A new realization of quantum geometry

[marcus]

I think this will probably turn out to be one of the most important QG papers this quarter. It is based on some very interesting and groundbreaking work by Dittrich and Geiller that has come out over the past year and a half. I'll give links to that after posting the main abstract:

http://arxiv.org/abs/1506.08571
A new realization of quantum geometry
Benjamin Bahr, Bianca Dittrich, Marc Geiller
(Submitted on 29 Jun 2015)
We construct in this article a new realization of quantum geometry, which is obtained by quantizing the recently-introduced flux formulation of loop quantum gravity. In this framework, the vacuum is peaked on flat connections, and states are built upon it by creating local curvature excitations. The inner product induces a discrete topology on the gauge group, which turns out to be an essential ingredient for the construction of a continuum limit Hilbert space. This leads to a representation of the full holonomy-flux algebra of loop quantum gravity which is unitarily-inequivalent to the one based on the Ashtekar-Isham-Lewandowski vacuum. It therefore provides a new notion of quantum geometry. We discuss how the spectra of geometric operators, including holonomy and area operators, are affected by this new quantization. In particular, we find that the area operator is bounded, and that there are two different ways in which the Barbero-Immirzi parameter can be taken into account. The methods introduced in this work open up new possibilities for investigating further realizations of quantum geometry based on different vacua.
72 pages, 6 figures

Here are a couple of papers they build on:

B. Dittrich and M. Geiller, “A new vacuum for loop quantum gravity”,
Class. Quant. Grav. (2014) http://arxiv.org/abs/1401.6441 .

B. Dittrich and M. Geiller, “Flux formulation of loop quantum gravity: Classical framework”,
Class. Quant. Grav. (2015), http://arxiv.org/abs/1412.3752 .

 

[marcus]

A question that immediately comes to mind is how will they incorporate the cosmological curvature constant?

Right now they are very close to Tullio Regge with a triangulation of flat simplexes

Dittrich has a paper in preparation that is titled something like "yet another vacuum" for QG. that may say something about Lambda
[56] B. Dittrich, “Yet another vacuum for loop quantum gravity”, to appear.

It came to be called "yet another" because last year she put out the one I already mentioned called
[17] B. Dittrich and M. Geiller, “A new vacuum for loop quantum gravity”, Class. Quant. Grav. (2014),arXiv:1401.6441 [gr-qc].

 

[marcus]

Some of us may remember 10 years or so ago when the "LOST" theorem came out, a uniqueness theorem for formulations of canonical LQG.
This was based on the Ashtekar-Lewandowski vacuum (AL vacuum for short).

Dittrich et al point out that the AL vacuum differs from the vacuum of BF-theory. on which the closely allied Spinfoam QG approach is based.
They have a way to get around the uniqueness result of Lewandowski, Okolow, Sahlmann, Thiemann and come up with a different version of canonical LQG. they refer to this as BF instead of AL and they call it the flux formulation of LQG.

As a reminder here's part of the abstract:
==quote ==
In this framework, the vacuum is peaked on flat connections, and states are built upon it by creating local curvature excitations. The inner product induces a discrete topology on the gauge group, which turns out to be an essential ingredient for the construction of a continuum limit Hilbert space. This leads to a representation of the full holonomy-flux algebra of loop quantum gravity which is unitarily-inequivalent to the one based on the Ashtekar-Isham-Lewandowski vacuum.
==endquote==
The idea of flat connections with local curvature excitations is interesting.
The inner product inducing a discrete topology on the gauge group raises questions at least for me.
The context of the two earlier papers may help us understand.

http://arxiv.org/abs/1401.6441
A new vacuum for Loop Quantum Gravity
Bianca Dittrich, Marc Geiller
(Submitted on 24 Jan 2014 (v1), last revised 5 May 2015 (this version, v2))
We construct a new vacuum and representation for loop quantum gravity. Because the new vacuum is based on BF theory, it is physical for (2+1)-dimensional gravity, and much closer to the spirit of spin foam quantization in general. To construct this new vacuum and the associated representation of quantum observables, we introduce a modified holonomy–flux algebra that is cylindrically consistent with respect to the notion of refinement by time evolution suggested in Dittrich and Steinhaus (2013 arXiv:1311.7565). This supports the proposal for a construction of the physical vacuum made in Dittrich and Steinhaus (2013 arXiv:1311.7565) and Dittrich (2012 New J. Phys. 14 123004), and for (3+1)-dimensional gravity. We expect that the vacuum introduced here will facilitate the extraction of large scale physics and cosmological predictions from loop quantum gravity.
11 pages, 5 figures. published in C&QG (May 2015)
http://inspirehep.net/record/1278728?ln=en The abstract is the published IOP one given at Inspire.

http://arxiv.org/abs/1412.3752
Flux formulation of loop quantum gravity: Classical framework
Bianca Dittrich, Marc Geiller
(Submitted on 11 Dec 2014)
We recently introduced a new representation for loop quantum gravity, which is based on the BF vacuum and is in this sense much nearer to the spirit of spin foam dynamics. In the present paper we lay out the classical framework underlying this new formulation. The central objects in our construction are the so-called integrated fluxes, which are defined as the integral of the electric field variable over surfaces of codimension one, and related in turn to Wilson surface operators. These integrated flux observables will play an important role in the coarse graining of states in loop quantum gravity, and can be used to encode in this context the notion of curvature-induced torsion. We furthermore define a continuum phase space as the modified projective limit of a family of discrete phase spaces based on triangulations. This continuum phase space yields a continuum (holonomy-flux) algebra of observables. We show that the corresponding Poisson algebra is closed by computing the Poisson brackets between the integrated fluxes, which have the novel property of being allowed to intersect each other.
60 pages, 13 figures. published C&QG(June 2015)
http://inspirehep.net/record/1333848?ln=en

 

[naima]

I read once in the blog of a defender of strings theory that LQG has nothing to say about particles and their attributes. Is this still true?

 

[marcus]

I would say that talking about matter is OFF TOPIC in this thread because the topic is this new paper and new realization of quantum geometry.
Geometry is the place in which matter can be defined. The idea of quantum geometry is to provide a more realistic framework which can accommodate whatever the preferred matter model ---e.g. the Standard Model---happens to be.

For example the existing framework for Quantum Field Theory and the Standard Model is flat Minkowski space of 1905 Special Rel. It does not even expand! And it does not curve so the effects of gravity cannot be realized. It is unreal. So obviously QFT and SM need a better geometric framework.
And whatever geometry framework is proposed, it should be quantum and it should not be fixed in advance. It has to be dynamic and interactive the way real geometry is.

So inventing quantum geometry is interesting. But doesn't have to say much of anything about matter. If it solves a few problems by providing natural cutoff scale, or regulator, for the matter theory, so much the better. that is just a nice bonus, an extra. Or if it solves the problem mentioned here, for example:
http://arxiv.org/abs/1506.08794
No fermion doubling in quantum geometry
Rodolfo Gambini, Jorge Pullin
(Submitted on 29 Jun 2015)
In loop quantum gravity the discrete nature of quantum geometry acts as a natural regulator for matter theories. Studies of quantum field theory in quantum space-times in spherical symmetry in the canonical approach have shown that the main effect of the quantum geometry is to discretize the equations of matter fields. This raises the possibility that in the case of fermion fields one could confront the usual fermion doubling problem that arises in lattice gauge theories. We suggest, again based on recent results on spherical symmetry, that since the background space-times will generically involve superpositions of states associated with different discretizations the phenomenon may not arise. This opens a possibility of incorporating chiral fermions in the framework of loop quantum gravity.

But solving problems for the particle physicists is not the main purpose. Eventually they will have to REBUILD their QFT, their model of the various matter fields, over in the new context of a more realistic (quantum and dynamic) geometry.

 

[marcus]

Just as a reminder, we don't want to stray too far off topic. Anyone who wants to discuss something different is very welcome to start a different thread!
This is the topic here, this new paper:

http://arxiv.org/abs/1506.08571
A new realization of quantum geometry
Benjamin Bahr, Bianca Dittrich, Marc Geiller
(Submitted on 29 Jun 2015)
We construct in this article a new realization of quantum geometry, which is obtained by quantizing the recently-introduced flux formulation of loop quantum gravity. In this framework, the vacuum is peaked on flat connections, and states are built upon it by creating local curvature excitations. The inner product induces a discrete topology on the gauge group, which turns out to be an essential ingredient for the construction of a continuum limit Hilbert space. This leads to a representation of the full holonomy-flux algebra of loop quantum gravity which is unitarily-inequivalent to the one based on the Ashtekar-Isham-Lewandowski vacuum. It therefore provides a new notion of quantum geometry. We discuss how the spectra of geometric operators, including holonomy and area operators, are affected by this new quantization. In particular, we find that the area operator is bounded, and that there are two different ways in which the Barbero-Immirzi parameter can be taken into account. The methods introduced in this work open up new possibilities for investigating further realizations of quantum geometry based on different vacua.
72 pages, 6 figures

 

[marcus]

On the second quarter's MIP poll, I have to say, Garrett Lisi's paper (Lie Group Cosmology) in which the universe is a giant Lie group is the clear favorite so far.
But ...there are are three runners-up and, significantly I think, this Bahr-Dittrich-Geiller paper is one of the three.
https://www.physicsforums.com/threa...rter-2015-mip-most-important-qg-paper.821473/

One reason the BDG paper is important, I think, is that it provides a new ontology. A new idea of the quantum states of geometric reality.
States are built by local curvature excitations to a flat vacuum ground state. Another important thing is that it serves as a bridge from the older canonical type LQG to spinfoam QG which is the other kind of LQG.

 

[atyy]

Dittrich and Geiller's post on the CQG blog!

New realizations of quantum geometry
http://cqgplus.com/2015/06/18/new-realizations-of-quantum-geometry/

 

[HomogenousCow]

I would say that talking about matter is OFF TOPIC in this thread because the topic is this new paper and new realization of quantum geometry.
Geometry is the place in which matter can be defined. The idea of quantum geometry is to provide a more realistic framework which can accommodate whatever the preferred matter model ---e.g. the Standard Model---happens to be.

For example the existing framework for Quantum Field Theory and the Standard Model is flat Minkowski space of 1905 Special Rel. It does not even expand! And it does not curve so the effects of gravity cannot be realized. It is unreal. So obviously QFT and SM need a better geometric framework.
And whatever geometry framework is proposed, it should be quantum and it should not be fixed in advance. It has to be dynamic and interactive the way real geometry is.

So inventing quantum geometry is interesting. But doesn't have to say much of anything about matter. If it solves a few problems by providing natural cutoff scale, or regulator, for the matter theory, so much the better. that is just a nice bonus, an extra. Or if it solves the problem mentioned here, for example:
http://arxiv.org/abs/1506.08794
No fermion doubling in quantum geometry
Rodolfo Gambini, Jorge Pullin
(Submitted on 29 Jun 2015)
In loop quantum gravity the discrete nature of quantum geometry acts as a natural regulator for matter theories. Studies of quantum field theory in quantum space-times in spherical symmetry in the canonical approach have shown that the main effect of the quantum geometry is to discretize the equations of matter fields. This raises the possibility that in the case of fermion fields one could confront the usual fermion doubling problem that arises in lattice gauge theories. We suggest, again based on recent results on spherical symmetry, that since the background space-times will generically involve superpositions of states associated with different discretizations the phenomenon may not arise. This opens a possibility of incorporating chiral fermions in the framework of loop quantum gravity.

But solving problems for the particle physicists is not the main purpose. Eventually they will have to REBUILD their QFT, their model of the various matter fields, over in the new context of a more realistic (quantum and dynamic) geometry.

How exactly are researchers approaching this problem? Are they still using the EFE as the classical analogue and quantizing them or are they starting from a whole nother direction?