no Ohmic (real) resistance, nevertheless, the limit is real!

How can this be true? When you think about the physics you should be even more irritated: Neither capacitors nor inductances do any work and only Ohmic resistance produces heat. By adding together elements that do not produce heat. After meditating this fact for a while one settle with the explanation that one might say that the energy is carried down the infinite circuit and never returns and thus is gone and might be considered heat. But this is not really convincing.

So we should better study the maths to some more detail. What we have done was to consider a sequence of resistances and computed the possible fixed points of this iteration. If the resistance converges it certainly will converge to a fixed point. But who tells you it really converges?

So, let's again add one infinitessimal bit of chain. Let us use new variables z and x such that

Thus z is the fixed point resistance we computed above and x is the thing which we take to 0 (and thus we can work at O(x)). We do the above calculation again and find that the change in resistance is

We can view i(z^2-R^2)/z as a vector field in the complex R plane

We can see (either from the formula or the plot) that for purely imaginary R we will always stay on the imaginary axis and flow to complex infinity! Without Ohmic resistance we will not flow towards the fixed points.

But even if we start off slightly off the imaginary axis we do not spiral in to one fixed point as one might have thought: The vector field is holomorphic and thus Hamiltonian. Therefore there is a conserved quantity (although right now I am too tired to compute it).

Well, I thought I might not be too tired, got confused for two entire hours and with the help of Enrico (who suggested separation of variables) and Jan found the conserved quantity to be

UPDATE: When I wrote this last night I was too tired to correctly compute the real part of 1-R^2. Thus I got the real components of the vector field wrong and this explains why I had such trouble to find the correct conserved quantity. After one short night of sleep I noticed my error and indeed the conserved quantity I had calculated earlier was correct,

(setting w=z=1) but the vector field was wrong (plots are corrected as well). See for example

Lehrer Video

2007-11-28T13:54:00.000+01:00

I bet many of you out there love Tom Lehrer songs as much as I do. So I hope you enjoy this video showing the master himself performing some of his maths songs:

There are some more Lehrer songs on youtube and especially this superb performance/animation of New Math:

An example of examples: Series and limits (in German)

2007-11-25T17:55:00.001+01:00

As an example of how I would explain a concept in terms of examples and counter-examples let me cut and paste a text that I wrote for a mailing list which explains the notion of series and limits to ninth grader. That mailing list is in German so is this text. Sorry.

Ich versuche es mal mit einer Prosabeschreibung. Also erstmal, was ist eine Folge? Einfach gesagt ist das ein Liste von Zahlen, die nicht aufhoert, also zB

1, 2, 3, 4, 5 etc.

oder auch

1, 1, 1, 1, 1, 1 etc.

oder auch

1, 1/2, 1/3, 1/4, 1/5, etc

oder auch

3, 3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, etc

oder auch

1, -1, 1, -1, 1, -1, etc

Vornehm gesagt ist eine Folge nix weiter als eine Funktion von den natuerlichen Zahlen in eine (Zahlen)-Menge Deiner Wahl. D.h. fuer jede natuerliche Zahl n (die Position in der Folge) gibt es eine Zahl a_n. Im ersten Beispiel ist

a_n = n

im zweiten Beispiel

a_n = 1

im dritten Beispiel

a_n = 1/n

und im vierten Beispiel ist a_n die Zahl, die man erhaelt, wenn man von pi die ersten n Dezimalstellen nimmt. Die fuenfte Folge koennen wir schreiben als


          n
a_n = (-1)

(minus eins hoch n)

Soweit alles klar?

Von einer Folge kann es nun sein, dass sie gegen einen Grenzwert a kovergiert (sie "diesen Grenzwert hat"). Grob gesagt soll das heissen, dass sie 'auf lange Sicht' der Zahl a immer naeher kommt. Das muss man nun etwas formalisieren. Eine moegliche Definition ist, dass fuer alle offenen Intervalle, die a enthalten, hoechstens endlich viele Glieder der Folge nicht auch schon in diesem Intervall liegen, egal wie klein das offene Intervall ist (wenn es kleiner wird, liegen halt mehr Folgenglieder nicht drin, aber es bleiben immmer endlich viele).

Nehmen wir zB das dritte Beispiel a_n = 1/n . Davon ist offenbar 0 der Grenzwert. Wir koennen das ueberpruefen. Ueberleg Dir ein offenes Intervall, das die 0 enthalet, also zb ]l,r[ . Damit die 0 drin ist, muss l negativ und r positiv sein. Offenbar liegen nur die a_n fuer die n<1/r ist, nicht in dem Intervall, alle anderen liegen drin, also haben wir tatsaechlich nur endlich viele Ausnahmen, egal welches Intervall wir nehmen.

Das zweite Beispiel, a_n = 1, hat auch einen Grenzwert, naemlich natuerlich die eins. Ein offenes Intervall, das die 1 enthaelt, enthaelt auch alle Folgenglieder, es gibt also ueberhaupt keine Ausnahmen.

An den beiden Beispielen sehen wir auch, dass es volkommen egal ist, ob der Grenzwert selber in der Folge vorkommt.

Bei der Definition ist es aber wesentlich, dass wir nur offfene Intervalle zulassen. Sonst koennten wir fuer die 1/n Folge das geschlossene Interval [0, 0] nehmen, dieses enthaelt zwar die Null, aber kein einziges Folgenglied, damit liegen alle, also unendlich viele Folgenglieder nicht im Intervall. Ueberlege Dir selbst, welche Folgen konvergieren wuerden, wenn wir geschlossene Invervalle nehmen wuerden.

Das Beispiel mit den Dezimalstellen von pi ist auch konvergent und hat den Grenzwert pi.

Du kannst Dir auch leicht ueberlegen, dass eine Folge nicht mehrere Zahlen als Grenzwert haben kann: Haette sie zwei verschidene Grenzwerte, koenntest Du zwei offene Intervalle I1 und I2 benutzen, die jeweils nur einen der beiden Grenzwerte enthalten und deren Schnitt leer ist (gegebenfalls musst Du sie entsprechend verkleinern). Dann muessen alle bis auf endlich viele der Folgenglieder in I1 enthalten sein. Daraus folgt aber, dass unendlich viele Folgenglieder nicht in I2 sind. Also gibt es einen Widerspruch zu der Annahme, dass ein ein Grenzwert in I2 ist.

Die fuenfte Folge, die abwechselnd 1 und -1 ist, ist hingegen nicht konvergent, sie hat keinen Grenzwert: Als Grenzwert kaemen sowieso nur 1 und -1 in Frage. Schauen wir uns also das offene Intervall

] 1/2 , 1 1/2 [

an. Dann liegen da zwar unendlich viele Folgenglieder drin (naemlich jedes zweite), aber es liegen auch unenedlich viele Folgenglieder nich drin, naemlich die restlichen. Also kann 1 kein Grenzwert sein, denn es gibt ein offenes Intervall, das 1 enthaelt, aber unendlich viele Folgenglieder nicht.

Bleibt noch die erste Folge a_n = n. Wenn wir als Grenzwert nur 'normale' Zahlen zulassen, dann hat die Folge keinen Grenzwert, da die Folgenglieder aus jedem endlichen offenen Intervall herauslaufen. Wir koennen aber auch "unendlich" als Grenzwert zulassen, wenn wir es als Obergrenze fuer offene Intervalle erlauben. So soll etwa

] l, unendlich [

die Menge aller Zahlen, die groesser als l sind sein. Nun koennten wir definieren, dass eine Folge gegen unendlich konvergiert, wenn in allen solchen Intervallen bis auf endlich viele Ausnahmen alle Folgenglieder drin liegen. In diesem Sinn konvergiert die erste Folge dann gegen unendlich. Auf aehnlich weise kann man dann auch definieren, was es heissen soll, dass eine Folge gegen minus unendlich konvergiert.

Wenn ich das urspruengliche Beispiel von Lukas richtig verstanden habe, war da der Witz, dass seine Folge abwechselnd positive und negative Zahlen haben sollte, die im Betrag immer groesser werden. Diese Folge konvergiert dann aber weder gegen unendlich noch minus unendlich aus dem gleichen Grund, wie die 1, -1, 1, Folge nicht gegen 1 oder -1 konvergiert.

Soweit zum Grenzwert und Konvergenz. Das Beispiel mit 1, -1,... suggeriert aber noch die Definition eines aehnlichen Begriffs, der aber in einem gewissen Sinn schwaecher ist: des Haeufungspunkts. Eine Zahl a ist ein Hauefungspunkt einer Folge, wenn in jedem offenen Interval, egal wie klein, das a enthaelt, auch unendlich viele Folgenglieder drin sind. Hier wird aber nichts darueber gesagt, wieviele Folgenglieder nicht drin sein duerfen.

Du ueberlegst Dir schnell, dass einen Folge, die einen Grenzwert a, auch a als Haeufungspunkt hat (und keinen weiteren). Die Folge mit den 1ern und -1ern hat zwei Haeufungspunkte, naemlich 1 und -1. Im Gegensatz zum Grenzwert kann eine Folge also mehrere Haeufungspunkte haben.

Die Folge 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, etc hat zB fuenf Haeufungspunkte, die Folge

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, etc

hat alle natuerlichen Zahlen als Haufungspunkt. Mit einem kleinen Trick ('Cantorsches Diagonalverfahren') kann man sich auch eine Folge ueberlegen, die alle rationalen oder sogar alle reellen Zahlen als Haeufungspunkt hat.

Von besonderem Interesse sind manchmal noch der groesste und der kleinste Haeufungspunkt einer Folge, Limes Superior und Limes Inferior genannt. Es ist eine Eigenschaft der reellen Zahlen, dass jede Folge von reellen Zahlen mindestens einen Haeufingspunkt hat (wenn man auch minus unendlich und unendlich als Haeufungspunkte zulaesst). Dieser Satz ist unter dem Namen "Satz von Bolzano Weierstrass" bekannt (siehe Wikipedia). Fuer die rationalen Zahlen stimmt er nicht (eines der obigen Beispiele fuer Folgen ist ein Gegenbeispiel, welches?)

Unsere Feststellung von oben kann man aber auch umkehren: Wenn eine Folge (im reellen) nur einen Haufungspunkt hat, der groesste also gleich dem kleinsten Haeufungspunkt ist, ist dieser automatisch auch schon Grenzwert der Folge und die Folge ist konvergent. Kannst Du das selber beweisen?

Soweit mein kleiner Crash-Kurs zum Thema Konvergenz von Folgen.

An example for example

2007-11-06T16:34:00.000+01:00

Tim Gowers has two very interesting posts on using examples early on in a mathematical exposition of a subject. I can only second that and say that this is my favorite way of understanding mathematical concepts: Try to think through the simplest non-trivial example.
Of course, for a mathematician it could be enough just to state a definition or state a theorem (including a proof) but very often this leaves one without a proper understanding of the subject. Why this definition and not something else? Where and how can I use the theorem? Why do I have to make assumption x, y and z?

For my practical purposes I often want to see "the key example" that shows what's going on and then often the general theory is a more or less obvious generalisation or extension or formalisation or abstraction of this key example. This second step hopefully is then clear enough that one can come up with it by oneself and it's really the key example one should remember and not the formal wording of the definition/theorem etc.

I am not talking about those examples some mathematicians come up with when pressed for an example, like after stating the definition of a vector space giving {0} as the example. This is useless. I want to see a typical example, one that many (all) other cases are modeled on not the special one that is different from all other cases. And as important as examples are of course counter-examples: What is close but not quite according to the new definition (and why do we want to exclude it? What goes wrong if I drop some of the assumptions of the theorem?

I have already talked for too long in the abstract, let me give you some examples:

What's a sheaf and what do I need it for (at least in connection with D-branes)? Of course, there is the formal definition in terms of maps from open sets of a topological space into a category. The wikipedia article Sheaf reminds you what that is (and explains many interesting things. I think I only really understood what it really is after I realised that it's the proper generalisation of a vector bundle for the case at hand: A vector bundle glues some vector space to every point of a topological space and does that in a continuous manner (see, that's basically my definition of a vector bundle). Of course, once we have such objects, we would like to study maps between them (secretly we want to come up with the appropriate category). We know already what maps between vector-spaces look like. So we can glue them together point-wise (and be careful that we are still continuous) and this gives us maps between vector bundles. But from the vector-space case we know that then a natural operation is to look at the kernel of such a map (and maybe a co-kernel if we have a pairing). We can carry this over in a point-wise manner but, whoops, the 'kernel-bundle' is not a vector bundle in general: The dimension can jump! The typical example here is to consider as a one dimensional vector bundle over the real line (with coordinate x). Then multiplication in the fiber over the point x by the number x is trivially a fiber-wise linear map. Over any point except x=0 it has an empty kernel but over x=0 the kernel is everything. Thus, generically the fiber has dimension 0 but at the origin it has dimension one. Thus, in order to be able to consider kernels (and co-kernels) of linear bundle maps we have to weaken our definition of vector bundle and that what is a sheaf: It's like a vector bundle but in such a way that linear maps all have kernels and co-kernels.
When I was a student in Hamburg I had the great pleasure to attend lectures by the late Peter Slodowy (I learned complex analysis from him as well as representation theory of the Virasoro algebra, gauge theories in the principle bundle language, symplectic geometry and algebraic geometry). The second semester of the algebraic geometry course was about invariants. Without the initial example (which IIRC took over a week to explain) I would have been completely lost in the algebra: The crucial example was: We want to understand the space of matrices modulo similarity transformations . Once one has learned that the usual algebraic trick is to investigate a space in terms of the algebra of functions living on it (as done in algebraic geometry for polynomial functions, or in non-commutative geometry in terms of continuous functions) one is lead to the idea that this moduli space is encoded in the invariants, that is functions that do not change under similarity transformations. Examples of such functions are of course the trace or the determinant. It turns out that this algebra of invariants (of course the sum or product of two invariant functions is still invariant) is generated by the coefficients of the characteristic polynomial that is, by the elementary symmetric functions of the eigenvalues (eigenvalues up to permutations). So this should be the algebra of invariants and its dual the moduli space. But wait, we know what the moduli space looks like from linear algebra: We can bring any matrix to Jordan normal form and that's it, matrices with different Jordan normal forms are not similar. But both and have the same characteristic polynomial but are not related by a similarity transformation. In fact the second one is similar to any matrix
for any . This shows that there cannot be a continuous (let alone polynomial) invariant which separates the two orbits as the first orbit is a limit point of points on the second orbit. This example is supposed to illustrate the difference between which is the naive space of orbits which can be very badly behaved and the much nicer which is the nice space of invariants.
Let me also give you an example for a case where it's hard to give an example: You will have learned at some point that a distribution is a continuous linear functional on test-functions. Super. Linear is obvious as a condition. But why continuous? Can you come up with a linear functional on test-functions which fails to be continuous? If you have some functional analysis background you might think "ah, continuous is related to bounded, let's find something which is unbounded". Let me assure you, this is the wrong track. It turns out you need the axiom of choice to construct an example (as in you need the axiom of choice to construct a set which is not Lebesgue measurable). Thus you will not be able to write down a concrete example.
Here is a counter-example: Of course is the typical example of a real finite dimensional vectors space. But it is very misleading to think automatically of when a real vector space is mentioned. People struggled long enough to think of linear maps as abstract objects rather than matrices to get rid of ad hoc basis dependence!

I a sitting in the "Mathematical quantum mechanics" class of our TMP program. There, to my mind, Prof. Siedentop does a pretty good job of motivating the ideas (in terms of examples and 'engineer talk' --- he means physicists talk) behind the definitions and proofs that he presents rather than getting bogged down in the technical details. Let's hope our (the physicists) insistence on this background information does not make him lose so much time he does not get to where he wants in the end...

Flickr upload

2007-10-29T15:26:00.000+01:00

I had not used Flickr in 18 months and when I wanted to use it today my script didn't work anymore since they have changed their API significantly in the meantime. Simply updating the perl module was not enough and I found the documentation the the web to be rather cryptic. Thus as a service to the community, this is what finally worked for me:

#!/usr/bin/perl

use Flickr::API;
use Flickr::Upload;

# Path to pictures to be uploaded
my $flickrdir = '/home/robert/fotos/flickr';

my $flickr_key = 'PUT YOUR KEY HERE';
my $flickr_secret = 'PUT YOUR SECRET HERE';

my $ua = Flickr::Upload->new( {'key' => $flickr_key, 'secret' => $flickr_secret} );
$ua->agent( "perl upload" );

my $frob = getFrob( $ua );
my $url = $ua->request_auth_url('write', $frob);
print "1. Enter the following URL into your browser\n\n",
"$url\n\n",
"2. Follow the instructions on the web page\n",
"3. Hit when finished.\n\n";

<>;
my $auth_token = getToken( $ua, $frob );
die "Failed to get authentication token!" unless defined $auth_token;

print "Token is $auth_token\n";

opendir(FLICKR, $flickrdir) || die "Cannot open flickr directory $flickrdir: $!";
while(my $fn = readdir FLICKR){
next unless $fn =~ /[^\.]/;
print "$flickrdir/$fn\n";

$ua->upload(
'auth_token' => $auth_token,
'photo' => "$flickrdir/$fn",
'is_family' => 1
) or print "Failed to upload $fn!\n";

}

sub getFrob {
my $ua = shift;

my $res = $ua->execute_method("flickr.auth.getFrob");
return undef unless defined $res and $res->{success};

# FIXME: error checking, please. At least look for the node named 'frob'.
return $res->{tree}->{children}->[1]->{children}->[0]->{content};
}

sub getToken {
my $ua = shift;
my $frob = shift;

my $res = $ua->execute_method("flickr.auth.getToken",
{ 'frob' => $frob ,
'perms' => 'write'} );
return undef unless defined $res and $res->{success};

# FIXME: error checking, please.
return $res->{tree}->{children}->[1]->{children}->[1]->{children}->[0]->{content};
}

You need a key and a secret which you can generate here. Of course, you also need the module

perl -MCPAN -e shell
install Flickr::Upload

Then all you have to do is to copy (or link) all pictures to be uploaded to one directory (/home/robert/fotos/flickr in my case) and run the script. It gives you an URL you have to paste into your browser and then press ok and the upload begins.

Quantum Field lectures and some notes

2007-10-26T17:58:00.001+02:00

The past few weeks here were quite busy as now the semester has started (October 15th) and with it the master program "Theoretical and Mathematical Physics" has become reality with the first seven student (one of them attracted apparently via this blog) have arrived and are now taking classes in mathematical quantum mechanics, differential geometry, string theory, quantum electrodynamics, conformal field theory, general relativity, condensed matter theory and topology (obviously not everybody attends all these courses).

I have already fulfilled my teaching obligation by teaching a block course "Introduction to Quantum Field Theory" the two weeks before the semester. Even though we had classes both in the morning and the afternoon for two weeks there was obviously only a limited amount of time and I had to decide which small part of QFT I was going to present. I came up with the following

Leave out the canonical formalism completely. Many courses start with it as this was the historical development and students will recognize commutation relations from quantum mechanics classes. But the practical use of it is limited: As soon as you get to interacting theories it becomes complicated and the formalism is just horrible as soon as you have gauge invariance. Of course, it's still possible to use it (and it is the formalism of choice for some investigations) but it's definitely not the simplest choice to be presented in an introductory class.
Thus, I was going to use the path integral formalism from day one. I spend the first two days introducing it via a series of double (multi) slit (thought) experiments motivating that a sum over paths is natural in quantum mechanics and then arguing for the measure factor by demanding the correct classical limit in a saddle point approximation. This heuristic guess for the time evolution was then shown to obey Schrödinger's equation and thus equivalence with the usual treatment was established at least for systems with a finite number of degrees of freedom.
In addition, proceeding using analogies with quantum mechanics can lead to some confusion (at least it did for me when I first learned the subject): The Klein-Gordon equation is often presented as the relativistic version of Schrödinger's equation (after discarding an equation involving a square root because of non-locality). Later then it turns out, the field it describes is not a wave function as it cannot have a probability interpretation. The instructor will hope that this interpretation is soon to be forgotten because it's really strange to think of the vector potential as the wave function of the photon which would be natural from this perspective. And if the Klein-Gordon field is some sort of wave function, why does it again need to be quantised? So what kind of objects are the field operators and what do they act on? In analogy with first quantisation one would guess they act on wave functionals that map field configurations on a Cauchy surface to complex numbers which are in some functional integral sense square integrable. OK, Fock space does the job but again, that's not obvious.
All these complications are avoided using path integrals. At least if one gets his head around these weird infinite dimensional integrals and the fact that in between we have to absorb infinite normalisation constants. But then, only a little bit later, one arrives at Feynman rules and for example the partition function for the free field is a nice simple expressions and all strange integrals are gone (they have been performed in a Gaussian way).
So instead of requantising an already (pseudo) quantum theory, I introduced the Klein-Gordon equation just as any classical equation of motion of a system which happens to have a continuum of degrees of freedom (I did it via the continuum limit of some "balls with springs" model). Thus before getting into any fancy quantum business, we solved this field equation (including the phi^4 interaction) classically. Doing that perturbatively, we came up with Feynman rules (tree diagrams only of course) and a particle-wave duality while still being entirely classical. As I am not aware of a book which covers Feynman diagrams from a classical perspective I have written up some lecture notes of this part. They also include the discussion of kink solutions which were an exercise in the course and which suggest the limitations of the perturbative approach and how solitonic objects have to be added by hand. (To be honest, advertising these lecture notes is the true purpose of this post... Please let me know you comments and corrections!)
The other cut I decided to make was to restrict attention only to the scalar field. I did not discuss spinors or gauge fields. They are interesting subjects for themselves but I decided to focus on features of quantum field theories rather than representation theory of the Lorentz group. The Dirac equation is a nice subject by itself and discussing gauge invariance leading to a kinetic operator which is not invertible (and thus requiring a gauge fixing and eventually ghosts to make it invertible) would have been nice, but there was no time. But as I said, there is a regular course on QED this semester and there all these things will be covered.
These severe cuts allowed us to get quite deep into the subject: When I took a QFT course, we spend the entire first semester discussing only free fields (spin 0, 1/2 and 1). Here, in this course, we managed to get to interacting fields in only two weeks including the computation of 1 loop diagrams. We computed the self-energy correction and the fish graph (including Schwinger parameters, Feynman trick and all that) went through their dimensional regularisation and renormalisation (including a derivation of the important residues of the gamma function). The last lecture, I could even sketch the idea of the renormalisation group, running coupling constants and why nature seems to use only renormalisable theories for particle physics (as the others have vanishingly small couplings at our scales).

This was quite an ambitious program but I think it went quite well (and the exam shows that I managed to transmit at least some information).

As far as books are concerned: For the preparation, I used Ryder, my favorite QFT text for large parts (and Schulman's book for the path integrals in quantum mechanics introduction). Only later I discovered that Zinn-Justin's book has a very similar approach (at least if you ignore all material on fields other than spin 0 and all the discussions of critical phenomena). Only yesterday, a copy of the new QFT book by Srednicki arrived on my desk (thanks CUP!) and from what I read there so far, this looks also extremely promising!

For your entertainment, I have also uploaded the exercise sheets here:
1 2 3 4 5

PS: If instead of learning QFT in two weeks you want to learn string theory in two minutes check this out.Didn't know molecules were held togehter by the strong force, though...

The fun of cleaning up

2007-09-19T12:37:00.000+02:00

Since early childhood I hate cleaning up. Now that I am a bit older, however, I sometimes realise it has to be done, especially of other people are involved (visitors, flat-mates, etc). See however this and this.

And yes, when I'm doing DIY or installing devices/cables/networks etc I am usually satisfied with the "great, it works (ahem, at least in principle)" stage.

But today, (via Terry Tao's blog) I came across The Planarity Game, which might have changed my attitude towards tidying up... Have fun!

Not quite infinite

2007-09-18T16:09:00.000+02:00

Lubos has a memo where he discusses how physicists make (finite) sense of divergent sums like 1+10+100+1000+... or 1+2+3+4+5+... . The last is, as string theorists know, of course -1/12 as for example explained in GSW. Their trick is to read that sum as the value at s=-1 of and define that value via the analytic continuation of the given expression which is well defined only for real part of s>1.

Alternatively, he regularises as . Then, in an obscure analogy with minimal subtraction throws away the divergent term and takes the finite remainder as the physical value.

He justifies this by claiming agreement with experiment (here in the case of a Casimir force). This, I think, however, is a bit too weak. If you rely on arguments like this it is unclear how far they take you when you want to apply them to new problems where you do not yet know the answer. Of course, it is good practice for physicists to take calculational short-cuts. But you should always be aware that you are doing this and it feels much better if you can say "This is a bit dodgy, I know, and if you really insist we could actually come up with a rigorous argument that gives the same result.", i.e. if you have a justification in your sleeve for what you are doing.

Most of the time, when in a physics calculation you encounter an infinity that should not be there (of course, often "infinity" is just the correct result, questions like how much energy I have to put into the acceleration of an electron to bring it up to the speed of light? come to my mind), you are actually asking the wrong question. This could for example be because you made an idealisation that is not physically justified.

Some examples come to my mind: The 1+2+3+... sum arises when you try to naively compute the commutator of two Virasoro generators L_n for the free boson (the X fields on the string world sheet). There, L_n is given as an infinite sum over bilinears in a_k's, the modes of X. In the commutator, each summand gives a constant from operator ordering and when you sum up these constants you face the sum 1+2+3+...

Once you have such an expression, you can of course regularise it. But you should be suspicious that it is actually meaningful what you do. For example, it could be that you can come up with two regularisations that give different finite results. In that case you should better have an argument to decide which is the better one.

Such an argument could be a way to realise that the infinity is unphysical in the first place: In the Virasoro example, one should remember that the L_n stand for transformations of the states rather than observables themselves (outer vs. inner transformations of the observable algebra). Thus you should always apply them to states. But for a state that is a finite linear combination of excitations of the Fock vacuum there are always only a finite number of terms in the sum for the L_n that do not annihilate the state. Thus, for each such state the sum is actually finite. Thus the infinite sum is an illusion and if you take a bit more care about which terms actually contribute you find a result equivalent to the -1/12 value. This calculation is the one you should have actually done but the zeta function version is of course much faster.

My problem with the zeta function version is that to me (and to all people I have asked so far) it looks accidental: I have no expansion of the argument that connects it to the rigorous calculation. From the Virasoro algebra perspective it is very unnatural to introduce s as at least I know of no way to do the calculation with L_n and a_k with a free parameter s.

Another example are the infinities that arise in Feynman diagrams. Those arise when you do integrals over all momenta p. There are of course the usual tricks to avoid these infinities. But the reason they work is that the integral over all p is unphysical: For very large p, your quantum field theory is no longer the correct description and you should include quantum gravity effects or similar things. You should only integrate p up the scale where these other effects kick in and then do a proper computation that includes those effects. Again, the infinity disappears.

If you have a renormalisable theory you are especially lucky: There you don't really have to know the details of that high energy theory, you can subsume them into a proper redefinition of your coupling constants.

A similar thing can be seen in fluid dynamics: The Navier-Stokes equation has singular solutions much like Einstein's equations lead to singularities. So what shall we do with for example infinite pressure? Well, the answer is simple: The Navier-Stokes equation applies to a fluid. But the fluid equations are only an approximation valid at macroscopic scales. If you look at small scales you find individual water molecules and this discreteness is what saves you actually encountering infinite values.

There is an approach to perturbative QFT developed by Epstein and Glaser and explained for example in this book that demonstrates that the usual infinities arise only because you have not been careful enough earlier in your calculation.

There, the idea is that your field operators are actually operator valued distributions and that you cannot always multiply distributions. Sometimes you can, if their singularities (the places where they are not a function but really a distribution) are in different places or in different directions (in a precise sense) but in general you cannot.

The typical situation is that what you want to define (for example delta(x)^2) is still defined for a subset of your test functions. For example delta(x)^2 is well defined for test functions that vanish in a neighbourhood of 0. So you start with a distribution defined only for those test functions. Then, you want to extend that definition to all test-functions, even those that are finite around 0. It turns out that if you restrict the degree of divergence (the maximum number of derivatives acting on delta, this will later turn out to be related to the superficial scaling dimension) to be below some value, there is a finite dimensional solution space to this extension problem. In the case of phi^4 theory for example the two point distribution is fixed up to a multiple of delta(x) and a multiple of the d'Alambertian of delta(x), the solution space is two dimensional (if Lorentz invariance is taken into account). The two coefficients have to be fixed experimentally and of course are nothing but mass and wave function renormalisation. In this approach the counter terms are nothing but ambiguities of an extension problem of distributions.

I has been shown in highly technical papers, that this procedure is equivalent to BPHZ regularization and dimensional regularisation and thus it's save to use the physicist's short-cuts. But it's good to know that the infinities that one cures could have been avoided in the first place.

My last example is of slightly different flavour: Recently, I have met a number of mathematical physicists (i.e. mathematicians) that work on very complicated theorems about what they call stability of matter. What they are looking at is the quantum mechanics of molecules in terms of a Hamiltonian that includes a kinetic term for electrons and Coulomb potentials for electron-electron and electron-nucleus interactions. The position of the nuclei are external (classical) parameters and usually you minimise them with respect to the energy. What you want to show is that the spectrum of this Hamiltonian is bounded from below. This is highly non-trivial as the Coulomb potential itself alone is not bounded from below (-1/r becomes arbitrarily negative) and you have to balance it with the kinetic term. Physically, you want to show that you cannot gain an infinite amount of energy by throwing an electron into the nucleus.

Mathematically, this is a problem about complicated PDE's and people have made progress using very sophisticated tools. What is not clear to me is if this question is really physical: It could well be that it arises from an over-simplification: The nuclei are not point-like and thus the true charge distribution is not singular. Thus the physical potential is not unbounded from below. In addition, if you are worried about high energies (as would be around if the electron fell into a nucleus) the Schrödinger equation would no longer be valid and would have to be replaced with a Dirac equation and then of course the electro-magnetic interaction should no longer be treated classically and a proper QED calculation should be done. Thus if you are worried about what happens to the electron close to the nucleus in Schrödinger theory, you are asking an unphysical question. What still could be a valid result is that you show (and that might look very similar to a stability result) is that you don't really get out of the area of applicability of your theory as the kinetic term prevents the electrons from spending too much time very close to the nucleus (classically speaking).

What is shared by all these examples, is that some calculation of a physically finite property encounters infinities that have to be treated and I tried to show that those typically arise because earlier in your calculation you have not been careful and stretched an approximation beyond its validity. If you would have taken that into account there wouldn't have been an infinity but possible a much more complicated calculation. And in lucky cases (similar to the renormalisable situation) you can get away with ignoring these complications. However you can sleep much better if you know that there would have been another calculation without infinities.

Update: I have just found a very nice text by Terry Tao on a similar subject to "knowing there is a rigorous version somewhere".

Not my two cent

2007-08-16T12:09:00.000+02:00

Not only that theoretical physicists should be able to estimate any number (or at least its exponent), we feel that we can say something intelligent about almost any topic especially if it involves numbers. So today, I will give a shot at economics.

As a grad student, I had proposed to a friend (fellow string theory PhD student) that I would guess that with about three months study it should be possible to publish a research paper in economics. That was most likely complete hybris but I never made the effort (but I would like to point out that still my best cited paper(364 and counting) was written after only three months as a summer student stating from scratch in biophysics (but of course with great company who, however at that point were also biophysics amateurs)).

About the same time, I helped a friend with the math (linear inequalities mostly) his thesis in macro-economics only to find out his contribution to money market theory was the introduction of new variable to the theory which he showed to be too important to neglect but which unfortunately is not an observable... (it was about the amount of a currency not in its natural country but somewhere else which makes the central bank underestimate the relative change when they issue a certain absolute amount of that currency into the market. For example about two thirds of the US$760 billion are estimated to be overseas and the US dollar is even the official currency in a number of countries other than the US according to Wikipedia).

Economics is great for theoretical physicists as large parts are governed by a Schödinger equation missing an i (a.k.a. diffusion equation or Black-Scholes equation) and thus path integral techniques come in handy when computing derivative prices. However, it's probably the deviations from BS where the money is made as I learned from a nice book written by ex-physicists now making money by telling other people how to make money.

Of course this is a bit worrying: Why do consultants consult rather than make money directly? This is probably connected with my problem of understanding economic theory at stage one: All these derivations start out with the assumption that prices are fair and there cannot be arbitrage which is just a fancy way of saying that you cannot make profit or at least that prices immediately equalize such that you make the same profit with whatever you buy. If there is a random element involved it applies to the expectation value and the only thing that varies or that you can influence is the variance. This just means that you cannot expect to make profit. So why bother?

There are however at least four possibilities to still make profit:

You counsel other people how to make money and charge by the hour. Note that you get your money even if your advice was wrong. And of course it can be hard to tell that your advice was wrong: If you suggest to play Roulette and always put money on red and double when you lose most people will make (small) money following these instructions. Too bad a few people (assuming the limit is high enough) will have big losses. But in a poll many people will be happy with your advice. You don't even have to charge by the hour, you can sell your advice with full money back guarantee, in that way you participate in winnings but not in losses and that's already enough.

You could actually produce something (even something non-material) and convert your resources (including your time and effort) into profit. But that's surplus and old fashioned. Note that at least infinitessimally your profit at time t is proportional to the economic activity A(t), i.e. as long as there is demand the more sausages the butcher produces the more money he makes.

You trade for other people in the money market and receive a commission per transaction. As transactions are performed when the situation changes your will make profit proportional to the (absolute value) of the time derivative of A(t). Thus you have an interest that the situation is not too stable and stationary. This includes banks and rating agencies and many more.

Finally, there is the early bird strategy: You get hold of a commodity (think: share in a dot-com company or high-risk mortgages) and then convince other people that this commodity is profitable so they as well will buy it. The price goes up (even if the true value is constant or zero) and indeed the people early in the game make profits. Of course if the true value is zero these profits are paid by the people who join too late as in any other pyramid scheme or chain letter. The core of all these models of course is as Walter Kunhardt pointed out to me
Give me $100. Then you can ask two other people to give you $100.
Of course, people following strategy three above like it if there is some activity of this type going on...

Julius Wess 1934-2007

2007-08-09T14:32:00.000+02:00

Just got an email form Hermann Nicolai:

Dear All,

this is to inform you of the passing away of Julius Wess who
was a teacher and friend to many of us. His untimely death (at the age of 72) is particularly tragic in view of the fact that he
would have been a sure candidate for the Nobel Prize in physics if supersymmetry
is discovered at LHC. We will always remember him as a great physicist
and human being.

Giraffes, Elephants and other scalings

2007-08-06T19:15:00.000+02:00

It's not the first time I am blogging about the as I find amazing fact that with some simple scaling arguments you can estimate quite a number of things without knowing them a priori. You could even argue that this is a core competence of the theoretical physicist: If you consider your self as being one you should be able to guesstimate any number and at least get the order of magnitude right . I have been told of ob interviews for business consultant jobs where the candidates were asked how many bricks the Empire State Building was build from and it's the physicists who usually are quite good at this.

Today, on the arxiv, Don Page gives some more examples of such calculations which I find quite entertaining (even if Lubos argues that they are too anthropocentric and apparently does not understand the concept of order of magnitude calculation where one sets not only h=G=c=1 but 2=pi=1 (ever tried this in mathematica and done further calculations?) as well): Page aims to compute the size of the tallest land animals from first principles and get it basically right.

The basic argument goes like this: First you assume chemistry (i.e. the science of molecules) is essential for the existence and dynamics of the animals. Then the mass of the electron and the fine structure constant give you a Rydberg which is the typical energy scale for atoms (and via the Bohr radius and the mass of a proton gives you estimates for the density both of planets and animal bodies). Molecular excitation energies are down by a factor of proton over electron mass. This implies the typical temperature: It should not be so high that all molecules fly apart but still be warm enough that not all molecular dynamics freeze out.

From this and the assumption that at this temperature atmospheric gases should not at a large scale have thermal energies higher than the gravitational binding energies to the planet gives you an estimate on the size of a the planet and the gravity there. The final step is to either make sure that the animals do not break whenever they fall or to make sure the animals do not overheat when they move or that gravity can be overcome to make sure all parts of the body can be reached by blood (this is where the Giraffes come in).

Of course these arguments assume that some facts about animals are not too different from what we find here (and some assumptions do not hold if all happens within a liquid, the pressure argument and the argument about falling which is why whales can be much bigger than land animals), but still I find it very interesting that one can "prove" why we are not much smaller or larger.

There is a very entertaining paper which makes similar arguments just the other way round (the title is misleading, its really about physics rather than biology): It argues why things common in B movies (people/animals much too large or too small) would not work in real life: King Kong for example would immediately break all his bones if he made one step. On the other hand, if we were a bit smaller, we could fall from any height as the terminal velocity would be much smaller. But simultaneously the surface tension of water would pose severe problems with drinking.

I would recommend this paper especially to the author of an article in this week's "Die Zeit" about nano scale machines that reports amongst other things about a nano-car with four wheels made of bucky balls. Understanding how things change when you try to scale things down show how the whole concept of wheels and rolling does not make sense at very small scales: First of all Brownian movement poses real threats, and then roughness of surfaces at the atomic scale would make any ride very bumpy (the author mentions these two things). But what I think is much more important is that gravity is completely negligible as your nano car would either float in the air or be glued by electrostatic forces (which for example cause most of the experimental headaches to people building submillimeter Cavendish pendulums to check the 1/r law or its modifications due to large extra dimensions) to the surface both perspectives not compatible with wheels and a rolling.

So there are good reasons why we are between one and two meters tall and why our engines and factories are not much smaller.

Kids and Computers

2007-07-24T17:01:00.000+02:00

Via Mark Jason Dominus' blog I learned about this paper: The Camel has two humps. It's written computer science professors that wonder why independent of teaching method used (and programming language paradigm) there seems to be a constant fraction of students who after an introductory course are not able to program a computer.

They claim this is not strongly correlated with intellectual capacity or grades for example in math. However, what they present is that there is a simple predictor of success in an introductory course in computer programming. Even before the course starts and assuming that the students have no prior knowledge in programming you give a number of problems of the following type:

int a=20;
int b=30;

a=b;

What are the new values of a and b?

The important thing is how to analyse the answers: The students not having been taught the correct meaning of the progamming language have several possibilities: Either they refuse outright to answer these problems as they do not know the answer. Or they guess. Given the way the question is phrased they might guess the the equal sign is not a logical operator but some sort of assignment. Then still they do not know how it works exactly, but there are several possibilities: Right to left (the correct one), left to right, some kind of shift that leaves the originating variable 'empty' or some add and assign procedure. It doesn't matter for which possibility the students decide, what counts (and that they are not told) is that between problems they stick to one interpretation. According to the paper, the final grades of both groups, the consistent and the inconsistent students, both follow some Gaussian distribution but with the consistent students in the region of good marks and the inconsistent students in the fail region.

This brings me to my topic for this post: Different philosophies approaching a computer. Last week, I (for the first time in my life, me previous computer experiences being 100% self study and talking to more experienced users one to one) at to sit through a computer course that demonstrated the content management system (CMS) that LMU websites have to use. It started out with "you have to double click on the blue 'e' to go to the internet' but it got better from there. The whole thing took three hours and wasn't to painful (but net exactly efficient) and finally got me the required account so I can now post web pages via the CMS. The sad thing about this course was that obviously this is the way many people use computers/software: They are told in a step after step manner how to do things and eventually they can perform these steps. In other words they act like computers themselves.

The problem with this approach of course is that the computer will always stay some sort of black box which is potentially scary and you are immediately lost once something is not as expected.

I think the crucial difference comes once you have been programming yourself. Of course, it is not essential to have written your own little office suite to be able to type a letter in word but very often I find myself thinking "if I had written this program, how would I have done it and how would I want the user to invoke this functionality?". This kind of question comes especially handy in determining what kind of information (in the form of settings and parameters) I have to supply to the computer that it can complete a certain task. Having some sort of programming experience also comes handy when you need find out why the computer is not doing what you expect it to do, some generalised version of debugging, dividing the problem into small parts, checking if they work, trying alternative ways etc.

This I consider the most basic and thus most important part of IT literacy, much more fundamental than knowing how you can convert a table of numbers into a pie chart using Excel or formating a formula in TeX (although that can come close as TeX is Turing complete... but at least you have to be able to define macros etc). You cannot start early enough with these skills. When you are still a kid you should learn how to write at least a number of simple programs.

20 years ago that was simple: The first computer I had under my fingers (I didn't own one but my friend Rüdi did, mine came later as my dad had the idea of buying a home assembly kit for an early 68k computer that took months to get going) greeted you with "38911 BASIC BYTES FREE" when you turned it on. Of course you could play games (and many of my mates entirely did that) but still the initial threshold was extremely low to start out with something along the lines of

10 PRINT "HELLO WORLD": GOTO 10

With a computer running windows this threshold is much higher: Yes you have a GUI and can move the mouse but how can you get the stupid thing to do something slightly non-trivial?

For Linux the situation is slightly better: There the prompt comes natural, and soon you will start putting several commands in a file to execute and there is your first shell script. Plus there is C and Perl and the like preinstalled you already have it and the way to the first "Hello world" is not that long.

So parents, if you read this: I think you really do your kids a big favour in the long run if you make sure they get to see a prompt on their computer. An additional plus is of course that Linux much better runs on dated (i.e. used) hardware. Let them play games, no problem, just make sure programming is an option that is available.

And yes, even C is a language that can be a first programming language although all those core dumps can be quite frustrating (of course Perl is much better suited for this as you can use it like the BASIC of the old days). My first C compiler ran on my Atari ST (after my dad was convinced that with the home build one we didn't get very far) which then (1985) had only a floppy drive (10 floppies in a pack for 90DM, roughly 50$) but 1MB RAM (much much more than the Commodore 64's of those days and nearly twice as much as PCs) so you could run a RAM disk. I had a boot disk that copied the C compiler (and editor and linker etc) into that ramdisk and off you went with the programming. The boot up procedure took up to five minutes and had to be repeated every time you code core dumped because you had gotten some stupid pointers wrong. Oh, happy days of the past...

Near encouters and chaotic spectral statistics

2007-06-28T15:00:00.000+02:00

Yesterday, Fritz Haake gave an interesting talk in the ASC colloquium. He explained why the observed statistics of energy levels characteristic for classically chaotic systems can be understood.

Classically, it is a characterisation of chaotic behaviour that if you start with similar initial conditions the distance will separate exponentially over time. This is measured by the Lyapunov exponent. Quantum mechanically, the situation is more complicated as the notion of paths is no longer available.

However it had been noticed already some time ago that if you quantise a classically chaotic system, the energy levels have a characteristic statistic: It's not the individual energy levels but you have to consider the difference between nearby levels. It the levels were random, the differences would be Poisson distributed (for a fixed density of states). However what one observes is a Wigner-Dyson distribution: It starts out with (E-E')^n for some small integer n (which depends on the symmetry of the system) before it falls of exponentially. This is just the same distribution that one obtains in random matrix theory (where n depends on the ensemble of matrices, orthogonal, unitary or symplectic). This distribution is supposed to be characteristic for chaos and does not depend (beyond the universality classes) on the specific system.

In the collqium now, Haake explained the connection between a positive Lyapunov exponent and level statistics.

Let us assume for simplicity that the hypersurfaces of constant energy in phase space are compact. This is for example the case for billards, the toy systems of chaos people: You draw some wall in hyperbolic space and study free motion with reflections at this wall. Now you consider very long periodic orbits (it's another property of chaotic systems that these exist). Because there is not too much room in the constant energy surface there will be a number of points where the periodic orbit nearly self-intersects (it cannot exactly self-intersect as the equation of motion in phase space is first order). You can think of the periodic orbit then as starting from this encounter point, doing some sort of loop coming back and leaving along the other loop.

Now, there is a nice fact about chaotic systems: For these self encounters there is always a nearby periodic orbit which is very similar along the loops but which connects the loops differently at the self encounter. Here is a simple proof of this fact: The strong dependence on initial conditions is just the same as stability of the boundary value problem: Let's ask what classical paths of the system are there such that x(t0)=x0 and x(t1)=x1. If you now vary x0 or x1 slightly, the solution will only very a tiny bit and the variation is exponentially small away from the endpoints x0 and x1! This is easy to see by considering a midpoint x(t) for t0<t<t1: The path has some position and velocity there. Because of the positive Lyapunov exponent, if you vary position or velocity at t, the end-points of the path will vary exponentially. Counting dimensions you see that an open set of varying position and velocity at t maps to an exponentially larger open set of x0 and x1. Thus, 'normal' variation at the end-points corresponds to exponentially small variation of mid-points.

Now you treat the point of near self encouter of the periodic orbit as boundary points of the loops and move them a bit to reconnect differnetly and you see that the change of the path in the loops is exponentially small.

Thus for a periodic orbit with n l-fold self-encounters, there are (l!)^n nearby periodic orbits that nearly differ only be reconnections at the self-encounters. This was the classical part of the argument.

On the quantum side, instead of the energy difference between adjacent levels (which is complicated to treat analytically) one should consider the two-point correlation for the density of states. This can be Fourier transformed to the time domain and for this Fourier transform there is a semiclassical expression coming from path integrals in terms of sums over periodic orbits. Now, the two point correlation receives contributions from correlations between two periodic orbits. The leading behaviour (as was known for a long time) is determined between the correlation between one periodic orbit and itself.

The new result is that the sub-leading contributions (which sum up to the Wigner Dyson distribution) can be computed by looking at the combinatorics of the a periodic orbit and its correlation with the other periodic orbits obtained by reconnecting at the near encounter points.

If you want to know the details, you have to look at the papers of Haake's group.

Another approach to these statistics is via the connection of random matrix theory to non-linear sigma models (as string theorists know). Haake claims that the combinatorics of these reconnections is in one to one correspondence to the Feynman diagrams of the NLSM perturbation theory although he didn't go into the details.

BTW, I just received a URL for the videos from Strings 07 for us Linux users which had problems with the files on the conference web page.

Shameless promotion

2007-05-24T09:26:00.000+02:00

Update: Due to some strange web server configuration at LMU, people coming from a blogspot.com address were denied access to the TMP web pages. This should be fixed now.

By now, I have settled a bit in Munich (and yes, I like beer gardens) , found a flat to move into in a week and started my new job as scientific coordinator of a new graduate course in theoretical and mathematical physics. There are many new things to learn (for example how to interact with the university's lawyer to come up with documents defining examination and application procedures for the course which both satisfy the scientists and the legal department) and it's quite exciting. The only downside is that right now as we have to get things going I have not actively done any physics in the past three weeks.

But today, Wolfgang, my collaborator from Erlangen and former office mate from Bremen comes to visit for two days and we hope to put some of the finishing touches on our entropy project. And yes, Jiangyang, I have not forgotten you and our non-commutative project and will restart working on it very soon. Promise!

What I wanted to talk about is that yesterday, the web page for the Elite Graduate Course in Theoretical and Mathematical Physics went on-line. A lot of things there are still preliminary but we wanted to get as much information out as soon as possible as the deadline (July 15th) for applications for the course starting in fall is approaching fast.

So if your are interested in theoretical physics (including quantum field theories and strings but not exclusively, there courses in condensed matter theory and statistical physics/maths as well) and looking for a graduate school you should definitely consider us.

Or if you know somebody in that situation, please tell him/her about our program!

I think, at least in Europe, this program is quite unique: It is a very demanding course offering classes in a number of advanced topics of current interest which in a very short time bring students up to the forefront of research. It hinges on the fact that the Munich area with its two universities (LMU and TUM) and several Max Planck institutes plus Erlangen university has an exceptional large number of leading researchers who teach courses in their area of specialisation. The program is run jointly by the physics and math department and several classes will be taught jointly by a mathematician and a physicist so students can obtain a wide perspective on topics on the intersection of these disciplines.

In addition to the courses on the web page which are scheduled on a regular basis, there will be
a large number of smaller courses on topics of recent interest or more specialised subjects to be decided on close to the time when they will be given.

Furthermore, it is planned (and there are reserved slots in the schedule) to have lectures given by visiting scientists adding expertise complementing the local one. Thus, if you are reading this and are further in your career to apply for a graduate course but have an idea for an interesting lecture course (like for example it could be given on a summer school) you could teach and would fancy visiting Munich (I mentioned the beer gardens above) please do get in touch with me! We do have significant money to make this possible.

Packing again

2007-04-27T12:01:00.000+02:00

I started this blog two and a half years ago when I had just moved to Bremen and discovered (not too surprisingly) that IUB is not as busy physics wise as DAMTP had been. I wanted to have some forum to discuss whatever crossed my mind and what I would have bored the other people at the morning coffee/tea meetings in Cambridge with.

Now my time here is up and once again I have put nearly all my life into moving boxes. I am about to return my keys and tomorrow I will be heading to Munich where next week I will start my new position as "Scientific Coordinator" of a new (to be started in autumn) 'elite' master course in theoretical and mathamatical physics chaired by Dieter Lüst.

This promises to be quite an exciting and attractive course which teaches many interesting subjects of choice ranging from condensed matter theory to QFT/particles and string theory. It will be run by math and physics departmens from both Munich univerisities joint by other places like Erlangen and people from the Max Planck Institut fü Physik (Heisenberg institute).

So if you are a student about to graduate (or obtain a Vordiplom) with strong interests in theoretical and mathematical physics you should seriously consider applying there!

I will be taking care of all kinds of organisational stuff and admin of this course and still hopefully have some time for actual physics (as I was promised). In any case, joining the big Munich string comunity (parts of which I know from earlier times like in Berlin) I am looking forward too and hope that moving to another (this time: high price) city will turn out well.

Causality issues

2007-04-04T16:03:00.000+02:00

Yesterday, I was reading a paper by Ellis, Maartens and MacCallum on "Causality and the Speed of Sound" which made me rethink a few things about causality which I thought I new and now would like to share. See also an old post on faster than light communication.

First of all, there is the connection between causality and special relativity: It is a misconception that a relativistic theory is automatically causal. Just because you contracted all Lorentz indices properly does not mean that in your theory there is no propagation faster than light. There is an easy counter-example: Take the Lagrangian . where f is a smooth function (actually quadratic is enough for the effect) of the usual kinetic term of a scalar phi. I have already typed up a brief discussion of this theory but then I realised that this might actually be the basis of a nice exam problem (hey guys, are you reading this???) for the QFT course I am currently teaching. So just a sketch at this point: The equation of motion allows for solutions of the form and when you now expand small fluctuations around this solution you see that they propagate with an adjustable speed of sound depending on f and V.

Obviously, this theory is Lorentz invariant, it's only the solution which breaks this invariance (as most interesting solutions of field theories do).

The next thing is how you interpret this result: For suitably chosen V and f you can communicate with space-time points which are space-like to you. So is that really bad? If you think about it (or read the above mentioned paper) you find that this is not necessarily so: You really only get into trouble with causality if you have the possibility to call yourself in the past and tell you the lottery numbers of a drawing in the future of your past self.

If you can communicate with space-points, this can happen: If you send a signal faster than the speed of light to a point P which is space like to you, then from there it can be sent to your past, part of which is again space-like to P. If the sender at P (a mirror is enough) is moving the speed of communication (as measured by the respective sender) has to be only infinitesimally faster than the speed of light (if the whole set-up is Lorentz invariant).

In the theory above, however, this cannot happen: The communication using the fluctuations of the field phi is always to the future as defined by the flow of the vector field V (which we assume to be time-like and future directed). Thus you cannot send signals to points which are upstream in that flow and all of your past is. And using light (according to the usual light-cones) does not help either.

This only changes if you have two such field with superluminus fluctuations: Then you can use one field's fluctuations to send to P (which has to be downstream for that field) and the other field to send the signal from P to your past. So strictly speaking, only if you have two such fields, there is potential for sci-fi stories or get rich fast (or actually: in the past) schemes. But who stops you to have two such fields if one is already around?

At this point, it might be helpful to formalise the notion of "sending signals" a bit further. This also helps to better understand the various notions of velocity which are around when you have non-trivial dispersion relations: As an undergrad you learn that there is the phase velocity which is and that there is the group velocity but at least to me nobody really explained why the later one is important. It was only claimed that it is this velocity which is responsible for signal propagation.

Anyway, what you probably really want is the following: You have some hyperbolic field equation which you solve for some Cauchy data. Then you change the Cauchy data in a compact region K and solve again. Hopefully, the two solution differ only in the region causally connected to K. For this, it is the highest derivative term in the field equation (the leading symbol) which matters and if you Fourier transform you see this is actually the group velocity.

Formulating this "sending to P and back" procedure is a bit more complicated. My suspicion is that it's like the initial value problem when you have closed time-like loops: Then not all initial data is consistent: If my time fore example is periodic with a period of one year I should only give initial data which produces a solution with the same periodicity. But how exactly does this work for the two superluminal fields?

There is one further complication: If gravity is turned on and I have to give initial data for it as well, things get a lot more complicated as the question of a point with given coordinates is space-like to me depends on the metric. But my guess would be that also changes in the metric propagate only with maximally the speed of light in the reference metric.

And finally, there is the problem that the theory above (for non-linear f) is a higher derivative theory. Thus the initial value problem in that theory is likely to require more than phi and its time derivative to be given on the Cauchy surface.

Generalised Geometries and Flux Compactifications

2007-02-20T23:03:00.000+01:00

For two days, I have attended a workshop at Hamburg on Generalised Geometries and Flux Compactifications. Even though the talks have been generally of amazingly high quality and with only a few exceptions have been very interesting I refrain from giving you summaries of the individual contributions. If you are interested, have a look at the conference website and check out the speakers most recent (or upcoming) papers.

I still like to mention two things though: Firstly there are the consequences of having wireless network links available in lecture halls: By now, we are all used to people doing their email during talks if one is less interested in what is currently going on on stage. Or alternatively, you try not to be so nosy as to try to read the emails on the laptop of the person in the row in front of you. But what I have encountered for the first time is that the speaker attributes a certain construction to some reference and then somebody from the audience challenging that reference to be the original source of that construction and than backing up that claim with a quick Spires HEP search.

In this case, it was Maxim Zabzine talking and Martin Roceck claiming to know an earlier reference to which Maxim replied "No, Martin, don't look it up on your laptop!". But it was already too late....

The other things is more technical and if you are not interested in the details of flux compactifications you can stop reading at this point. I am sure, most people are aware of this fact and also I had read about it but in the past it had never stroke me as so important: In traditional compactifications without fluxes on Calabi-Yaus say, the geometry is expressed in terms of J and Omega which are both covariantly constant and which fulfill J^3=Omega Omega-bar = vol(CY). Both arise from fierzing the two covanriantly constant spinors on the CY. Now, it's a well defined procedure to study deformations of this geometry: For a compact CY and as the Laplacian (or the other relevant operator to establish a form is harmonic) is elliptic, the deformations can be thought of to come from some cohomology class which is finite dimensional. So, effectively one has reduced the infinite dimensional spaces of forms on the CY to a finite dimensional subspace one in the end arrives at a finite number of light (actually massless) fields in the 4d low energy theory.

Or even more technical: What you do is to rewrite the 10d kinetic operator (some sort of d'Alambertian) as the sum of 4d d'Alambertian and a 6d Laplacian. The latter one is the elliptic operator and one can decompose all functions on the CY in terms of eigenfunctions of this Laplacian. As a result, the eigenvalues become the mass^2 of the 4d field and since the operator is elliptic, the spectrum is discrete. Any function which is no harmonic has a KK-mass which is parametrically the inverse linear dimension of the CY.

If you now turn on fluxes, the susy conditions on the spinors is no longer that they are covariantly constant (with respect to the Levi-Civita connection) but that they are constant relative to a new connection where the flux appears as torsion, formally Nabla' = Nabla + H. As a consequence one only has SU(3) structure: One can still fierz the spinors but now the resulting forms J and Omega are no longer harmonic. Thus it no longer makes sense to expand them (and their perturbations) in terms of cohomology classes. Thus the above trick to reduce the deformation problem to a finite dimensional one now fails and one does no longer have the separation into massless moduli and massive KK-states. In principle, one immediately ends up with infinitely many fields of all kinds of uncontrollable masses (unless one does not assume some smallness due to the smallness of the fluxes one has introduced). This is just because there is no longer a natural set of forms to expand things in.

However, today, Paul Koerber reported on some progress in that direction for the case of generalised Kähler manifolds: He demonstrated that one can get in that direction by the consideration of the cohomology with respect to d+H, the twisted differential. But still, this is work in progress and one does not have these cohomologies under good control. And even more, quasi by definition these do no longer contain the 'used to be' moduli which now obtained masses due to flux induced superpotential. Those are obviously not in the cohomology and thus still have the same status as the massive KK-modes which one would like to be parametrically heavier to all this really make sense.

There were many other interesting talks, some of them on non-geometries, spaces pioneered by Hull and friends where one has to use stringy transformations like T-dualities when going from one coordinate patch to another. Thus at least they are not usual geometries but maybe as T-duality has a quasi-field theoretical description there, might be amenable to non-commutative geometry.

Trusting voting machines

2007-01-08T19:24:00.000+01:00

Before New Year I attended day one of the 26th Chaos Communication Congress (link currently down), the yearly hacker convention organised by the Chaos Computer Club. One of the big topics were voting machines especially after this and this and there being a strong lobby to introduce voting machines in Germany.

Most attendants agreed that most problems could be avoided by just not using voting machines but old school paper ballots. But there were also arguments in favour especially for elections with complicated voting systems: In some local elections in Germany, the voter can cast as many votes (70 IIRC) as there are seats in the parliament she is voting for with up to three votes per candidate. Obviously this is a night mare for a manual count. The idea behind these systems is to give voters rather than parties more influence on the composition of the parliament (while maintaining proportional vote) than in list voting system used most of the time: There, the parties set up sorted lists and the voters just vote for parties determining the number of seats for each party. Then these seats are filled with the candidates from the list from the top. This effectively means that the first list positions of the big parties are not really voted for in the general election as these people will go into parliament with nearly 100% probability and only the candidates further down the list are effectively decided about by the constituency.

The obvious problem with more complicated voting systems which might have the advantage of being fairer is that they are harder to understand and consequently they have the risk of being less democratic because too many voters fail to understand them. But voting systems should be a topic of a different post and have already been a topic in the past.

I would like to discuss what you can do to increase trust in voting machines if you decided to use them.

Note that nobody claims you cannot cheat in paper elections. You can. But typically, you can only influence a few votes with reasonable effort and it is unlikely that these will have big influences, say for order N voters you only influence O(1) votes. However with voting machines there are many imaginable attacks with influence possibly O(N) votes threatening the whole election.

The problem arises from the fact that there are three goals for an election mechanism which are hard to achieve all at the same time: a) the result should be check able b) the vote should be anonymous and c) the voter should not get a receipt of his vote (to prevent vote selling). If you drop either of these criteria the whole thing becomes much easier.

The current problems with voting machines mostly come from a): The machine just claims that the result is x and there is no way of challenging or verifying it. And in addition you have no real possibility to work out what software the machine is actually running, it could just have deleted the evil subroutines or it could be more involved.

A first solution would be that the voting machine prints out the vote on an internal printer and presents it to the voter so she could check it is what she really voted but the printout remains inside the voting machine and is a checkable proof of the vote. However now, you have to make sure you do not run into conflict with anonymity as it should not be possible to later find out what was for example the 10th vote.

Here comes my small contribution to this problem: Why not have the machine prove that what it claims is the result is correct? I would propose the following procedure:

For each voting possibility (party/candidate) there is one separate machine that cannot communicate with the other machines plus there is one further separate, open machine per polling station. Before the election, in some verifiable open and random procedure a hyperplane in an M dimensional projective space is determined (M has to be larger than the total number of voters), maybe with different people contributing different shares of information that go into the selection of that hyperplane. The idea behind this procedure is that any three points determine a plane in 3-space and it does not matter which 3 points you give (as long as they are not collinear).

Then each voter is presented a random point on that hyper surface (on some electric device or printout) and she votes by inputting that point into the machine that represents her vote.

After the election, each of the machines holds the information about as many points as votes have been cast for that party/candidate. Then it announces its result (the number of votes), N_i say. The new thing is it has to prove that it has really been given this N_i votes but it can do so demonstrating that it holds the information about N_i points. For example it does it by requesting further M-1-N_i points. After obtaining these it should know the hyper surface and can show this when given one further point with one coordinate missing: From the knowledge of the hyper surface it can compute the remaining coordinate and thus showing it really holds the information from the N_i votes it claimed it received. The nice thing about this procedure is that it does not matter which N_i points it received it is only the number that matters.

Of course, this is only the bare procedure and you can wrap it with further encryption for example to make the transition of the points from the polling to the counting computers safer. Furthermore, the dimensions sould really be larger so no points are accidentally linear dependant.

And of course, this procedure only prevents the machines from claiming more votes than they actually received. There is nothing which stops them from forgetting votes. Therefore, in this system, the individual machines should be maintained by the parties whose votes they count: These would have a natural interest in not losing any votes.

But at least, this procedure makes sure no votes from one candidate are illegally transferred to another candidate by evil code in a voting machine.

Approaching the holiday season

2006-12-21T10:10:00.000+01:00

It' there: I started my Christmas holiday. I decided to start it early this year, after I have spent three weeks managing the technicalities of merging contributions of 80 professors into a 200 page 100MB pdf document containing this year's research report of the school for engineering and science of IUB. At least we were using TeX and subversion otherwise the nightmare would have been complete. Submission deadline to the dean was on Monday, so after reporting on esearch I could do some actual research myself or..... go on holiday.

I picked option one for three quite productive days and then moved to option two. Just five minutes before I intended to leave I received an email from a journal editor that requested a referee report on some paper to be written until January 3rd. Just in case I get too bored
under the tree...

Now, I am at my parents' place and turned on the computer so check what has happened on the net over night and there is a message from Clifford asking me to My instructions:

1. Grab the book closest to you.
2. Open to page 123, go down to the fifth sentence.
3. Post the text of next 3 sentences on your blog.
4. Name of the book and the author.
5. Tag three people.

So, let's do it. 1. done 2. done. 3.:

500g Kastanien werden kruz angeröstet, dann von der äußeren und inneren Haut befreit und in Wasser weichgekocht; ich tropfe sie ab und streich sie durch ein feines Sieb. Dann rühre ich 150g Butter mit 150g Zucker und 2 Päckchen Vanillinzucker, einer Prise Salz sowie 3 ganzen Eiern recht schaumig und gebe den dicklichen Kastanienbrei dazu.

4. It's "Backe backe Kuchen mit Erna Horn" by Erna Horn, a recipe book for bakery. The quotation above is from recipe number 242 "Kastanientorte" a maroon cake. 5. is more difficult (given the exponential growth of these kinds of chain letter things). I tried to get out of the string theory circles by handing this over to Georg, Anna, and Amelie.

Effectiveness of Symmetry

2006-12-19T18:09:00.000+01:00

For some strange reasons, only today I had my copy of the November issue of "Physics Today" in my mailbox, a few days after the December issue arrived. It contains an opinion piece "Reasonably effective: I. Deconstructing a miracle" by Frank Wilczek (online available only to subscribers of Physics Today unfortunately).

He discusses the famous Wigner quote about the unreasonable effectiveness of Mathematics in the Natural Sciences and comes to the conclusion that it can be traced back to symmetries and locality. Then he writes

Since any answer to a "why" question can be challenged with a further "why," any reasoned argument must terminate in premises for which no further reason can be offered.

Let's nevertheless still take this argument a little bit further, just for the fun of it. Why is the world symmetric and local? We are used from string theory that at least continuous symmetries are always local and global symmetries arise only as asymptotic versions of local symmetries. But on the other hand, local gauge symmetries are not really symmetries but just redundancies of a convenient notation. It is possible to rewrite systems with gauge symmetries only in terms of invariant objects (like e.g. Wilson loops) although that formulation is not particularly simple. It's just that in terms of more fields (like longitudinal polarisations of gauge bosons) the formalism simplifies (e.g. becomes linear) and one has to use gauge transformations to get rid of the unphysical polarisations. Thus saying the world has lots of symmetries really means the best formalisation of the world that we know of has many redundancies.

Or turned the other way round: It's not really the symmetries which are properties of the theories. For example people used to point out that GR is diffeomorphism invariant, it keeps its form in any coordinate system. Thus the infinite dimensional group of diffeos make up the symmetries of GR. But this argument is wrong. For example Misner, Thorne and Wheeler spend the entire chapter 12 of the Telephone Book on demonstrating that you can formulate Newtonian gravity in a diffeomorphism invariant way. This is just a fancy way of expressing what every first year student knows: You are not forced to use Cartesian coordinates to discuss Newtonian gravity you can use spherical coordinates as well. Thus this theory is also coordinate invariant.

The real difference between Newton's gravity and GR is that in Newton's version there is a covariantly constant one form dt which is background in the sense that it is not determined by an equation of motion but it is just there. It is externally given. Therefore what is often called "many symmetries" really means absence of such background structure.

But still, why is the world symmetric? One possible answer is that amongst spin 1 fields only gauge potentials have normalisable interactions. Thus, it might well be that at some high, fundamental scale there are many more fields and gauge symmetries are not that preferred. However upon RG-flow these other fields decouple completely. Thus having only gauge interactions remaining really just comes from the fact that our energy scales are much lower than the Planck scale.

You could even try to put an anthropic twist on this: If observers require a large number of degrees of freedom being strongly coupled tuned tuned to critical values (think: neurons in your brain) it is not unreasonable to believe that you have to be quite far from the fundamental scale where all hell of quantum gravity breaks lose. If that were the case, you could argue that observers always see gauge interactions and chiral fermions first as only these survive the running to these scales necessary for observing subjects. Of course, there is still the hierarchy problem of why there is a Higgs. And this argument does not tell us why we observe non-abelian gauge theories, U(1)'s would do as well. For this we might have to invoke moduli trapping or similar mechanisms.

Farewell hep-th/yymmxyz

2006-12-05T08:27:00.000+01:00

In case you did not notice: The arXiv is going to change the naming scheme. This has become necessary as the current scheme only allows for 999 papers per month in each category and the mathematicians had reached 989 in November already. The new numbering will for example read

arXiv:0701.0001

and there is the possibility to explicitly refer to specific versions as in

arXiv:0701.0001v1

Note that the section (like hep-th) is no longer part of the identifier but it can be added as in

arXiv:0701.0001v1 [q-bio.CB] 1 Jan 2007

I would have liked a more conservative change (like just adding an eighth digit) as now I will have to change a number of regular expression in programs that are supposed to spot references to the arXiv like my seminar announcement web suite and my .bib file updater (which uses Spires to find out if a paper has appeared in print).

Coherent states

2006-11-27T15:52:00.000+01:00

If you are not working on quantum optics, you might be in a similar situation as I was a couple of days ago regarding coherent states: You had encountered them in a homework exercise on the harmonic oscillator where you had to prove that they are eigenstates of the creation operator and have minimal uncertainty. And you know that they are important to quantum optics. At least this was my state of knowledge until very recently. Since then, I have read this review (with which I do not agree in all parts) and have spent some thoughts on the subject I would like to share my current understanding.

Let's suppose we have two hermitean operators A and B and want to find states such that the uncertainty is minimal. To this, let's briefly go through the derivation of the uncertainty relation: You assume any state and from it form new states and similarly for B where is the expectation value of A. For these two new states, you use the Cauchy-Schwarz inequality (which basically says in the form ), expand and find

Finally, we use that the absolute value of the imaginary part of a number is less or equal to that number and realise that as A and be are hermitean the imaginary part of the left expectation value is to arrive at which is the usual uncertainty relation.

If you want to rest for a minute think about the following puzzle: Consider a particle on a circle (or on the interval with periodic boundary conditions). Take the wave function and compute that for this state while . This seems to clash with what we just derived. Where is the flaw? (Hint: see a previous post about quantum mechanics)

Back to the main argument. We want to find a state which saturates the inequality. To have that we have to saturate the inequality in the two places where used inequalities: The Cauchy Schwarz and the abs less Im parts of the argument. Cauchy Schwarz is saturated (the scalar product is maximal for vectors of fixed length) if the two vectors are proportional to each other, that is if there is a complex number such that in our case . We can rearrange that to that is has to be an eigenvector of the operator . Furthermore, for the absolute value of a number to be equal to its imaginary part, the number has to be purely imaginary. In our case, this means has to be purely imaginary which can only be if is purely imaginary.

So we found that the uncertainty of operators A and B in a state is minimal if the state is an eigenvector of an operator with real .

So much for the general theory. Now, we can specialise to the usual case A=x and B=p and conclude that states of minimal uncertainty are eigenstates of for some real . Note that so far we have not talked about the harmonic oscillator at all. We have just picked two operators and asked for states in which they have minimum uncertainty. This was a question at the level of Hilbert space operators and we did not specify any sort of dynamics.

Thus, coherent states are not about the harmonic oscillator at all. It just happens that they are eigenstates of annihilation operators for some harmonic oscillator. Above any real does the job and this translates directly to the frequency of the oscillator: What people call "squeezed states" are just coherent states for a different that can in a similar way be related to the annihilation operators at different frequencies.

This so far is my current understanding. In the above mentioned review there is another generalisation which does involve dynamics which I do not yet fully understand. It somehow splits a Hamiltonian into sums of products of 'elementary' operators and then considers the Lie algebra generated by these elementary operators upon commutators. Then you exponentiate this algebra to a group and consider the orbit of the ground state of that Hamiltonian under the action of this group. The part I do not yet understand is how physical this is and how the different choices on the way (the set of elementary operators for example) influence the result.

Science and arXiv

2006-11-15T09:42:00.000+01:00

I was made part of the team that assembles our school's research report involving contributions from all faculty members including references to their publications. My job is mainly merging all contributions into a single LaTeX document and managing the references. We decided to do them in BibTeX so we asked all faculty to provide a list of their papers in BibTeX format. The idea was of course that only a tiny part of this data would have to be typed as most literature databases (such as spires for our field) provide data in this format and other programs like Endnote can export BibTeX as well. Thus with a bit of cut&paste the job would be easy.

I had not expected the amount of computer illiteracy amongst science professors. OK, I knew that most biologists do not use TeX for their papers. But I must admit that I was impressed receiving an MS Word document containing BibTeX entries but for example having all the title="..." fields set in italics. That is not to speak of the many complaints I received about people having to retype their references. Plus the concept of separating content and layout is completely alien to many.

But what I really wanted to talk about is this: I learned during one of these discussions with an experimental surface physicist (who by the way keeps his references typed in a Word document) why he does not submit his preprints to the arXiv: He told me Science does not accept papers which are in electronic archives others than their own! I find this completely ridiculous and could not believe it since at least my only science paper (btw my first paper at all and still the one with most citations although not in high energy)had a preprint on the arXive. But it seems he is right, at least as far as the current policy is concerned.

Please, please, somebody tell me this interpretation is not true and the greed of the AAAS is not in the way of good scientific practices.

Paper cranes

2006-11-10T14:23:00.000+01:00

Yesterday, I served as jury member for the exciting physics competition which is part of the WellenWelten ('wave worlds') physics exhibition in Bremen.

Children (possibly in teams) from 10 to 19 could choose from six construction tasks (announced two months earlier) and present their result in the Congress Centre.

I had to judge paper cranes. The task was to use only paper, glue, sand and twine to build a crane. It should only touch the table in an area of A4 size and be able to hold a 400g weight 40cm above the table and 25cm in front of the base. Furthermore, the crane had to be stable both with and without the weight. With these constraints the task was to build the crane as light as possible.

We had to judge the cranes not only on their stability and weight but also on design, presentation and construction. The level of the 15 submissions was very high and it was not easy to determine the winners. In the end we settled for this crane

which was constructed by two girls from 9th grade (13 years). The three runners up are

.

You find all the pictures here (two pages). Red t-shirts indicate participants and their teachers and dark blue is the jury.

Seminars while you drive

2006-11-08T17:26:00.000+01:00

If I drive longer distances and get bored of listening to the radio I love audio books. Too bad they are usually quite expensive. But i discovered an alternative: Listen to seminars.

A good place to start is the KITP, for example their Blackboard Lunches.

The audio formats they offer are realaudio and ipod. As I do not own one of these white gadgets, I have to use the other option. My car radio can play mp3's (from its USB port or CDs). So I have to convert the .rm files to mp3. Here is how you can do that (so you don't have to spend as much time on it as I did):

In case it is not done already, install mplayer.

When you call it like

mplayer -quiet -vo null -vc dummy -af volume=0,resample=44100:0:1   -ao pcm:waveheader http://online.itp.ucsb.edu/download/bblunch/dine.rm

it will create a file audiodump.wav which in turn you can convert to an mp3 with bladeenc.

After you've done that for a couple of talks, use

mp3burn *mp3

to write the seminars to a CD and you can hit the road, Jack!

Update: Of course you don't use mp3burn as that would produce an ordinary audio CD with maximally 72 Minutes playing time. You want a data disk with the mp3's on it so you really use a program like xroastcd.

Two Sudoku Problems

2006-11-07T17:50:00.000+01:00

Here are two Sudoku meta-problems I have been thinging about for a while.

The first is about the normal form of a sudoku. The rules of sudoku have a huge symmetry group of order (3!)^8 x 2 x 9! = 1.22E12. It is generated by permutations of rows in a group of three rows, permutations of these groups, same thing for columns, transposition of the grid and permutations of the numbers 1,...,9 which are just labels for nine different things. So for each grid, there are 1.22E12 grids which are basically the same. Is there an easy way to determine if two puzzles are related by the symmetry and even better, is there a normal form (a distinct element for each orbit)?

There is a trivial solution to this problem: You just write out all 10^12 puzzles you obtain by acting with the symmetry group and then sort them according to some lexicographic ordering. This gives a normal form.

What I am asking for is a more direct construction. One which sounds more like: Use the 9! permutations of the symbols to make the first row read 123456789. Then use row permutations to make the square below the 1 as small as possible. Then use row permutations to make the next squares under that square as small as possible... Unfortunately, starting to permute colums screws up what was achieved with this ordering prescription. So how would a better algorithm read?

The other problem is how to rate the difficulty of a puzzle? Question one really applied after the puzzle is solved, this question is about puzzles to be done. Newepaper which publish these puzzles often give ratings such as "simple", "intermediate", "hard". But I found these ratings differ significantly between papers (what Zeit online considers hard is much much easier compared to what The Guardian calls a hard sudoku) but are also not consistent amongst themselves.

Earlier, I have talked about the perl program I wrote to solve sudokus. It recursively figures out for all squares which numbers are still allowed and then takes the square with the least number and tries to put these numbers there. If there is an empty square with no allowed numbers remaining it backtracks.

Thus the search can be represented by a tree where each node represents a square to be filled out and there are as many branches from that node as there are numbers which are not yet rules out. What I am looking for is a numerical rating for a puzzle which is a predictor of how hard I find to do the puzzle and for example correlates with the time it takes me to solve it. Even if I use a different strategy when doing these puzzles by hand I would expect the information could be obtained form the tree. Do you have any good idea for such a function from trees to the reals, say? Obviously the trees all have a depth given by the number of empty squares in the puzzle and each node can have at most nine branches but typically has much less (even for "hard" puzzles most of the nodes have only one branch).

An easy guess is of course the number of nodes or the number of leaves but I found those at least not be proportional to my manual solution time. To give you an idea: Today's hard puzzle from Die Zeit

..573.864
.4...8...
.83.....1
71....2..
3..6921.7
4...7..9.
....4.97.
..6..5..8
.......1.

has 52 nodes (four times the program encounters situtations with two possibilities, all others are unique or dead ends, manually it took me exactly 6:30) while

.98......
....7....
....15...
1........
...2....9
...9.6.82
.......3.
5.1......
...4...2.

has 2313 nodes and took me well over an hour some months ago.

Of course, if early on you have several possibilities and learn only much later which ones do not work this is much worse than having many possibilities which are ruled out immediately.

UPDATE: In case anybody is interested, I put up the decision tree for the difficult puzzle.

Cheap quantum cryptography

2006-10-11T18:43:00.000+02:00

Quantum information theory is a fascinating subject. By applying the simple computational rules of quantum mechanics it is often possible to process information much better than with just classical devices. A famous example being Shor's algorithm for the factorisation of integers relevant for breaking many popular public key encryption schemes such as RSA. The speed up is not exponential but "only" by a square root but this can already be substantial. Formal computer scientists amuse themselves by investigating how the many complexity classes change if you have access to some quantum computations, the Complexity Zoo gives a nice overview.

A beatiful introduction into the subject are the lecture notes by John Preskill. A more formal aspect (investigating which quantum machines can be constructed, how does the impossible quantum copier differ from possible devices and that in the language of operator algebras) is treated in the notes by Reinhard Werner.

However, all this, much like string theory, is only theory and does not have real world applications. As far as real experiments go, IBM has been able to factor 15 with a quantum computer in 2001.

Another potentially applicabel area of application is cryptography: It is possible to construct quantum channels that are immune to eavesdropping: If somebody in the middle listens in the information does not reach the intended recipient anymore. This has been demonstrated in a real experiment by Anton Zeilinger: He managed to transmit the keys of a classical encryption scheme via entangled photons (with a bitrate of 400-800bit/s and 3% error).

This still involves a considerable experimental set-up and remember: You only transmit the keys, as the quantum channel is my far too slow to transmit real data (you probably read much faster than 400bit/s). But there are cheap alternatives (not in the theoretical but in the practical sense) which are as well impossible to crack: One-time pads. Assume I have 1GB of data which I want to transmit to you securely. All I need are 1GB of random numbers which I share with you beforehand and then I xor the data with the random numbers transmit the result (which is just noise for anybody in between) and xor the encrypted message to recover the original data.

This is not elegant as we have to share the random numbers beforehand, we have to share as many bits as we want to transmit. But for many practical applications this is easily possible (headquater of some company who want to transmit construction plans to factory, a spy who wants to phone home, you name it): Cheap small harddrives have more room than all the information you would like to transmit secretly in all you life. When you construct the factory you just bring there the harddrive in a sealed box and practially all future communication is secure, the spy carries a sealed usb-stick and has more shared randomness than all the secret pictures he is goint to take and all the reports he has to send. This is not elegant but dead cheap and efficient.

And if you like you can even produce the randomness using quantum physics to make it physically safe: For example you could sample the timing of the ticks of a Geiger counter and have pure quantum randomness.

Here is a small homework: Take t1, t2,...,tN to be the times of N clicks. By themselves, they are not pure random numbers but the time difference follows a Poisson distribution. Assume that the ti are specified with B bits each, so the possible time resolution is dt and the decay constant of the probe is lambda. How many bits of pure randomness can you extract and how do you do it?

Admitting my ignorance

2006-09-26T16:00:00.000+02:00

I don't know what is you attitude towards rigorous functional analysis but mine could be summarised as "I know it exists. There are subtleties but they don't bite as long as you are not asking for it. So, for everyday quantum mechanics, it's enough to remember that operators are not just matrices and there might be convergence issues (otherwise taking the trace would show that [x,p]=i cannot work)".

I knew that most of the time we are dealing with unbounded operators which are thus not continuous and mathematicians might be worried about their domains of definition (which can only be a proper subset of the Hilbert space) but if you do the natural things (and implicitly work on the proper dense subset of the Hilbert space), you will be ok. Furthermore, the spectrum of an operator can be a bit tricky as everybody knows that the 'eigenfunctions' for example of the momentum operator are plane waves which are not square integrable and similarly, eigenfunctions of x are 0 as elements in L^2. But every child knows that the proper definition of the spectrum of A are those z for which (A-z) is not invertible and any physics argument involving eigenfunctions can be made precise using wave packets which are not exactly eigenfunctions but if one wanted to one could control the error and after a long and messy argument you could prove what the physicist had known right from the beginning.

But, as I have learned, sometimes the subtleties are also physically relevant: The first time I realised this was in my oral diploma exam: I was asked to discuss the particle in a piecewise constant potential (and compute reflection and transmission coefficients etc). I was asked why I picked particular boundary conditions of my wave function at the jumps of the potential. Luckily, instead of parroting what I had read in some textbook ('the probability current has to be continuous so no probability gets lost') I had one of my very few bright moments and realised (I promise I came up with this myself, I had not heard or read it before) that this comes from requiring the Hamiltonian (esp. the kinetic term) to be self-adjoined: If you check this property, you have to integrate by parts and the boundary terms vanish exactly if you assume the appropriate continuity conditions of the wave function.

More recently I learned when the distinction between continuous and point spectrum is physically important: Long ago, in some advanced quantum mechanics class, we were shown some strange, seemingly unmotivated calculation with a random potential which after some time showed that that the eigenfunctions of the Hamiltonian have exponential decay. And "thus, even with arbitrarily small randomness the conductor turns into an isolator." I had never quite understood how this calculation was supposed to imply this conclusion. Only a few months ago, I understood in a seminar by Hajo Leschke that what was really meant was "with probability 1 the spectrum is a pure point spectrum and thus there are no scattering states". For more information check out a PhysRept by Fröhlich and Spence or, if you are particularly brave, the discussion of the RAGE-theorem in Reed Simon vol. III.

But this is not what I came to tell you about. I came to tell you that yesterday over lunch I was reading quant-ph/0609163 by H. Nicolic about myths and facts about quantum mechanics. I could comment on many sections but one particular argument stroke me. It goes back to Pauli and shows that if your Hamiltonian is bounded from below there is no time operator.

Here I will give you a slightly modified version: Consider quantum mechanics on the half line 0}">. In the zeroth approximation you would take 0})"> as your Hilbert space. Obviously, in this space x is a positive operator. From the above reasoning it follows that you probably want to ask your wave functions to vanish at 0 as otherwise p is not symmetric:.

Now take some wave function
such that the expectation value of x is finite, say . Now, you can convice yourself that by applying a translation operator
you produce a new state for which x has the negative expectation !

How did that happen, wasn't x supposed to be a positive operator? The solution can be found in chapter 2.5 of Thirring's text book vol. 3 (no link from Amazon) as pointed out to me by Wolfgang Spitzer. The solution comes really from the functional analysis fine print: p is only symmetric but not self-adjoined. The domain of definition of is strictly larger as it does not require the vanishing condition at 0! There is no self-adjoined extension of p which is still hermitean. And therefore you cannot form the translation operator: It is not defined.

Another way to see this is to realise that for
you need all powers of p. And those are only symmetric (vanishing boundary terms) if actually all derivatives of the wave function vanish at 0. And as the translation operator works nicely only on analytic functions (after all it's just the Taylor series) that requirement does not leave us with too many functions.

Therefore you really have to worry about the finer points of functional analysis not to translate wave packets to where they should not be!

While you wait: web 2.0

2006-09-15T13:18:00.000+02:00

There is some physics in the pipe but unfortunately not yet in a state to be discussed at large. So while you wait (hopefully), let me point out an article in this week's Zeit (in German of course) which I find a nice summary of implications of web 2.0. And this wouldn't be web 2.0 if I would not offer you some music videos from youtube to listen to while you read (mainly Michael Brecker, Mike Stern and Keith Jarret).

Ahrenshoop update

2006-08-31T14:40:00.000+02:00

As said earlier, I would like to talk about a few talks here at the Ahrenshoop conference. Let's see how far I get before the afternoon sessions start.

The first talk I would like to mention is Matthias Gaberdiel's about closed string moduli influencing open string moduli. As an example consider strings on a circle. Generically, you can have D0 and D1 branes. D0 sit at a point on the circle and correspond to Dirichlet boundary conditions of open strings while D1 branes wrap the circle and correspond to Neumann conditions. However, if the circle has exactly the self dual radius (fixed under T-duality), the generic U(1) symmetry is enhanced to SU(2) (at level 1 to be specific), thus there is a full SU(2) worth of D-branes. A similar thing happens if the radius is rational in string units R=M/N R_sd say. Then, there besides the genereic branes there are SU(2)/Z_M x Z_N branes.

Thus the spectrum of branes depends critically on R. But R itself is a closed string modulus! You can change it by exciting closed string fields and the obvious question is what happens to the additional branes if you tune the radius away from the special values. Matthias and friends worked out the details and found that because of a bulk boundary 2 point function, in the presence of the special D-brane the operator changing the radius is no longer marginal. Thus changing the radius kicks of an RG flow which they can in fact integrate and show that the special brane decays into either a D0 or D1 brane depending on whether the radius is increased or reduced. They can fill all this prose with calculations which are quite neat and do more general cases. So, go and read their paper!

The next talk I would like to report on was by Niklas Beisert about the spin chain/integrability business. I must admit, in the past I was not following these developments closely and was quite confused. People wrote papers and gave talks reporting that they had done more and more loops for larger and larger subgroups and compared that to many different stringy calculations. But I was lost and had no real idea about where the real progess was happening.

Now Niklas seems to have cleared up a lot of the supergroup theory and the dust has settled considerably. He presented the situation as follows: Both the gauge theory and the stringy side of dilatations operators seem to be integrable in the sense that the S-matrix factorises into products of two particle S matrices. As both sides have N=4 Susy the superalgebra SU(2,2|4) is a symmetry and it seems to restrict this 2 particle S-matrix considerably: The dispersion relation with the square root and the sin is completely fixed by the symmetry and the S-matrix is determined up to a scalar function (diagonal in flavour space). Thus, everything except this function is kinematics and the function contains all the dynamics.

The gauge and the string side of things are different expansions of this function (one from the weak and one from the strong coupling side). On the gauge side, the function to all perturbative orders that have been worked out vanishes while on the other side, the function vanishes at low orders but is non-zero from coupling^3 on. This explains that up to two loops the matching worked (it just tested the kinematics) and why there are discrepancies from 4 loops (where the function starts to matter). I should add that this is not leathal to AdS/CFT since you should not expect a functions expanded around two different regimes to look the same.

Chairwoman just clapped hands, have to go.

Short update:
Internet connectivity is a bit tricky here since all connections go through two ISDN lines and some people use it to do skype effectively stopping connectivity for everybody else. But let me just add to Niklas talk that the reason for the strong conclusions he can draw from the group theory can be traced back to the unusual fact that for that supergroup it happens that the tensor product of two fundamental representations is itself irreducible thus there is no branching. I should also have mentioned that Niklas and friends have a guess for the exact form of that dressing function involving Gamma functions and Betti numbers.

Second update: The paper by Gaberdiel and friends is out.

Re: Re:

2006-08-30T09:27:00.000+02:00

I am currently attending the 38th Ahrenshoop Symposium, a conference with quite a history, as in cold war times it was the possibility to GDR physicists to invite western collegues and discuss high energy physics with them. There have already be a number of very interesting talks but those might be covered in a later post.

Right now (while I should better listen to Yaron Oz telling us the latest about pure spinors) I feel a certain need to say one or two words about hep-th/0608210 which comments on Guiseppe's and my paper of two years ago.

Thomas accuses us to draw wrong physical conclusions from a correct mathematical calculation. He refers to our discussion of the harmonic oscillator in the polymer Hilbert space. It is not about the fact that there only the ground state is stationary or that time evolution is not continious or that formally that state is a state of infintie temperature. All these things still hold true.

All these might look a bit formal. So, how can you determine if a different version of an oscillator ist physically different from the usual one? You might say "I simply check the spectrum". But that does not work as the alternative does not have the operator you would like to compute the spectrum of. But I hear you cry "that show that it's screwed, I can observe that spectrum for example as optical absorbtion spectrum of molecular vibrations". Unfortunately, that's not true if you just have the oscillator, you would have to couple it to the radiation field and thus the full system is interacting and much more complicated. Thus we didn't take that route in our paper.

Our alternative was to define a family of operators H_e such that you formally would have the Hamilton operator as H_0 if that limit existed (as of course it does not in the polymere case) and show that it has unsusual properties as e goes to zero (for all e the expectation value is 0 but the variance goes like 1/e^2 for almost all states).

So, what does Thomas now say about this? He proposes to restrict attention to a finite subspace of the Hilbert space (the 'nonrelativistic' states), say of dimension n. In this subspace, there are only n^2 independant observables (a finite number!), given by the n x n herminean matrices. Then you compute the expectation values of these n^2 observables in the original Fock space. Finally you employ a theorem that tells you that in any Hilbert space you can find a density matrix that for a finite list of observables gives you expectation values not further off than a given delta.

In other words, if I tell you which finite number of observations I am going to do and which values I expect then you can cook up a state in any Hilbert space that gives these values to any precission.

Note however that the state is chosen after I tell you which observations I am going to make. If I only do one unplanned observation you will get different answes or you have to readjust the state.

Thus, Thomas argues that if I tell you beforehand what I plan to observe, he can prepare any Hilbert space that it looks like my favourite Hilbert space.

OK, we could proceed along those lines. Mankind will only make a finite number of observations (including for example various clicking patterns in particle detectors and the temperature in you office), thus all we need is a finite list of numbers. Thus, in the end any theory of everything just boils down to this list of numbers. All the rest (Lagrangian, branes etc) is just mumbo jumbo!

As always, make up your own mind!

I would really like to hear other ideas of mathematical representaitons of observations that show that we know what a harmonic oscillator looks like!

Before I forget: All this does not touch the main part of the paper: In exactly the critical dimension you don't have to rely on these weakly discontinious representatins of the operator algebra because exactly there there are continious representations in terms of the usual Fock space even if that breaks half of the diffeos spontaneaously and those have to be represented in a non-trivial way. We just suggest that for 'good' theories this should be possible and then try to work out physical consequences you have to face otherwise.

Scaling of price of margarine

2006-08-17T16:44:00.000+02:00

Often people think that physicists have to remember a lot of formulas like one for how to compute the resistance if you know the current and the voltage and another one for how to compute the voltage from the resistance and the current. If they are slightly more educated they realise that you only have to memorise R=U/I and algebra does the rest.

But actually, even that is not true. The way to think about Ohm's law is really to realise that for an Ohmian resistor the current is proportional to the voltage. And if you want, you can call the constant of proportionality resistance (or conductivity if you think in the opposite way). This is the important part of Ohm's law just like it's the 1/r^2 dependence of Newton's law (at least in 3D and at bit later you realise that this is just an expression of the analogue of Gauss' law) or that in string units the radius of the M-Theory circle is proportional to g_s (keeping alpha' fixed) as the mass of a D0 is proportional to 1/g_s. To know how things scale is enough in most cases rather than the knowledge of a formula.

So, let's apply this to an everyday situation. I am slightly worried about my weight so I want to buy Lätta margarine in the supermarket. It comes in two package sizes 250g and 500g. Let's take the prices from here, so you pay 0.85 Euro for 250g and 1.35 for 500g. Obviously, I buy the bigger package as I pay less than twice the money for twice the margarine.

But wait, can we compute how this price comes about? Let's assume the price consists of a price for the package and the price of the actual margarine. Of course, the price for margarine is proportional to the amount M of margarine. The price of the package is likely to be proportional to the surface of the margarine, so it scales like M^(2/3). Thus the total price is something like

P = M value + M^(2/3) package

Plugging in the two prices for the two sizes we can solve for "value" and "package". We find that the price of the margarine is -18.7 cents per kg. That's right, it has a negative price, just like for example nuclear waste. This opens up great possibilities, for example we can work out that 1.76 metric tons of margarine together with its package is exactly for free. Or, if I accept to take ten tons, Unilever will pay me 821.33 Euros! I see another get rich quickly scheme coming up.

Finite Group of Order Two

2006-08-15T08:13:00.000+02:00

In case you have not yet seen this:

by The Klein Four via Alien Ted.

FAZ

2006-08-01T08:10:00.000+02:00

The Frankfurter Allgemeine Zeitung has an article on Peter Woit's book but starts out with a portrait of Lubos. Not too bad and entertaining to read (in German).

Where have all the trackbacks gone?

2006-07-26T17:27:00.000+02:00

I just tried to send trackback pings for the previous posts. First I realised that there are no more trackback links at the ArXiV (because of this discussion?) and then golem.ph.utexas.edu explained to me (via Haloscan)

Problem: Server said 'You are not allowed to send TrackBack pings.'

Too bad.

Mastering anomalies?

2006-07-26T15:45:00.000+02:00

After a long and not really fruitful discussion over at Jacques' I had a look a Thomas Thiemann's (with K. Giesel) latest opus magnum (with two further parts). I did not get very far in the introduction as already on page four he mentions an interesting trick: The Master Constraint.

Before I say what it is, let be introduce a bit of the background. We are in the context of theories with gauge invariances which are as everybody knows redundancies in the degrees of freedom. One might think that in quantising the theory one should directly work with the gauge invariant observables but this is often not the case since the description with gauge invariances often has ma much simpler structure, e.g. the space of connections is affine, as discussed elsewhere.

The price one has to pay are is the gauge invariance one drags around and which one has to mod out after the quantization. This can turn out to be impossible and for chiral theories it generically is. Of course I have just described the fact that the theory is anomalous.

Let me discuss this in a concrete example, the bosonic string in which the Virasoro algebra plays the role of the gauge invariance. In this example, the modes of everything are labeled by integers rather than by continuous variables (as for example for the axial anomaly) so there are fewer pitfalls from integrals etc one has to avoid. Plus we have discussed this case in detail in our paper so I don't have to repeat myself too much (for this discussion ignore all the parts on polymer states and the LQG way of doing things, just focus on the mathematical description of what one usually does (Fock space, Gupta Bleuler etc)).

The task in quantization is to turn the classical algebra of observables (functions on phase space) into a quantum algebra with representations on Hilbert spaces. The problem is that the classical algebra is a Poisson algebra with two multiplicative structures, the usual (pointwise) product of functions and the Poisson bracket. Both are supposed to map into in single product in the quantum algebra such that the Poisson bracket becomes the commutator for that product. Already in quantum mechanics of a single degree of freedom you know that this does not work exactly but only "up to higher order h-bar terms".

What does this mean in practice? The usual procedure is to take a subset of observables (typically coordinates of the phase space, or x and p and 1, or the field and its canonical momentum) which have simple Poisson brackets and which generate the classical algebra in terms of the pointwise product. Then one 'promotes' them to operators such that the Poisson bracket goes to commutator rule holds exactly. For all the other observables, one fixes a way of wring them in terms of the simple ones (aka one fixes an operator ordering prescription) and uses this and the product in the operator algebra to define their quantum versions.

Now, what about the gauge symmetry? In the classical theory, Noether's theorem tells us, that all symmetries are inner, that is, for each symmetry transformation, there is a function of phase space which generates it via Poisson brackets. Furthermore, the group relations for the transformations map to Poisson brackets of the generators.

In the quantum theory, you now have to deal with the gauge symmetry. There are two slightly different ways of saying what goes on: The first is quite abstract and is the one we used in the LQG string paper: You take your quantum algebra as above and have your symmetry act on it by an automorphism. Now, in a representation on a Hilbert space, you demand that this automorphism is implemented by unitary operators U(S) (for a gauge transformation S). There is no direct way to obtain these, educated guessing is probably best. The property you demand is that when A is in the algebra and p is the representation you have .

This implies that the U(S) nearly implement the group law: where is a phase. Of course, the above constistency condition for U(S) does not change if you change U(S) by a phase. The question is if you can find a consistent assignment of phases for all U(S) such that all the go away. If this is impossible, you have an anomaly.

In the other approach you use your knowledge of the classical symmetry generators and quantise them as all the other functions on phase space. Often as in the case of the bosonic string, they are quadratic in the basic fields which you quantised directly. This implies that the ordering ambiguity is just a complex number (imaginary for anti-hermitean generators). Again, the difficult step is to find an assignment of these such that the group law holds in terms of commutators.

If you don't succeed you could subtract the left hand side from the right hand side of your expression of the commutator and have the physical state condition that this anomaly (a complex non-zero number) annihilates physical states. This condition of course immediately empties your physical Hilbert space and you are left with nothing.

So the upshot of all this is: In this canonical quantisation approach, the way the anomaly manifests itself is in the inability to get the quantum symmetry algebra to work.

Now we come to the Master Constraint Trick: Assume, we write all our symmetry generators as C_i for i in some index set (let's not worry for a second that this will be infinite in the examples and thus one should worry about existence of the sums). Then form for some positive a_i.

As you can see, M annihilates a state exactly iff all C_i annihilate the state. Thus this constraint contains all the other constraints! Even better, as we only have one constraint, the algebra is trivial and for obvious reasons it also holds in the quantised version.

One is of course not yet done as again the kernel of M could be empty and thus the spectrum of this positive operator could be bound away from zero. But Giesel and Thiemann instruct as what to do:

This can be cured by subtracting from the Master Constraint the minimum of the spectrum provided of course that it is finite and vanishes as so that the modified constrain still has the same classical limit as the original one. One then defines the physical Hilbert space as the (generalised) kernel of the Master Constraint,...

Great, now we finally know how to get rid of these stupid anomalies!

Everything solved!

2006-07-04T13:23:00.000+02:00

While I was away for a wonderful vacation in southern France, I nearly missed gr-qc/0606121. In the first paragraph we are reminded

The issue of the dynamics is perhaps the central problem in canonical quantization approaches to totally constrained theories like quantum general relativity. There are three salient aspects of the problem that have prevented from advancing in the quantization. The first one is how to construct a space of physical states for the theory that are annihilated by the quantum constraints and that is endowed with a proper Hilbert space structure. The second issue is related to the introduction of a correspondence principle with the classical theory, in particular to check the constraint algebra at a quantum level. The third problem is how to address the ``problem of time'' that is, to introduce a satisfactory picture for the dynamics of the theory in terms of observable quantities.

Then come three pages of semi-technical stuff (finite number of degrees of freedom models, Legendre transformations) and eventually

Summarizing, the method of uniform discretizations allows to tackle
satisfactorily the three central problems of the dynamics of quantum
general relativity and provides new avenues for studying numerically
classical relativity as well.

Well done, guys! Now we can stop worrying about quantum gravity and spend all your energies to cheer up Klinsi's Jungs!

Sorry, I didn't have anything more intelligent to say.

Higher order stuff

2006-05-29T23:27:00.000+02:00

It has been over a month since my last post. And as I will be on vacation in a few days which will probably take me off line for a while I thought I should send some sort of ping before disappearing to Provence.

We had been busy finishing our paper on WMAP multipole vectors and soon after got busy with Wolfgang, my office mate, thinking about entanglement entropy. The latter project is not yet in a stage to be discussed in detail but I think this potentiallly quite interesting.

Instead, today I would like to mention a book that I have been reading recently: It's "Higher Order Perl" by Mark Jason Dominus. This is the most interesting computer book I have read for years. It has the potential to change my thinking about programming as much as the first time I learned about Perl.

If you have formal trainig in computer science and talk Lisp each day, this is will not be too interesting to you. But if you like me learned programming the street style there are a few things to note.

When I have to explain why I like Perl so much I could say it's because you can write nice short effective programs and it's very easy to communicate to the compiler what you want. Plus you have regular expressions and don't have to worry about memory management and garbage collection. But usually, I explain how much I liked the idea (of course like all not unique to Perl) that if you have a collection of several things integers are natural labels in very few cases, thus an array is rarely what you really want. It's a bit lik e coordinates. One option is to call things by their name which gives you a hash ($lastname{Robert} is much more natural than $lastname[1]).

The other case is that you don't care that some element is the 4711th as long as you get all of them either at once or you get one after the other (and can iterate over those like in a foreach $element(@list) construct). This gives you lists. If you have a list, you can take the first element and the rest and you can add elements to the beginning and the end. No need to worry how many elements there are as long as there are more than zero.

The "higher order" in the title of the book refers to the possiblility to have functions that return functions (or rather references to functions. So what?, I hear you think. Well, for the mathematically inclined: This allows you to go to the dual space of your data! Instead of manipulating the data, you can now manipulate the ways to access the data. And you should know that the dual space can in general be of very different size. Think of your examples in functional analysis or about distributions: There you first restrict yourself to very nice functions, the "test functions" and then look at their dual space which gives you distributions, quite powerful objects that are more general than functions.

Now back to the programming examples: Think again of a list. I mentioned, the only thing you needed to do with lists is take them as a whole or access the next element. But this is really enough! It is enough to pretend you know the list if you know a way to always get hold of the next element.

For example, with this you can come up the the universal doubling function. It takes a list and returns a list that has each element doubled. But in fact, all you do is to answer the question for the next element by going to the original list, taking the next element from that, double it and return it. Or you can interlace two lists or concatenate them by using obvious strategies to return the next element of the resulting lists.

This way, you can of course handle infinite lists that don't fit into your memory like the list of all integers. This is where really the power of the dual space notion comes in. All you return is a function that computes the next element. From this you can obtain the list of even numbers by applying the doubling function or the list of primes by discarding integers that are not prime.

Isn't that neat? Go get this book and read it!

Fast strings

2006-04-19T17:44:00.000+02:00

When I explain strings to non-stringy physicist, I often start out by stating that a string is like a rubber band and has a potential energy which is proportional to its length (like a 'relativistic Hook law'). Then, all you have to do is to covariantize this statement and you arrive at the Nambu Goto action.

You can, of course read this backwards: A string with potential energy E has a length proportional to E. Now, you can often read this as an explanation of why hard string scattering behaves much better than hard scattering of particles: At high energies, the strings in fact expand and thus the interaction delocalizes.

This, however, is only semi-true: I would think of what you have in hard scattering are strings which have been accelerated so that they have large centre of mass energy. But the centre of mass energy is decoupled from the internal energy of the oscillators and thus a boosted short string will still be short although it has large (kinetic) energy. On the other hand, you can have a long string with zero kinetic energy which just happens to be very heavy. So, in general, heavy strings are long, not fast ones.

So far, this is just kinematics, but can we see this in practice? What happens to a string that runs through a linear accelerator? So, the set-up would look as follows: You start with a string which is in a low mass state which is charged under some U(1) (maybe in a KK type theory, D-branes are welcome as well). Now it feels a electric field strength (of some cavity say). This electric field is a condensate of low energy (given by the normal frequency of the cavity) photons. So, you have to compute the scattering of the string with lots and lots of low energy strings in the vector field state.

Question: Even if the individual photon has energy much less than 1/sqrt(alpha'), does this scattering excite any of the higher oscillator modes (which make the string grow)? A Feynman diagram would look somewhat like


e----x-----x-----x-----x- ... -x-----h
     |     |     |     |  ...  |
     A     A     A     A       A

where e is the charged low energy state, A is the gauge field and h is a heavy state. Has this been done before?

Eurostrings

2006-04-10T15:44:00.000+02:00

I just came back from one week of Eurostrings at DAMTP, Cambridge which was a combination of a network meeting of the EU String Network then turning into a celebration of Michael Green's 60th birthday. So, before anything else:

Happy Birthday, Michael!

This was a particularly nice event and quite different from other european meetings there was also a large number of people from the Americas attending which I assume was due to a) celebrate Michael and b) that this year's Strings '06 conference in Beijing is not too attractive for a number of people for various reasons.

Once more, I was surprised how many people actually read this blog and came to me during the conference mentioning some of my entries in the past.

As here was no wireless network operational during the conference and unlike the Loops '05 I did not feel the strong urge to report. Victor Rivelles already has and Peter Woit has as well.

There were no really big surprises, just look at the titles of the talks and you get a pretty good idea what was going on; there are online proceedings for those who want more details. Looking through my notes reminds me of a few that are worth mentioning never the less: There were talks by Damour, West and Kleinschmidt about the relations between M-Theory and hyperbolic Kac-Moody algebras. By now, it becomes clear how this works dynamically (at least at low levels). The KMA structure even fixed numbers like the coefficient of the CS term in 11d sugra which is usually determined by supersymmetry. I would really like to see worked out how this works in detail, it would not be the first time, there is a relation between exceptional Lie algebras and susy.

A number of people talked about the relations between spin chains and N=4 SYM and strings and another theme discussed by several speakers was the relation between black holes and topological strings (known under the names of OSV). Especially, Strominger gave a nice derivation on the blackboard of the mysterious square formula.

Seiberg gave two talks both quite interesting, the first on a paradox if you apply T-duality in the euclidean time direction to relate high and low temperature physics and how this is related to the Hagedorn transition and the second on his findings in N=1 theories which even if they have a vanishing Witten index often have a non-susy meta-stable state at the origin of scalar field space. This is potentially very interesting phenomenologically as it provides a mechanism of dynamical susy breaking but unfortunately I understand too little of N=1 gauge theories to give you more information. But you can read it all in the paper.

Finally, I would like to point out a little triviality in elementary quantum mechanics which seems not to be generally appreciated. Imagine that some degrees of freedom are not accessible to you, maybe because they are behind a curtain or even the horizon of a black hole. Formally, you write your Hilbert space as a tensor product . The whole system is in a state described by a density matrix which could well be a pure state . As you see only part of the degrees of freedom, you observe only the partial trace , where the trace is over .

The time evolution is is given by the Heisenberg equation and this implies that the entropy does not change with time. Especially, a pure state (with entropy 0) cannot evolve to a mixed state and vice versa.

This however is not true for the reduced state . It evolves unitarily as only if the total Hamiltonian is a tensor product that is if the two tensor factors of the Hilbert space do not interact.

Otherwise, for example if you throw stuff behind the curtain (or horizon) the time evolution of is more complicated and will change in time. This means, if we only observe part of the Hilbert space, a state that was pure in our part of the Hilbert space can become mixed by interactions with the other degrees of freedom.

This is of course well known to people working on decoherence but somehow not so much amongst people thinking about quantum cosmology.