Penrose's argument that q.g. can't remove the Big Bang singularity

[marcus]

Not sure what a squeezed state is. Something to do with the saturated uncertainty relation?

No idea! "Squeezed" was a new one on me when I encountered it in the LQG literature recently. What do you think of Kiefer's paper above, and the idea of treating ALL states as density matrices*, just having different degrees of purity (as measured by the trace of their squares)?

*trace class operators, to put it more generally

 

[marcus]

The Kiefer Schell paper (as it says in the abstract) is about superpositions gradually becoming indistinguishable from a classical mixture, so it should interest several of us here. there is a nice figure on page 8 which shows the purity factor of a state decline from 1 to zero over the course of "internal time". I'm not sure what internal time means here.

==quote from page 8 of Kiefer Schell==
The iteration at each time step starts with the calculation of s₀ and s₁, as described above. As a constraint on the numerical evolution, we normalize s₀ and s₁ such that trρ_red = 1 is always preserved. Since the initial state is unentangled, trρ_red² is initially equal to one. As the inner time variable increases, the total state becomes entangled, and the purity factor decreases—the gravitational variables are in a mixed state, and decoherence becomes more and more efficient. The result can be plotted as a function of the inner time variable φ, see Fig. 1.
==endquote==

Figure 1 is what i was just talking about.
When I say state (in this discussion, since thread concerns the entropy of quantum states) I mean trace class operator on the Hilbert space. IOW loosely speaking "density matrix".

 

[Demystifier]

I'm also used to seeing peaked semiclassical states in LQG, are these also pure?

Not necessary, but the specific peaked states you have seen probably are.

My take on it is that when Kiefer or others start with the U in a pure state and have it progressively decohere, this does not mean that in reality the U would necessarily have to start pure. The analysis just shows how it could start in a purER state and become LESS pure.
The analysis is a fortiori. It is just a convenient simplification to imagine that the system starts in a pure state, the important thing is progressive decoherence starting from whatever level of (im)purity or mixedness. I can imagine you might disagree.

Actually, I agree.

 

[Demystifier]

And mixed states are probabilistic superpositions of pure states, are they not?
What I'm unsure about is the meaning of "squeezed" states. Do you have a good brief explanation, or a link for that?

http://en.wikipedia.org/wiki/Mixed_state_(physics)#Mixed_states
http://en.wikipedia.org/wiki/Squeezed_state

 

[marcus]

...My take on it is that when Kiefer or others start with the U in a pure state and have it progressively decohere, this does not mean that in reality the U would necessarily have to start pure. The analysis just shows how it could start in a purER state and become LESS pure.
The analysis is a fortiori. It is just a convenient simplification to imagine that the system starts in a pure state, the important thing is progressive decoherence starting from whatever level of (im)purity or mixedness. I can imagine you might disagree.

Actually, I agree.

I'm glad we agree on that! Thanks for the pointer to the "quantum state" article. It's well-written and clears up some confusion on my part. The C* algebra approach seems (a bit abstract but) interesting. This paragraph was helpful (and might be to others besides myself):
==quote==
A pure quantum state is a state which can be described by a single ket vector, as described above. A mixed quantum state is a statistical ensemble of pure states (see quantum statistical mechanics). Equivalently, a mixed-quantum state on a given quantum system described by a Hilbert space H naturally arises as a pure quantum state (called a purification) on a larger bipartite system H ⊗ K, the other half of which is inaccessible to the observer.
A mixed state cannot be described as a ket vector. Instead, it is described by its associated density matrix (or density operator), usually denoted ρ. Note that density matrices can describe both mixed and pure states, treating them on the same footing.
...
A simple criterion for checking whether a density matrix is describing a pure or mixed state is that the trace of ρ² is equal to 1 if the state is pure, and less than 1 if the state is mixed.[4] Another, equivalent, criterion is that the von Neumann entropy is 0 for a pure state, and strictly positive for a mixed state.
==endquote==

This seems to afford the right context in which to look at the issue of entropy in LQC bounce. I'll bring forward the Kiefer Schell details from a few posts back. It's not about bounce (but about geometry settling into an orientation) nevertheless I think it shows how one might set the problem up.

http://arxiv.org/abs/1210.0418
Interpretation of the triad orientations in loop quantum cosmology
Claus Kiefer, Christian Schell
(Submitted on 1 Oct 2012)
Loop quantum cosmology allows for arbitrary superpositions of the triad variable. We show here how these superpositions can become indistinguishable from a classical mixture by the interaction with fermions. We calculate the reduced density matrix for a locally rotationally symmetric Bianchi I model and show that the purity factor for the triads decreases by decoherence. In this way, the Universe assumes a definite orientation.
12 pages, 1 figure

[As the wikiP that Demy linked points out] purity and mixedness are not absolute properties but are on a range. Maybe all states should be thought of as a density matrix rho and the degree of purity would be the trace of the square of rho.
==quote page 7 Kiefer Schell==
A measure for the purity of the total state (15) is the trace of ρ_red², which is equal to one for a pure state and smaller than one for a mixed state; it is directly related to the linear entropy S_lin = 1 − ρ_red² [5]. One could also discuss the von Neumann entropy −k_Btr (ρ_red ln ρ_red), but for the present purpose it is sufficient to restrict to S_lin.
==endquote==

 

[marcus]

Maybe during the (repellent gravity) phase of the bounce all horizons are destroyed and all information becomes accessible to the observer. So the statistical quantum state of the prior classical phase is driven to purity. This could be a way of addressing the issue raised by Finbar.

...
So maybe I'll do a u-turn and tentatively buy the "bounce" cosmology. We could think that the state of the universe at the bounce is a pure state. Then the universe evolves to the current day at which point each observer can only see a finite amount of the universe which will be described as a mixed state. Finally the universe then collapses at which point all the universe comes back together and a pure state is again recovered. Now the issue is why the final pure state would look anything like the initial pure state.

Just a note: Commonly in LQC modeling they include Lambda and there is no re-collapse--our classical phase just keeps expanding in the future. They do also study repeated bounce models (zero Lambda) but it isn't necessary. When you run the model back in time it bounces and expands (IOW you see a collapsing prior classical U). So we can rephrase your puzzle and it is just as puzzling put this way---with a single bounce.

I guess one still has to wonder what sort of thing that could be considered an observer could survive through a bounce, and maintain its integrity/identity. But let's set that question aside and assume everything is well-defined. The puzzle that won't go away is how a mixed state in the prior collapsing phase (where lots of information starts out being inaccessible to the observer) can become pure.

"In a moment, in the twinkling of an eye..."

 

[marcus]

On reflection, I've concluded that the way people are going understand these issues will likely be to go back to the June 1994 Connes-Rovelli paper.
http://arxiv.org/abs/gr-qc/9406019
Von Neumann Algebra Automorphisms and Time-Thermodynamics Relation in General Covariant Quantum Theories
A. Connes, C. Rovelli
(Submitted on 14 Jun 1994)
We consider the cluster of problems raised by the relation between the notion of time, gravitational theory, quantum theory and thermodynamics; in particular, we address the problem of relating the "timelessness" of the hypothetical fundamental general covariant quantum field theory with the "evidence" of the flow of time. By using the algebraic formulation of quantum theory, we propose a unifying perspective on these problems, based on the hypothesis that in a generally covariant quantum theory the physical time-flow is not a universal property of the mechanical theory, but rather it is determined by the thermodynamical state of the system ("thermal time hypothesis"). We implement this hypothesis by using a key structural property of von Neumann algebras: the Tomita-Takesaki theorem, which allows to derive a time-flow, namely a one-parameter group of automorphisms of the observable algebra, from a generic thermal physical state. We study this time-flow, its classical limit, and we relate it to various characteristic theoretical facts, as the Unruh temperature and the Hawking radiation. We also point out the existence of a state-independent notion of "time", given by the canonical one-parameter subgroup of outer automorphisms provided by the Cocycle Radon-Nikodym theorem.
25 pages

A unified framework for spacetime geometry, quantum theory, and thermodynamics seems to be needed. The vN-algebra approach seems to provide it. I had the luck to be exposed to C*-algebras in grad school around the time Dixmier's book first came out in English (1977). I guess we should say "von Neumann algebra". How is Loop gravity going to be rebuilt in vN-algebra terms? All states of the universe are mixed, with different degrees of purity. What equation drives the state to high levels of purity at or around the bounce?

It must have to do with the dissipation of horizons. They must shrink to nothing or burst, during collapse to the extreme density. How does one formulate the concept of horizon in the vN-algebra setting? The "purification" bipartite factorization of the hilbertspace might be used: H⊗K, described in the article Demy pointed to, with K standing for information "inaccessible to the observer".