Algebraic Quantum Gravity (Thiemann and Giesel)

[marcus]

selfAdjoint called attention to a new series of papers by Thomas Thiemann (AEI-Golm) and Kristina Giesel (Perimeter Institute)
https://www.physicsforums.com/showthread.php?p=1045334#post1045334
in post #503 of the non-string QG bibliography thread.

I am printing off the first in the series which sA flagged.

http://www.arxiv.org/abs/gr-qc/0607099
Algebraic Quantum Gravity (AQG) I. Conceptual Setup

"We introduce a new top down approach to canonical quantum gravity, called Algebraic Quantum Gravity (AQG):The quantum kinematics of AQG is determined by an abstract *-algebra generated by a countable set of elementary operators labelled by an algebraic graph. The quantum dynamics of AQG is governed by a single Master Constraint operator. While AQG is inspired by Loop Quantum Gravity (LQG), it differs drastically from it because in AQG there is fundamentally no topology or differential structure. A natural Hilbert space representation acquires the structure of an infinite tensor product (ITP) whose separable strong equivalence class Hilbert subspaces (sectors) are left invariant by the quantum dynamics. The missing information about the topology and differential structure of the spacetime manifold as well as about the background metric to be approximated is supplied by coherent states. Given such data, the corresponding coherent state defines a sector in the ITP which can be identified with a usual QFT on the given manifold and background. Thus, AQG contains QFT on all curved spacetimes at once, possibly has something to say about topology change and provides the contact with the familiar low energy physics. In particular, in two companion papers we develop semiclassical perturbation theory for AQG and LQG and thereby show that the theory admits a semiclassical limit whose infinitesimal gauge symmetry agrees with that of General Relativity. In AQG everything is computable with sufficient precision and no UV divergences arise due to the background independence of the undamental combinatorial structure. Hence, in contrast to lattice gauge theory on a background metric, no continuum limit has to be taken, there simply is no lattice regulator that must be sent to zero."

 

[marcus]

does anyone understand the notion of a "generic embedding"?

this occurs on page 5---two sentences before the start of section 1.5 (about 2/3 the way down the page).
They say it will be explained later---but I have not yet come to where it is explained.

as a general rule Thiemann and Giesel like to work in an embedding-free way. the graph is abstract and is not imagined to be immersed in a continuum in any particular fashion.

but in order to define the VOLUME operator, they must embed the graph, and so they have a special mechanism for doing that in a generic way when absolutely unavoidable.

when anyone finds where they explain that, on what page, please tell me. I will also look.

 

[marcus]

I found where "generic embedding" is defined.
it is the footnote 11 on page 12

"11. The possible embeddings of an algebraic graphs fall into diffeomorphism equivalence classes. An embedding is called generic if a random embedding results with non vanishing probability in an embedded graph of the same equivalence class. If there is more than one possibility then we must pick one. For our cubic graph to be considered later we consider half – generic embeddings in the sense that there is a neighbourhood of each vertex and a coordinate system in which the graph looks like..."

I am uncertain about the details and consequences of this definition, but the exerpted footnote gives the general idea and flavor of what it is.

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BTW in case anyone else is looking at the paper, or glancing at this thread about it,
the term "algebraic" does not seem to work for me in this context.

I see how the approach could be called abstract, or graphtheoretical, or combinatorial, in its basic character.

that is to say it gets away from differential geometry (for better or worse)

but as far as I can see so far it does not get to a place where it uses conventional algebraic structures that I associate with usual algebraic approaches. Maybe that will come later. OK I see now, it is clear from the abstract what the algebraic angle is. There is a *-algebra or a C*-algebra. I guess I confuse those things with functional analysis

 

[f-h]

AQG looks like a simplified version of LQG. No graph topology? Smolin won't like that ;)

 

[Pietjuh]

but as far as I can see so far it does not get to a place where it uses conventional algebraic structures that I associate with usual algebraic approaches. Maybe that will come later. OK I see now, it is clear from the abstract what the algebraic angle is. There is a *-algebra or a C*-algebra. I guess I confuse those things with functional analysis

The C*-algebra has a lot to do with functional analysis because we can concretely define it as a complex algebra of linear operators on a Hilbert space and which satisfies the following two properties:
- It is closed in the norm topology of operators
- it is closed under the operation which takes adjoints.

You can ofcourse study it's properties from an algebraic viewpoint since it's simply an algebra so that is probably the reason this theory is called AQG :)