Taking Bisognano-Wichmann into Loop

[marcus]

http://arxiv.org/abs/1408.0121
Thermally correlated states in Loop Quantum Gravity
Goffredo Chirco, Carlo Rovelli, Paola Ruggiero
(Submitted on 1 Aug 2014)
We study a class of loop-quantum-gravity states characterized by (ultra-local) thermal correlations that reproduce some features of the ultraviolet structure of the perturbative quantum field theory vacuum. In particular, they satisfy an analog of the Bisognano-Wichmann theorem. These states are peaked on the intrinsic geometry and admit a semiclassical interpretation. We study how the correlations extend on the spin-network beyond the ultra local limit.
11 pages, 4 figures

The Bisognano-Wichmann theorem (originally proved for quantum fields on Minkowski space) is an intriguing and powerful result---it underlies Unruh's discovery of a radiation bath experienced by an accelerated observer, and when extended to curved spacetimes can be shown to underlie Hawking radiation. I think there are connections with KMS condition, Tomita flow or thermal time. Here the authors are setting out to extend the BW idea to Loop spacetime geometry.

I went looking for sources on BW and found this by Christoph Solveen
http://math.mit.edu/~eep/CFTworkshop/Tuesday4-30pmChristophEdited.pdf
and this by Jens Mund
http://arxiv.org/abs/hep-th/0101227

 

[marcus]

I noticed in this 2010 paper by Jens Mund
http://arxiv.org/abs/1012.1454
there is an appendix, "Tomita-Takesaki Theory and the Bisognano-Wichmann Theorem"
starting on page 14

 

[atyy]

Chirco, Rovelli and Ruggiero explicitly mention stringy work as one of their motivations!

However, Chirco et al's work is intrinsically loopy: "Notice that the main hypotheses of the Bisognano-Wichmann theorem are positivity of energy and Lorentz invariance. The last is a dynamical property in the sense that a boost generates the change of a state from a given (spacelike) plane to a boosted one. Therefore the Bisognano-Wichmann property captures aspects of a state's evolution. As we will see, this is reflected in the states we define below: their definition depends on the (covariant [25-30]) definition of the loop quantum dynamics. Therefore they can also be viewed as a step towards fully physical dynamical quantum gravity states."

 

[marcus]

I agree. It's clearly and explicitly a Loop paper. But it was a friendly gesture of them to mention in passing the papers by Mark van Raamsdonk et al which do some interesting (but essentially unrelated) things with Bis-Wich in AdS/CFT context. The key reference, though, in that motivation paragraph on page 1 is to 1401.5262 Spacetime Thermodynamics Without Hidden Degrees of Freedom by "CHRR" Chirco, Haggard, Riello, Rovelli.

I think the CHRR paper is one of the most far-reaching QG papers to come out this year because it explains how Jacobson's derivation of GR from Thermodynamics actually works. And it explains the context for the paper in question here. I think it is the best introduction to what they are doing with Bis-Wich that I've been able to find. It can serve as a preparation for understanding the current paper.

So I'm going to focus, for the time being, on what they say about Bisognano-Wichmann in the CHRR paper. It is spelled out more clearly there.

 

[marcus]

As background, referring to the CHRR paper
http://arxiv.org/abs/1401.5262
Spacetime thermodynamics without hidden degrees of freedom
the important point is its deeper grasp of Jacobson's 1995 result. His argument does not need to refer to statistical entropy. So there is actually no hint of a need for hidden d.o.f.
The article is understandably written which is a big help. Look at section II "Statisticial and Entanglement Entropy". They make the distinction very clearly. Both are defined for a quantum system and can be written in the vonNeumann form, but they are different.

==final paragraph of section IV on page 4==
The important point of this discussion is that Jacobson’s result can be split into two parts. The second part being the step from (30) to the Einstein equations. This is a nice piece of differential geometry, but has nothing to do with thermodynamics. The first step is to get to equation (30). As we have seen, this equation comes from considering the entanglement entropy across the horizon plus a universality assumption. Again, here statistical considerations play no role, and nothing points to underlying degrees of freedom, or to a reading of the Einstein equations as equations of state.
==endquote==