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==quote from Dah-Wei Chiou's latest paper==

In the research of loop quantum gravity (LQG), the sum-over-histories formulation is an active research area that goes under the name “spin foam models” (SFMs) (see [9] and references therein for LQG and SFMs). In particular, over the past years, SFMs in relation to the kinematics of LQG have been clearly established [11–14]. However, the Hamiltonian dynamics of LQG is far from fully understood, and although well motivated, SFMs have not been systematically derived from any well-established theories of canonical quantum gravity. Meanwhile, loop quantum cosmology (LQC) has recently been cast in a sum-over-histories formulation, providing strong support for the general paradigm underlying SFMS [15, 16]. In this paper, the timeless path integral is systematically derived from the canonical formalism of relativistic quantum mechanics, and we hope it will shed new light on the issues of the interplay between LQG/LQC and SFMs.

==endquote==

I like Chiou's work. It was his July paper on Unimodular Loop Quantum Cosmology that prompted me to start the "Unigrav" thread (about Unimodular Gravity) that has been quite active recently. Here is Chiou's Timeless Path Integral paper:

http://arxiv.org/abs/1009.5436

Dah-Wei Chiou

30 pages

(Submitted on 28 Sep 2010)

"Starting from the canonical formalism of relativistic (timeless) quantum mechanics, the formulation of timeless path integral is rigorously derived. The transition amplitude is reformulated as the sum, or functional integral, over all possible paths in the constraint surface specified by the (relativistic) Hamiltonian constraint, and each path contributes with a phase identical to the classical action divided by [tex]\hbar[/tex]. The timeless path integral manifests the timeless feature as it is completely independent of the parametrization for paths. For the special case that the Hamiltonian constraint is a quadratic polynomial in momenta, the transition amplitude admits the timeless Feynman's path integral over the (relativistic) configuration space."

Almost all the first 16 references are to LQG community stuff. You can see the references peppered through the paragraph I quoted at the start of this post. What he is doing seems highly relevant.

In the research of loop quantum gravity (LQG), the sum-over-histories formulation is an active research area that goes under the name “spin foam models” (SFMs) (see [9] and references therein for LQG and SFMs). In particular, over the past years, SFMs in relation to the kinematics of LQG have been clearly established [11–14]. However, the Hamiltonian dynamics of LQG is far from fully understood, and although well motivated, SFMs have not been systematically derived from any well-established theories of canonical quantum gravity. Meanwhile, loop quantum cosmology (LQC) has recently been cast in a sum-over-histories formulation, providing strong support for the general paradigm underlying SFMS [15, 16]. In this paper, the timeless path integral is systematically derived from the canonical formalism of relativistic quantum mechanics, and we hope it will shed new light on the issues of the interplay between LQG/LQC and SFMs.

==endquote==

I like Chiou's work. It was his July paper on Unimodular Loop Quantum Cosmology that prompted me to start the "Unigrav" thread (about Unimodular Gravity) that has been quite active recently. Here is Chiou's Timeless Path Integral paper:

http://arxiv.org/abs/1009.5436

**Timeless path integral for relativistic quantum mechanics**Dah-Wei Chiou

30 pages

(Submitted on 28 Sep 2010)

"Starting from the canonical formalism of relativistic (timeless) quantum mechanics, the formulation of timeless path integral is rigorously derived. The transition amplitude is reformulated as the sum, or functional integral, over all possible paths in the constraint surface specified by the (relativistic) Hamiltonian constraint, and each path contributes with a phase identical to the classical action divided by [tex]\hbar[/tex]. The timeless path integral manifests the timeless feature as it is completely independent of the parametrization for paths. For the special case that the Hamiltonian constraint is a quadratic polynomial in momenta, the transition amplitude admits the timeless Feynman's path integral over the (relativistic) configuration space."

Almost all the first 16 references are to LQG community stuff. You can see the references peppered through the paragraph I quoted at the start of this post. What he is doing seems highly relevant.

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