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).

Where have all the trackbacks gone?

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?

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!