The String Coffee Table

The Mathematics Department of the University of Hamburg has a postdoctoral position available in the area of Algebra, Conformal Field Theory and String Theory which is part of the Collaborative Research Centre 676 “Particles, Strings and the Early Universe: the Structure of Matter and Space-Time” funded by the German Science Foundation (DFG).

The position starts in the fall of 2007 and is for a period of 2 years with the possibility of an extension for an additional year. The candidate is expected to do research at the interface of Algebra, Conformal Field Theory and String Theory.

Applicants must have a PhD in Theoretical Physics or Mathematics.

See the full announcement.

Further links:

Prof. Ch. Schweigert’s homepage

Some entries discussing the group’s work:

The FRS Theorem on RCFT

FRS Reviews

Unoriented Strings and Gerbe Holonomy

some projected research

The same day as the “Lessons from the LQG string” appeared on hep-th, there was another paper by Corichi, Vukasniac and Zapata crosslisted from gr-qc discussing the loopy oscillator and coming to conclusions which at first sight comes just to the opposite conclusions. Their abstract starts out with “In this paper, a version of polymer quantum mechanics, which is inspired by loop quantum gravity, is considered and shown to be equivalent, in a precise sense, to the standard, experimentally tested, Schroedinger quantum mechanics.” while I derived that at high frequencies the absorption spectrum of the polymer oscillator is quite distinct from the usual Fock/Schrödinger version. How could this be?

Luckily, their paper is written in a very clear manner and free from the notational ballast which makes many LQG papers hard to read. With only a brief read one can find the resolution: The two papers are doing different things. That’s not too surprising. Let me spell this out in a bit more detail. In one sentence: I tried to take the polymer oscillator literally and work out the conclusions from what I am given while they apply some limiting/regularisation/renormalisation procedure to the system to finally end up with the usual Fock description.

As a warning I should say that what I am going to present here is probably some kind of caricature of their paper. I have had some email exchange with the authors from Mexico and they have been very helpful and responded to many questions and I am extremely thankful. What I write here is the result of my learning process but might not be the way they would summarise their paper. So: All the errors in this presentation are mine!

The first difference is that the two papers start from different polymer Hilbert spaces: In my case, I have a basis labelled by points in phase space and the Weyl operators by translations in the $x$ and $p$ directions. Because of the singular scalar product these actions are not continuous and neither $X$ nor $P$ exist as operators, only $e^{iaX}$ and $e^{ibP}$ (for real $a,b$). This has the advantage that the classical time evolution translates directly to a unitary operator in that Hilbert space: It just rotates the phase space by an angle proportional to $t$.

They start with a Hilbert space where a basis is labelled by points on the real line and there is an $X$ operator which even has normalisable eigenfunctions. However, there is still no $P$ operator but what would be $e^{ibP}$ acts by translations by $b$. This Hilbert space does not come with a nice time evolution of the oscillator but we will see below what they do instead. However, this difference is I think only technical and does not really matter in the following.

Then they go through some mathematically involved (projective) limiting procedure and play the “go to the dual space”-game several times. The result is that they pick a squence of subspaces of countable dimension, namely at stage $n$ they consider only the span of vectors over points of the form $m/2^n$ for integer $m$. These are in one to one correspondance to characteristic functions of the interval $(m/2^n,(m+1)/2^n)$ in the usual Hilbert space $L^2(R)$. This mapping however is not in isometry: In $L^2(R)$ these characteristic functions have a norm given by their length, i.e. $2^{-n}$ while they have norm 1 in the polymer space. Now comes the trick: You redefine (“renormalise”) the norm on the polymer side by copying the norm form the $L^2(R)$ side. When doing this you should remind yourself that the norm/scalar product is where the choice of state showed up in the GNS construction. So, by redefining the norm you effectively revise your choice of state. And the two descriptions (Fock and polymer) only differed by the choice of state…

At each finite stage of this regularisation procedure, you have broken most translation operators $e^{ibP}$, only the ones for which ${2^n}b$ is an integer survive. But you can use those to come up with finite difference versions $P_{fd}$ for what would be the $P$ operator and use it to define a regularised oscillator Hamiltonian $H=\frac{1}{2}(P_{fd}^2+X^2)$.

Finially you take the $n\to\infty$ limit everywhere. To nobodies’ surprise you end up with the usual Hamiltonian in the usual Fock space. Strictly speaking, you have only defined the operators $e^{ibP}$ for those rational $b$ which have a denominator which is a power of two. But as you are taking limits anyway, you can use these and continuity to define them for all $b$. Of course, as Drs Stone and von Neumann have told you long ago, there is no other continious choice of representation of Weyl operators than the standart one.

So what do we learn? As Giuseppe put it to me (of course with his better manners in more polite words): Both papers agree that the polymer representation sucks. In my paper, I show that how much it sucks and in their paper they show how you can redefine it away and proceed to the usual Fock space.

But I should warn you, dear reader: The original motivation for considering polymer representations at all (not so much for the oscillator but for gauge theories and gravity) was that it gives an easy (trivial) implementation of diffeomorphism symmetries. This is a central part of all this “background independance” stuff.

But the procedure these people suggest is to introduce a regulator (the $1/2^n$ equal partitioning), do the calculation and then remove the regulator. This regulator is nothing but a background! And it breaks many of the nice symmetries you wanted to maintain.

So there are two obvious questions: 1) In systems more involved than the harmonic oscillator (which is just the free theory in 0+1 dimensions), is it possible to renormalise scalar products and operators in a way that the limit exists? This question is like the continuum limit for a lattice regularisation: In nice theories (like QCD) it exists, in other cases there is no good continuum limit like for example QED, because the theory is not asymptotically free. And in the case of gravity I would be worried that the well known non-renormalisability (in the usual treatment) shows up when you try to remove the regulator and find the whole thing exploding.

But let’s assume for a second this problem does not occur or you have found a way to solve it. Then there is still question 2), the anomalies: The regularisation has broken many essential symmetries. Thus it is non-trivial that these reappear in the continuum limit. And we know: In general they don’t. The polymer state didn’t have this problem as it preserved the symmetries. But now they are explicitly broken. So you are thrown back to the situation of the conventional treatment (with a UV cut-off say). If you don’t believe this, you are welcome to upgrade the content of the paper to the case of the bosonic string and show how Diff($S^1$) reappears in the continuum limit.

It’s now two years, that Giuseppe and I have put out out our paper comparing the usual quantisation of the bosonic string to Thiemann’s loop inspired version. A bit to my surprise, that paper was of interest to a number of people and the months afterwards I was lucky to tour half of Europe to give seminars about it (in that respect it was my most successful paper ever; the only talk I have given more often is my popular science talk “Phaser, Wurmloch, Warpantriebe” about physics with a Star Trek spin prepared for the Max Planck society public outreach).

That paper had quite a resonance in the blogosphere as well, but its results have not always been presented in a way we intended them. This might also be because the paper was in large parts quite technical and some of the main messages were burried in mathematical arguments.

So I thought it might be a good idea to put out a “mainly prose” version of the argument which leaves out the technicalities to bring home the main messages. This I did and you should be able to find it on hep-th as you read this.

Remember the philosophy of this investigation: The loopy people always insist that diffeomorphism invariance is so central to gravity that it is important to build it into a theory of quantum gravity right from the beginning and all the problems one has with perturbatively quantising GR are due to ignoring this important symmetry or at least not building it into the formalism but expanding around some background.

As GR is a complicated interacting theory it is easy to get lost in the technical difficulties and one should consider simpler examples to test such claims.

The world sheet theory of the bosonic string is such an example as it is extremely simple being a free theory but still has an infinite dimensional symmetry of diffeomorphisms of the lightcone coordinates. It is thus the ideal testbed for approaches to diffeomorphism invariant theories where one can compute everything and check if it makes sense.

The first part of today’s paper explains all this and shows that the difference in the treatments can be summarised by saying that the usual Fock space quantisation of the string uses a Hilbert space built upon a covariant state whereas the loopy approach insists on invariance of that state which is a much stronger requirement.

My point is that covariance is the property which is physically required (and in fact states in the classical field theory are covariant but not invariant) and thus statements like the LOST theorem have too strict assumtions.

If you insists on invariance you end up with a Hilbert space representation which is not continuous as this is what LOST like theorems tell you. The question now is if this discontinuity makes your theory useless as a quantum theory. Well, everybody is free to set up the rules of the game they call “quantisation” and in the end only theories which do not disagree with experiments are good theories. But as we are all well aware, there are not too many experiments performed today which study properties of quantum gravity or bosonic string and thus this test is not available for the time being.

A weaker test would be to apply your rules of quantisation to other systems which are available for experimentation and see what they give there. Thus the second part (as in the original paper with Giuseppe) deals with a loop inspired quantisation of the harmonic oscillator. The old paper was criticised for providing a solid argument that it is observationally possible to distinguish the loopy oscillator from the Fock oscillator.

The second part of the new paper I think provides such an argument: It couples the oscillator to an electromagnetic radiation field and computes the absorption spectrum. Remember that usually the absorption for a transition between states $|m\rangle$ and $|m'\rangle$ goes like

$$\frac{1}{(\Omega-\omega_m+\omega_{m'})^2}\, .$$

Here, $\Omega$ is the frequency of the radiation. Now, the loopy result is proportional to

$$\frac{1}{\sin((\Omega-m+m')/N)^2}$$

where $N$ is a large natural number characterising the states. Thus if $\Omega\ll N$ the two expressions agree but for large $\Omega$ they don’t (don’t worry about an overall constant).

Thus if I am only allowed to measure within a finite frequency band for $\Omega$ the states can be made similar by choosing $N$ large enough. But once that $N$ is chosen the experimenter can reveal the difference by studying the behaviour at large frequencies.

So are they the same or not? Well, that’s a long story for which you have to read the paper.

After you’ve done that, you can come back here and comment.

I’m a little reluctant to post much on the master constraint program because I haven’t read much on it. But I thought I’d post this if others want to comment on the subject.

My initial question is how does the master constraint program work in classical mechanics? In particular, say we are given some symplectic manifold and some set of constraints. The master constraint is $$M = K^{IJ} C_I C_J .$$ Using this, how does one obtain the constrained phase space?

Or is this the wrong question to ask?

I’ve been trying to understand Thomas Thiemann’s riposte to the papers of Nicolai, Peeters and Zamaklar, Nicolai and Peeters and Helling and Policastro. I’m fairly busy right now with a paper of my own and moving, so I’ll concentrate on the part where he describes how LQG-quantization replicates the usual quantization of the harmonic oscillator. Maybe later, I can get to trying to understand the master constraint program.

I’ve already posted some comments at Christine Dantas’s blog, but I thought I might also try to post them here. I really would love to see some sort of discussion on these points. One of the things I think any scientist should be able to do is to get in front of a chalkboard and be able to communicate a pretty good idea about what’s going on with their work. Think of this as a long distance chalkboard, and I’m the skeptical visitor.

Anyways, I will try in this post to summarize my understanding of the construction in Thiemann. I hope people will correct me if I get it wrong. (And I hope I get the algebra correct….)

The physics blog-wars have hit the world of publishing with Peter Woit’s Not Even Wrong and Lee Smolin’s The Trouble with Physics. Having given up blogging long ago, I still seem to have spent an inordinate amount of time in these internet trenches. Since Dr. Woit was kind enough to send me a review copy of his book, once more into the breach I suppose. Here is my contribution to the chorus of reviews that will surely be appearing. Like the book, it is aimed at the general public rather than towards physicists.

Review of Not Even Wrong

Any comments and corrections are greatly appreciated. The first paragraph follows after the jump.

A new group blog has been created.

The $n$-Category Café.

It’s hosted by John Baez, David Corfield and myself.

The café is supposed to be the right place for the sort of discussion of mathematical physics that you know from John’s This Week’s Finds in Mathematical Physics, maybe from some of the stuff that I have been posting here, hopefully close to the constructive style that is practised on David’s blog.

We are very glad to be able to use Jacques Distler’s sophisticated blog technology.

I will probably move much of my activity from the coffee table to the café.

On the occasion of the availability of the new edition of Anders Kock’s book on synthetic differential geometry ($\to$) I want to go through an exercise which I wanted to type long time ago already.

I’ll redo the derivation of the transition laws for 2-connections ($\to$) using synthetic language. This greatly simplifies the derivation, to the extent that the equations in terms of differential forms become almost identical to the diagrammatic equations that we derive them from.

As explained for instance in

Richard J. Szabo
Superconnections, Anomalies and Non-BPS Brane Charges
hep-th/0108043

a special case of Quillen’s concept of superconnections can be used to elegantly subsume both the gauge connection as well as the tachyon field on non-BPS D-branes into a single entity.

Assuming that this is not just a coincidence, one might ask what it really means. What notion of functorial parallel transport ($\to$) is encoded in these superconnections?

I’ll give an interpretation below. With hindsight, it is absolutely obvious. But I haven’t seen it discussed before, and - trivial as it may be - it deserves to be stated.

Before finishing the last entry I should review some basic facts about K-theory and D-branes, beyond of what I had in my previous notes ($\to$).

Apart from the Brodzki-Mathai-Rosenberg-Szabo paper ($\to$) I’ll mainly follow

T. Asakawa, S. Sugimoto, S. Terashima
D-branes, Matrix Theory and K-homology
hep-th/0108085

which is based in part on

Richard J. Szabo
Superconnections, Anomalies and Non-BPS Brane Charges
hep-th/0108043

and

Jeffrey A. Harvey, Gregory Moore
Noncommutative Tachyons and K-Theory
hep-th/0009030.

I have begun reading

Jacek Brodzki, Varghese Mathai, Jonathan Rosenberg, Richard J. Szabo
D-Branes, RR-Fields and Duality on Noncommutative Manifolds
hep-th/0607020 .

This is a detailed study of the concepts appearing in the title, using and extending the topological and algebraic machinery known from “topological T-duality” (I, II, III, IV, V). The motivation is to formulate everything in $C^*$-algebraic language, in order to get, both, a powerful language for the ordinary situation as well as a generalization to noncommutative spacetimes.

Warning: I am still editing this entry.

A few entries ago, I was claiming that other people are implicitly claiming that the field content of $D=11$ supergravity encodes precisely a 3-connection taking values in a certain Lie 3-algebra ($\to$). In my first attempt to make a couple of remarks on that, I ran out of time ($\to$). Here is the second attempt.

Yesterday, Kentaro Hori gave a talk on (unpublished) joint work with Manfred Herbst and David Page, another version of which I had heard a while ago in Vienna ($\to$), on

K. Hori, M. Herbst
Phases of $N=2$ theories in $1+1$ dimensions with boundary, I

Quantizing abelian self-dual $p$-form connections on $(2p+2)$-dimensional spaces gives rise to quantum observable algebras which are Heisenberg central extensions of the group of gauge equivalence classes of these connections, with the cocycle given by the Chern-Simons term in $2p+1$ dimensions (I, II, III ).

In

Kiyonori Gomi
Projective unitary representations of smooth Deligne cohomology groups
math.RT/0510187

the author spells out the technical details of the construction of unitray representations for a certain (“level 2”) cases of these central extensions (compare the discussion in II), effectively generalizing the construction of positive energy reps of Kac-Moody groups $\hat LU(1)/\mathbb{Z}_2$ (corresponding to $p=0$) to higher $p$.

These reps should be the Hilbert spaces of states of the quantum theory of self-dual $p$-form fields. Their irreps would correspond to the superselection sectors.

In the 4th (and probably last) session of our seminar Birgit Richter talked in more detail about

$\;\;$ 0) elliptic curves and formal groups

$\;\;$1) “classical” elliptic cohomology (according to Landweber, Ochanine and Stong)

and a tiny bit about

$\;\;$2) topological modular forms (due mainly to Hopkins)

and ran out of time before talking about

$\;\;$3) other forms of elliptic cohomology (e.g. Kriz-Sati) ,

complementing my rough outline last time with more technical details.