The 2009 Engle Han Thiemann paper (written while Engle was postdoc at Erlangen) was intended to be the first of a series, “Canonical path integral measures for Holst and Plebanski gravity. I."
However number II of the series never appeared.
This August 2013 paper of Engle Shirazi could be considered the sequel--the continuation of the 2009 work--shifting the focus to spin foams.
Here's the 2009 paper's abstract:
Canonical path integral measures for Holst and Plebanski gravity. I. Reduced Phase Space Derivation
Jonathan Engle, Muxin Han, Thomas Thiemann
(Submitted on 17 Nov 2009)
An important aspect in defining a path integral quantum theory is the determination of the correct measure. For interacting theories and theories with constraints, this is non-trivial, and is normally not the heuristic "Lebesgue measure" usually used. There have been many determinations of a measure for gravity in the literature, but none for the Palatini or Holst formulations of gravity. Furthermore, the relations between different resulting measures for different formulations of gravity are usually not discussed.
In this paper we use the reduced phase technique in order to derive the path-integral measure for the Palatini and Holst formulation of gravity, which is different from the Lebesgue measure up to local measure factors which depend on the spacetime volume element and spatial volume element.
From this path integral for the Holst formulation of GR we can also give a new derivation of the Plebanski path integral and discover a discrepancy with the result due to Buffenoir, Henneaux, Noui and Roche (BHNR) whose origin we resolve. This paper is the first in a series that aims at better understanding the relation between canonical LQG and the spin foam approach.
27 pages
==quote page 1 of EHT 2009==
Richard Feynman, in the course of his doctoral work, developed the path integral formulation of quantum mechanics as an alternative, space-time covariant description of quantum mechanics, which is nevertheless equivalent to the canonical approach [1]. It is thus not surprising that the path integral formulation has been of interest in the quantization of general relativity, a theory where space-time covariance plays a key role. However, once one departs from the regime of free, unconstrained systems, the equivalence of the path integral approach and canonical approach becomes more subtle than originally described by Feynman in [1]. In particular, in Feynman’s original argument, the integration measure for the configuration path integral is a formal Lebesgue measure; in the interacting case, however, in order to have equivalence with the canonical theory, one cannot use the naive Lebesgue measure in the path integral, but must use a measure derived from the Liouville measure on the phase space [2].
Such a measure has yet to be incorporated into spin-foam models, which can be thought of as a path-integral version of loop quantum gravity (LQG) [3, 4]. Loop quantum gravity is an attempt to make a mathematically rigorous quantization of general relativity that preserves background independence — for reviews, see [8, 6, 7] and for books see [9, 10]. Spin-foams intend to be a path integral formulation for loop quantum gravity, directly motivated from the ideas of Feynman appropriately adapted to reparametrization-invariant theories [4, 5]. Only the kinematical structure of LQG is used in motivating the spin-foam framework. The dynamics one tries to encode in the amplitude factors appearing in the path integral which is being replaced by a sum in a regularisation step which depends on a triangulation of the spacetime manifold. Eventually one has to take a weighted average over these (generalised) triangulations for which the proposal at present is to use methods from group field theory [3]. The current spin foam approach is independent from the dynamical theory of canonical LQG [11] because the dynamics of canonical LQG is rather complicated. It instead uses an apparently much simpler starting point: Namely, in the Plebanski formulation [14], GR can be considered as a constrained BF theory, and treating the so called simplicity constraints as a perturbation of BF theory, one can make use of the powerful toolbox that comes with topological QFT’s [12]. It is an unanswered question, however, and one of the most active research topics momentarily1, how canonical LQG and spin foams fit together. It is one the aims of this paper to make a contribution towards answering this question.
==endquote==
==quote pages 1,2 of Shirazi Engle
http://arxiv.org/abs/1308.2946 ==
In the path integral approach to constructing a quantum theory, the integrand of the path integral has two important parts: a phase part given by the exponential of i times the classical action, and a measure factor. The form of the phase part in terms of the classical action ensures that solutions to the classical equations of motion dominate the path integral in the classical limit so that one recovers classical physics in the appropriate regime. The measure factor, however, arises from careful canonical analysis, and is important for the path integral to be equivalent to the corresponding canonical quantum theory. In most theories, this means that it is important, in particular, in order for the path integral theory to have such elementary properties as yielding a unitary S-matrix that preserves probabilities. The importance of having the correct measure factor is thus quite high.
Spin-foams are a path integral approach to quantum gravity in which one does not sum over classical gravitational histories, but rather quantum histories arising from canonical quantization. Specifically, in spin-foams, one sums over histories of canonical states of loop quantum gravity. These histories possesses a natural 4-dimensional space-time covariant interpretation, whence each can be thought of as a quantum space-time. This approach allows one to retain the understanding gained from loop quantum gravity, such as the discreteness of area and volume spectra, while at the same time formulating the dynamics in a way that makes space-time symmetries more manifest [1].
The starting point for the derivation of spin-foams is the Plebanski-Holst formulation of gravity [2–6], in which the basic variables are a connection ω and what is called the Plebanski two-form, Σ. However, in the final spin-foam sum, the connection ω is usually not present, and one sums over only spins and intertwiners, which determine certain eigenstates of Σ alone. Because of this, the continuum path integral most directly related to the spin-foam sum is a Plebanski-Holst path integral in which only Σ appears, and in which the connection has been integrated out. We call such a path integral purely geometric because Σ directly determines the geometry of space-time.
Because of the quantum mechanical nature of the histories used in spin-foams, ensuring that the summand has the required phase part and measure factor is not completely trivial. Only within the last couple years has the correct phase part been achieved [7, 8]. Regarding the measure factor, a first step has been carried out in the work [5], where the correct measure factor is calculated for the Plebanski-Holst path integral with both ω and Σ present. However, until now, the measure factor for the path integral with Σ alone, necessary for spin-foams, had not yet been calculated. To carry out this calculation is the main purpose of the present paper. In order to be sure about all numerical factors, we do this in two different ways: (1.) by starting from the path integral in [5] and then integrating out the connection degrees of freedom, and (2.) by starting from the ADM path integral and introducing the necessary variables from there. We find that these two ways of calculating the measure factor exactly match, as must be the case, as the canonical measure factor ultimately descends from the Liouville measure on the reduced phase space, which is independent of the formulation of gravity used [5].
The path integral derived in this paper is ready to be discretized and translated into a spin-foam model, a task which will be carried out in forthcoming work. Furthermore, when this is accomplished, we would like to emphasize that, because both primary and secondary simplicity constraints are already incorporated in the continuum path integral [5,9], they will be automatically incorporated in the resulting spin-foam model as well.
==endquote==
On the other hand, what do I know? Maybe you will like this idea of Freidel.)
