Loop Quantum Cosmology: Bojowald's "Dynamical Coherent States

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In summary: This raises questions about the relevance of models with such a time for the real universe."In summary, Bojowald, Cartin, and Khanna introduce a new class of loop quantizations for anisotropic models which includes features seen in inhomogeneous situations. This new effect leads to dynamical difference equations with non-constant step-size and presents new mathematical challenges. The authors also discuss the implications of these models for semiclassical behavior and show that stability conditions can heavily constrain these models. Ultimately, they question the relevance of models with a discrete internal time for the real universe.
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For me, the Bojowald "Dynamical coherent..." paper that appeared 29 March signaled a change in the Loop picture, which I will try to sketch out in this thread. First here is the abstract of that particular paper, which BTW Jal already flagged.

Dynamical coherent states and physical solutions of quantum cosmological bounces
Martin Bojowald
30 pages, 3 figures
IGPG-07/3-5, NSF-KITP-07-55

"A new model is studied which describes the quantum behavior of transitions through an isotropic quantum cosmological bounce in loop quantum cosmology sourced by a free and massless scalar field. As an exactly solvable model even at the quantum level, it illustrates properties of dynamical coherent states and provides the basis for a systematic perturbation theory of loop quantum gravity. The detailed analysis is remarkably different from what is known for harmonic oscillator coherent states. Results are evaluated with regard to their implications in cosmology, including a demonstration that in general quantum fluctuations before and after the bounce are unrelated. Thus, even within this solvable model the condition of classicality at late times does not imply classicality at early times before the bounce without further assumptions. Nevertheless, the quantum state does evolve deterministically through the bounce."

I'm going to try to say what i think is going on with this paper and some related ones i have in mind. If anyone wants to volunteer, I'd be glad of any help reading this one of Bojowald. Busy now but may be able to get started this evening or tomorrow.
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==exerpts from Bojowald "Dynamical Coherent States" paper==

[From page 3]

The model is paradigmatic for background independent quantum gravity obtained from a loop quantization where the usual free field theory basis is not available. Since the bounce model is exactly solvable, it can provide a perturbative basis for quantum gravity including all possible interactions and degrees of freedom. Thus, the form of coherent states determined here is relevant not only for the model itself but for quantum gravity in general.

[From page 27]

We end by repeating that any physical statements derived from a single model have to be confirmed by a perturbation analysis around the model. This is feasible in our case, as it is for perturbations around any solvable model, but still requires detailed work which is now in progress. Only such an analysis could justify the transfer of results from single models to our own universe.

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  • #3
We should get to know Hans Kastrup of Aachen a little better. We can't know him personally but we can try to understand his perspective some.

Kastrup was advisor to both Thiemann and Bojowald. I don't know if he was Thiemann's official PhD thesis advisor (as he was Bojowald's) but they co-authored Thiemann's first paper in 1993--he was like a PhD advisor whether officially or not. Interestingly, it was hearing a seminar talk by Renate Loll about the Ashtekar formulation of GR that got them started them writing the paper.

Kastrup advisor-lineage goes back to Gauss
Carl Friedrich Gauss -> Christian Ludwig Gerling -> Julius Plücker
->C. Felix Klein
->C. L. Ferdinand von Lindemann (Erlangen, 1873)
->Arnold Sommerfeld (Königsberg, 1891)
->Werner Heisenberg (1923, Munich)
->Hans Kastrup
->Martin Bojowald (Aachen, 2000)

this is according to the Wikipedia Nobel geneology page
also from that page one sees other familiar and not-so-familiar descendents of Gauss, such as:

C. F. Gauss -> Christoph Gudermann -> Karl Weierstraß -> Carl Runge
->Max Born (Berlin, 1880) -> Victor Weisskopf (Göttingen, 1931)
->Murray Gell-Mann (MIT, 1951) -> Sidney Coleman -> Lee Smolin

Here, according to what Kastrup and Thiemann say in their 1993 paper, is how they got interested in trying a certain approach to Quantum Gravity.
"We thank R. Loll for a seminar which stimulated our interest in Ashtekar’s formulation of general relativity and we are indebted to A. Ashtekar for helpful discussions during the spring meeting of the German Physical Society in Berlin, where the main results of this paper were presented..."

This is from the Acknowledgments section of:
Canonical Quantization of Spherically Symmetric Gravity in Ashtekar's Self-Dual Representation
T. Thiemann, H.A. Kastrup

what I mean by "advisor lineage" is that Bojowald's PhD thesis advisor was Kastrup, whose advisor was Heisenberg, whose advisor was Sommerfeld, and so on back to Carl Gauss. A lot of the coaching and training in how to do important original theory research appears to come from this relation/experience and so it turns out that greatness tends to run in lines. See the Wikipedia page (by Alejandro Rivero) to see where other familiar favorites (like Richard Feynman) come in this "family tree".
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  • #4
Related paper by Bojowald came out today

Lattice refining loop quantum cosmology, anisotropic models and stability
Martin Bojowald, Daniel Cartin, Gaurav Khanna
24 pages

"A general class of loop quantizations for anisotropic models is introduced and discussed, which enhances loop quantum cosmology by relevant features seen in inhomogeneous situations. The main new effect is an underlying lattice which is being refined during dynamical changes of the volume. In general, this leads to a new feature of dynamical difference equations which may not have constant step-size, posing new mathematical problems. It is discussed how such models can be evaluated and what lattice refinements imply for semiclassical behavior. Two detailed examples illustrate that stability conditions can put strong constraints on suitable refinement models, even in the absence of a fundamental Hamiltonian which defines changes of the underlying lattice. Thus, a large class of consistency tests of loop quantum gravity becomes available. In this context, it will also be seen that quantum corrections due to inverse powers of metric components in a constraint are much larger than they appeared recently in more special treatments of isotropic, free scalar models where they were artificially suppressed."

Originally Loop Quantum Cosmology (LQC) was constructed assume a simplified universe----assuming the isotropy and homogeneity that cosmologists usually assume.

The LQC program is now being extended by gradually relaxing those restrictions.

Isotropy essentially means "looks the same in all directions". But the real universe, while approximately isotropic as far as we know, deviates slightly from perfect isotropy. E.g. the CMB temperature map is not perfectly uniform. So in this paper Bojowald is working on the anisotropic case.

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