The String Coffee Table

More polymer oscillators

Posted by Robert H.

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.