|Nov11-12, 03:33 PM||#52|
Penrose's argument that q.g. can't remove the Big Bang singularity
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 trρred = 1 is always preserved. Since the initial state is unentangled, trρred2 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.
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".
|Nov12-12, 02:46 AM||#53|
Blog Entries: 19
|Nov12-12, 02:51 AM||#54|
Blog Entries: 19
|Nov12-12, 12:14 PM||#55|
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. Another, equivalent, criterion is that the von Neumann entropy is 0 for a pure state, and strictly positive for a mixed state.
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.
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 ρred2, 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 Slin = 1 − ρred2 . One could also discuss the von Neumann entropy −kBtr (ρred ln ρred), but for the present purpose it is sufficient to restrict to Slin.
|Nov12-12, 12:53 PM||#56|
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.
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..."
|Nov14-12, 11:14 AM||#57|
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.
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.
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".
|Similar Threads for: Penrose's argument that q.g. can't remove the Big Bang singularity|
|Penrose Singularity Theorem||Advanced Physics Homework||0|
|Singularity theorems (Hawkign & Penrose)||Special & General Relativity||2|
|Roger Penrose - Before the Big Bang||Beyond the Standard Model||6|
|On Penrose's argument against density operators||General Physics||5|
|Penrose's new ideas on the big bang||General Physics||20|