Relational Observables in LQG: Gauge Invariance and Incorporation of Matter

[atyy]

It's often said that gauge invariant local observables in quantum gravity must be relational. In classical gravity, relational observables are constructed with matter. LQG for the most part has not had matter, yet it has been said to have observables such as area or volume. Are these non-local, or not gauge invariant? There have also been attempts to incorparate matter in LQG. Do these have gauge invariant local observables?

 

[marcus]

... There have also been attempts to incorparate matter in LQG. Do these have gauge invariant local observables?

For concreteness let's see how matter has been included in recent papers--say that appeared since 2009. Loop gravity and cosmology have been evolving rapidly so it's good to stick to recent papers. If you follow the LQG/LQC literature at all you may be able to jog my memory and suggest recent matter-including papers that I missed.

Which ones were you thinking of? There was an Ashtekar paper that included Fock space, I think it came out this year. There was Rovelli's paper on including fermions in LQG, that came out in 2010 as I recall. Were you thinking of these? I'll go fetch some links.

http://arxiv.org/abs/1012.4719
Spinfoam fermions
Eugenio Bianchi, Muxin Han, Elena Magliaro, Claudio Perini, Carlo Rovelli, Wolfgang Wieland
(Submitted on 21 Dec 2010)
We describe a minimal coupling of fermions and Yang Mills fields to the loop quantum gravity dynamics. The coupling takes a very simple form.
8 pages

You might also look at Ashtekar's very recent paper, the section starting on page 27 where they define creation and annihilation operators and construct a Fock Hilbertspace they call H₁. It's not clear to me whether the Fock space represents inhomogeneous perturbations due to matter or to something else. It's suggestive but not the paper I was thinking of so I will look some more. FWIW here is that link as well:
http://arxiv.org/abs/1211.1354
An Extension of the Quantum Theory of Cosmological Perturbations to the Planck Era
Ivan Agullo, Abhay Ashtekar, William Nelson
(Submitted on 6 Nov 2012)
Cosmological perturbations are generally described by quantum fields on (curved but) classical space-times. While this strategy has a large domain of validity, it can not be justified in the quantum gravity era where curvature and matter densities are of Planck scale. Using techniques from loop quantum gravity, the standard theory of cosmological perturbations is extended to overcome this limitation...
50 pages

Here's a related paper by other authors that also came out this year:
http://arxiv.org/abs/1205.1917
Hybrid quantization of an inflationary universe
Mikel Fernández-Méndez, Guillermo A. Mena Marugán, Javier Olmedo
(Submitted on 9 May 2012)
We quantize to completion an inflationary universe with small inhomogeneities in the framework of loop quantum cosmology. The homogeneous setting consists of a massive scalar field propagating in a closed, homogeneous scenario. We provide a complete quantum description of the system employing loop quantization techniques. After introducing small inhomogeneities as scalar perturbations, we identify the true physical degrees of freedom by means of a partial gauge fixing, removing all the local degrees of freedom except the matter perturbations. We finally combine a Fock description for the inhomogeneities with the polymeric quantization of the homogeneous background, providing the quantum Hamiltonian constraint of the composed system. Its solutions are then completely characterized, owing to the suitable choice of quantum constraint, and the physical Hilbert space is constructed. Finally, we consider the analog description for an alternate gauge and, moreover, in terms of gauge-invariant quantities. In the deparametrized model, all these descriptions are unitarily equivalent at the quantum level.
16 pages

 

[marcus]

In LQG it's assumed that the areas volumes etc are DETERMINED by some physical entity. As I recall Rovelli has somewhere discussed that an area could be determined by something material such as a desktop. But also as in the case of a BH the gravitational field itself can determine geometric quantities--an horizon area, the direction of a gravitational plane-wave...

I'm not an expert and can't answer all your questions about the technicalities here (maybe Francesca or f-h will help out) but I'll just say what I think. I think whatever material or process is imagined to determine these things, a surface is DEFINED as a collection of spinnetwork links (e.g. those cut by the desktop surface, or by the BH horizon, or whatever material).

A volume "is" a certain set of nodes, an area "is" a certain set of links and the corresponding LQG operators in effect measure the volume or the area which is defined by those nodes or those cut links.

 

[atyy]

Hmm, the comprehensive review of LQG in an issue of SIGMA includes an article by Tambornino on this issue.

 

[atyy]

Which ones were you thinking of?

Maybe http://arxiv.org/abs/1206.3807 ?

Thanks for the LQC references. I hadn't been following those. Can they get all the predictions of inflation, like the cmb temperature anisotropy angular power spectrum?

 

[marcus]

Maybe http://arxiv.org/abs/1206.3807 ?

Thanks for the LQC references. I hadn't been following those. Can they get all the predictions of inflation, like the cmb temperature anisotropy angular power spectrum?

That's the point, isn't it?
My impression is that LQC is a leader in that respect.
The CMB features observed so far are well-explained by a standard inflation. Other cosmic models have come under suspicion that they need fine tuning or else make adequate inflation quite unlikely. LQC has the virtue that it robustly predicts an adequate inflation--and consequently makes the observed power spectrum etc features highly probable.

There are some fine points--more subtle power spectrum features. What really matters there, I think, is what FUTURE observation shows. When CMB is mapped with higher resolution will this confirm what the LQC people expect? Here are some LQC phenomenology-related papers. Sometimes this link is slow or times out, other times it works.

http://www-library.desy.de/cgi-bin/spiface/find/hep/www?rawcmd=FIND+%28DK+LOOP+SPACE+AND+%28QUANTUM+GRAVITY+OR+QUANTUM+COSMOLOGY%29+%29+AND+%28GRAVITATIONAL+RADIATION+OR+PRIMORDIAL+OR+inflation+or+POWER+SPECTRUM+OR+COSMIC+BACKGROUND+RADIATION%29+AND+DATE%3E2008&FORMAT=www&SEQUENCE=citecount%28d%29

I just tried it and it came up with 68 papers that appeared 2009 or later, but the link took about one minute, it was slow. These are mainly concerned with the angular power spectrum (which can show the imprint of gravitational waves frozen in at recombination time, among other things.)

In case the link fails when you try it, or isn't helpful, this paper is old (March 2011) but shows the general thrust of some recent LQC research.
http://arxiv.org/abs/1103.2475
Probability of Inflation in Loop Quantum Cosmology
Abhay Ashtekar, David Sloan
(Submitted on 12 Mar 2011)
Inflationary models of the early universe provide a natural mechanism for the formation of large scale structure. This success brings to forefront the question of naturalness: Does a sufficiently long slow roll inflation occur generically or does it require a careful fine tuning of initial parameters? In recent years there has been considerable controversy on this issue. In particular, for a quadratic potential, Kofman, Linde and Mukhanov have argued that the probability of inflation with at least 65 e-foldings is close to one, while Gibbons and Turok have argued that this probability is suppressed by a factor of ~ 10^-85. We first clarify that such dramatically different predictions can arise because the required measure on the space of solutions is intrinsically ambiguous in general relativity. We then show that this ambiguity can be naturally resolved in loop quantum cosmology (LQC) because the big bang is replaced by a big bounce and the bounce surface can be used to introduce the structure necessary to specify a satisfactory measure.
The second goal of the paper is to present a detailed analysis of the inflationary dynamics of LQC using analytical and numerical methods. By combining this information with the measure on the space of solutions, we address a sharper question than those investigated in the literature: What is the probability of a sufficiently long slow roll inflation WHICH IS COMPATIBLE WITH THE SEVEN YEAR WMAP DATA? We show that the probability is very close to 1. ...

 

[atyy]

I do wonder about neglecting backreaction, especially in the Planckian regime, but it does seem they get a definite prediction out (Fig 1 of http://arxiv.org/abs/1209.1609). Do any existing missions have enough finesse to test their prediction?