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).