Gauge Field Theory Coherent States (GCS) : III. Ehrenfest Theorems

TL;DR

This work proves a precise semiclassical limit for gauge-field coherent states built from Hall’s heat-kernel construction, showing that for states labeled by a classical phase-space point $(A,E)$ the expectation values of elementary operators approach their classical values and the scaled commutators reproduce Poisson brackets to zeroth order in the classicality parameter $t$. The analysis reduces to a single edge, computes explicit matrix elements for momentum and holonomy operators, and demonstrates that polynomials of elementary operators, as well as certain non-polynomial functions (via the Hamburger moment problem), recover their classical counterparts in the $t o 0$ limit. By establishing Ehrenfest properties for these coherent states, the paper provides a rigorous link between quantum gauge theories and classical general relativity, and supports the claim that the infinitesimal quantum dynamics mirrors classical GR at leading order. These results are essential for validating the classical limit of quantum gravity constructions based on diffeomorphism-invariant gauge theories and coherent-state techniques. The approach hinges on a combination of heat-kernel coherent states, projective-limit kinematics, and moment-method techniques to manage non-polynomial observables like the volume operator.

Abstract

In the preceding paper of this series of articles we established peakedness properties of a family of coherent states that were introduced by Hall for any compact gauge group and were later generalized to gauge field theory by Ashtekar, Lewandowski, Marolf, Mourão and Thiemann. In this paper we establish the ``Ehrenfest Property'' of these states which are labelled by a point (A,E), a connection and an electric field, in the classical phase space. By this we mean that i) The expectation value of {\it all} elementary quantum operators $\hat{O}$ with respect to the coherent state with label (A,E) is given to zeroth order in $\hbar$ by the value of the corresponding classical function O evaluated at the phase space point (A,E) and ii) The expectation value of the commutator between two elementary quantum operators $[\hat{O}_1,\hat{O}_2]/(i\hbar)$ divided by $i\hbar$ with respect to the coherent state with label (A,E) is given to zeroth order in $\hbar$ by the value of the Poisson bracket between the corresponding classical functions $\{O_1,O_2\}$ evaluated at the phase space point (A,E). These results can be extended to all polynomials of elementary operators and to a certain non-polynomial function of the elementary operators associated with the volume operator of quantum general relativity. It follows that the infinitesimal quantum dynamics of quantum general relativity is to zeroth order in $\hbar$ indeed given by classical general relativity.

Theorems & Definitions (16)

• Definition 2.1
• Theorem 3.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Theorem 3.2
• Lemma 3.6
• Theorem 3.3