1+1D LQG is really about boundary states
Posted by Urs Schreiber
Over on Peter Woit’s blog we might have the chance to continue some of the discussion (I, II, III) about alternative and non-perturbative quantizations of gravity with Lee Smolin. I am looking forward to seeing his replies.
Meanwhile, I would like to maybe help focus the discussion by, following the important suggestion by Hermann Nicolai, again concentrating on a simple toy example of full 3+1d gravity, namely 1+1 dimensional gravity coupled to scalar matter.
As readers of this blog will remember, it was Thomas Thiemann’s paper hep-th/0401172 which made a couple of people aware of a crucial, in principle very well-known but rarely noticed technicality in the LQG approach:
This is the postulate that ‘to quantize’ a theory does not require to have an operator representation of canonical coordinates and canonical momenta.
This plays a big role in LQG because there the reparameterization constraints are implemented not by quantizing the generators of reparameterizations (as is done, for instance, in the standard quantization of 1+1 dimensional gravity which leads to the string worldsheet theory and where the crucial and characteristic quantum effects come precisely from operator ordering effects in the operator representation of the generators) but by constructing operators on a non-seperable Hilbert space which represent the classical symmetry.
While some theorists feel that this step should be uncontroversial, for instance Josh Willis argued that ‘relaxed canonical quantization’ is nothing to be worried about, it should be noted that when it is applied to systems whose standard (e.g. path integral or BRST) quantization we do understand, like the free particle or 1+1 dimensional gravity, the results are blatantly different from the standard results. Recall that these standard results for well understood systems like the free particle is what we experimentally know to be correct.
There is one paper which tries to address how this difference could disappear in some sort of limit. This is gr-qc/0207106, where the concept of ‘Shadow States’ was introduced as a means to get back from a ‘shadow state’ in the non-seperable LQG-like Hilbert space to that of the corresponding ordinary Hilbert space in standard quantization. But a closer inspection of that paper (and in particular the lower half of p. 14) showed that this is only made to work by including information obtained in the standard quantization. But this is not available for systems where the standard canonical quantization is not available, like for 3+1D gravity.
In the discussion Thomas Thiemann conceded that hence the use of ‘relaxed canonical quantization’ is a step based on the hope that experiment will (or would) find that at the Planck scale we have to modify standard quantum theory. Now, quantum gravity in general is a highly speculative business, but evidence that non-standard quantum theory will be a key ingredient are rather rare.
Since this sort of discussion seems to have been delayed by the general complexity of quantum gravity in more than 2+1 dimensions, I find it very helpful to discuss all these conceptual questions in lower dimensions, preferably in 1+1 of them, where standard quantum gravity is under full control, while still being non-trivial.
In particular, the reason why I decided to warm up the former discussion again is that I would like to point out one particular aspect of LQG quantization in 1+1 dimensions, which previously did not receive any attention: That’s its formal relation to boundary state formalism in string theory.
Recall that in LQG in 3+1 dimensions people construct a space of states that are invariant under spatial reparameterizations, and that they then try to construct a Hamiltonian constraint operator (but as far as I am aware this object is still elusive) to act with it on the space of reparameterization invariant states.
Furthermore recall that when 1+1 dimensional gravity is quantized as in string theory, physical states are not spatially reparameterization invariant. In the standard mode notation the reparameterization invariance assumed in LQG would amount to requiring that physical states be annihilated by for all integer . But, as is very well known, physical states can be annihilated at most by half of these generators.
So if we still insist to follow the LQG prescription of restricting to rep-invariant states we find non-physical states, in the sense of OCQ, namely the boundary states. They become BRST closed when multiplied by an appropriate ghost sector and are in this sense again physical, but the important point is that they are not annihilated by the worldsheet Hamiltonian constraints.
Remarkably, this is precisely the situation in current 3+1 dimensinal LQG: The space of spatially rep-invariant states can be written down, but the operator version of the Hamiltonian constraint on this space is not available.
Still, boundary states in string theory play an important role and are interesting all by themselves.
Since Thomas Thiemann has originally suggested that Pohlmeyer invariants serve as observables for the LQG-like quantization of the string it is maybe interesting to observe that Pohlmeyer invariants, do play a role in the standard quantization of (super-)string as boundary state deformation operators, which turn on non-abelian gauge fields on D-branes.
Posted at August 23, 2004 11:13 AM UTC