Admitting my ignorance

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 . In the zeroth approximation you would take 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

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

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:

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

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

In case you have not yet seen this:



by The Klein Four via Alien Ted.

FAZ

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).