Quant. stat. mech. based on general covariance (BHR's paper)

[marcus]

Bianchi Haggard Rovelli just posted a landmark paper showing that QSM rises automatically from the GR requirement of general covariance. Yesterday in another thread Atyy identified their paper as especially interesting. I agree.

http://arxiv.org/abs/1306.5206
The boundary is mixed
Eugenio Bianchi, Hal M. Haggard, Carlo Rovelli
(Submitted on 21 Jun 2013)
We show that Oeckl's boundary formalism incorporates quantum statistical mechanics naturally, and we formulate general-covariant quantum statistical mechanics in this language. We illustrate the formalism by showing how it accounts for the Unruh effect. We observe that the distinction between pure and mixed states weakens in the general covariant context, and surmise that local gravitational processes are indivisibly statistical with no possible quantal versus probabilistic distinction.
8 pages, 2 figures

I believe the point is that with general covariance you don't have an independent time variable and you don't have "initial" and "final" as realistic predicates. You can't have transition amplitudes between initial and final states. So you have to recast quantum theory in boundary formalism, using a bounded region of space-time.

Then the Hilbertspace is associated with the entire boundary (not merely with initial and final slabs) and the amplitudes depend on boundary geometry.

Hence entanglement with the outside must blur any distinction between pure and mixed states and statistical mechanics is so to say intrinsic--inherent to the situation--comes with the territory

Earlier we had a thread or two about "Tomita flow" time. A very beautiful kind of time that is born from star-algebra. Something I like very much about the BHS paper is that after they do away with time by adopting the boundary formalism, at the very end time reappears as the Tomita flow---"saving the day". This is so nice. Please read the paper and say if you find it interesting too (as do Atyy and I)!

 

[marcus]

I guess intuitively you could say that this checkmates the quantum gravity "problem of time", so we want to look for any possible weak spot, or sensitive spot needing more reinforcement.

I think (just on brief acquaintance with the paper, which appeared on arxiv yesterday) that a possible soft spot is the concept of MEAN GEOMETRY which appears in the paragraph on page 5 right before the start of "Section V Unruh Effect".

That is where they have a reference to [15] which is Rovelli's 2012 paper on General Relativistic Statistical Mechanics. http://arxiv.org/abs/1209.0065.

In that September 2012 paper, the definition of mean geometry is on page 3 right before equation (11). That is the definition applied in a classical GR setting, before considering the quantum theory.

Then the concept of mean geometry is extended to the quantum theory on page 4 right before equation (22). For the quantum version to exist we invoke the Tomita flow associated with a state ρ (positive functional on the *-algebra of observables.) Then we must exhibit a classical spacetime M, with a foliation, and a classical metric [STRIKE]g[/STRIKE], where the foliation time-evolution MIMICS the Tomita flow. This, to me, is the hard part. To be able to say that the mean geometry [STRIKE]g[/STRIKE] exists, we have to be able to get a classical grip on the Tomita flow of the observable algebra.

As I see it, there is room here for a general theorem to be proven about the existence of a [STRIKE]g[/STRIKE]. When does one exist? or not exist? On brief acquaintance and shallow understanding, I fantasize that this could be an interesting theorem.

In the meantime, on page 4 there is a footnote 10 which points to literature [34-39] in which the existence of [STRIKE]g[/STRIKE] is taken for granted, or implicitly assumed. Works by Ashtekar, Freidel, Livine, Speziale, Magliaro, Perini, Sahlmann, Thiemann, and others. So maybe it is not so controversial after all. But the author is here being very careful and every time saying "if the mean geometry exists". All these other people seem to think it does, but still it would be nice to have a theorem.

 

[atyy]

I think this could be very interesting. I've never understood how one chooses a "boundary" of the universe, so it makes sense to me that the boundary should be mixed. Also because a pure state with entanglement results in subsystems appearing mixed. They do comment at the end on entanglement, and the Bianchi-Myers ideas, which draw on the same roots as the Ryu-Takayanagi formula, and MERA/AdS.

I am skeptical of the Bianchi-Myers ideas because MERA/AdS would suggest that the geometry is coarse, and that additional ingredients (large N, quantum expanders) are needed to get fine geometry. But this feels like a kick in a good direction, even if it's not head on.

 

[marcus]

I really appreciate your insight and perspective here, Atyy. It is like having another pair of eyes. My superficial reaction to your mention of "boundary of the universe" is that that the boundary formalism may actually represent giving up on quantum cosmology.

That's not necessarily bad, it's just a choice one makes. One can choose to study subsystems, like microscopic geometry and like black holes and whatever else one can put a boundary around.

Or one can choose to model the whole cosmos. Both of those are good things to do. And if you model the whole cosmos there is a preferred time---Friedmann time also called "universe time". I think at some point it was shown (by Rovelli maybe) to agree with Tomita flow time in the appropriate setup. But there one is not working in the boundary formalism. Forgive me for being a little vague about this.

 

[atyy]

I really appreciate your insight and perspective here, Atyy. It is like having another pair of eyes. My superficial reaction to your mention of "boundary of the universe" is that that the boundary formalism may actually represent giving up on quantum cosmology.

That's not necessarily bad, it's just a choice one makes. One can choose to study subsystems, like microscopic geometry and like black holes and whatever else one can put a boundary around.

Or one can choose to model the whole cosmos. Both of those are good things to do. And if you model the whole cosmos there is a preferred time---Friedmann time also called "universe time". I think at some point it was shown (by Rovelli maybe) to agree with Tomita flow time in the appropriate setup. But there one is not working in the boundary formalism. Forgive me for being a little vague about this.

I'm only kidding, but maybe it'd be good if it got rid of the multiverse:)

I guess we can still have quantum cosmology in the limited sense, like the quantum perturbation theory plus inflation that has made such amazingly correct predictions so far.

Is Tomita time a classical or quantum concept?

 

[marcus]

Is Tomita time a classical or quantum concept?

Quantum in the highest sense. It is a C* algebra or von Neumann algebra concept.

Instead of a Hilbert space you think about the OBSERVABLES as part of an abstract algebra with a socalled involution (a star operation taking A to A*).

Then a STATE ρ is a positive linear functional defined on the algebra.

And then the Tomita flow is defined on the algebra. It makes one observable flow into another. As they would by the passage of time.

The Tomita flow is derived from the positive linear functional ρ (the state of the world, that assigns numbers to observables, i.e. to elements of the algebra) together with the * operation (which corresponds to taking the adjoint or conjugate transpose of a matrix).
============

Basically this was a formulation of Quantum Mechanics that von Neumann came up with. He generalized the idea of the operator algebra---the algebra of operators on a Hilbert space.
He said what if the algebra of operators is actually more basic than the Hilbert space?

It was found that you could actually recover the Hilbert space from the algebra. So nothing is lost by going up one level of abstraction. You can come back down again and recover specifics, after a fashion.

So then Tomita found a clever way to finesse a time flow out of the algebra (given a state rho defined on the algebra). Have to go, back later.