This paper seems to me especially interesting:(adsbygoogle = window.adsbygoogle || []).push({});

http://arxiv.org/abs/1308.2946

Purely geometric path integral for spin foams

Atousa Shirazi, Jonathan Engle

(Submitted on 13 Aug 2013)

Spin-foams are a proposal for defining the dynamics of loop quantum gravity via path integral. In order for a path integral to be at least formally equivalent to the corresponding canonical quantization, at each point in the space of histories it is important that the integrand have not only the correct phase -- a topic of recent focus in spin-foams -- but also the correct modulus, usually referred to as the measure factor. The correct measure factor descends from the Liouville measure on the reduced phase space, and its calculation is a task of canonical analysis.

The covariant formulation of gravity from which spin-foams are derived is the Plebanski-Holst formulation, in which the basic variables are a Lorentz connection and a Lorentz-algebra valued two-form, called the Plebanski two-form. However, in the final spin-foam sum, one sums over only spins and intertwiners, which label eigenstates of the Plebanski two-form alone. The spin-foam sum is therefore a discretized version of a Plebanski-Holst path integral in which only the Plebanski two-form appears, and in which the connection degrees of freedom have been integrated out. We call this a purely geometric Plebanski-Holst path integral.

In prior work in which one of the authors was involved, the measure factor for the Plebanski-Holst path integral with both connection and two-form variables was calculated. Before one discretizes this measure and incorporates it into a spin-foam sum, however, one must integrate out the connection in order to obtain the purely geometric version of the path integral. To calculate this purely geometric path integral is the principal task of the present paper, and it is done in two independent ways. Gauge-fixing and the background independence of the resulting path integral are discussed in the appendices.

21 pages

Shirazi gave a talk on it which is available as online Pirsa video. http://pirsa.org/13070086 (advance to minute 64:00) The talk is just 20 minutes and helps one get an intuitive appreciation of their result. The measure factor they get seems likely to be right because they calculated it from two different approaches and go the same answer.

Engle has been notably active in perfecting the EPRL (Engle-Pereira-Rovelli-Livine) path integral. Here's a bit of background information: https://www.physicsforums.com/showthread.php?t=706775

The usual form of loop gravity depends on two main variables, a connection ω and atetradoften denoted by e^{i}which can be imagined as a moving frame, or as a Lorentz algebra valued one-form. That means e∧e is an algebra-valued two-form. (Technically called "Plebanski two-form." These authors work with a two-form X^{IJ}which is a constant multiple of e∧e (see equation 7 on page 4).

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# A purely geometric path integral for gravity

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