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Gauge Field Theory Coherent States (GCS) : I. General Properties

Thomas Thiemann

TL;DR

This paper develops a general framework for constructing diffeomorphism-covariant gauge-theory coherent states by a complexifier-based transform, comparing three families of constructions and endorsing the heat-kernel complexifier (Option 2) for practical semiclassical analysis. It establishes a rigorous kinematical setup using holonomy-flux variables, spin-network kinematics, and a Segal-Bargmann–type representation, and outlines strategies to build gauge- and diffeomorphism-invariant coherent states. The work also presents a concrete 2+1 Euclidean gravity model to illustrate gauge-invariant coherent states and their semiclassical behavior, laying groundwork for connecting non-perturbative quantum gravity with low-energy Standard Model physics. Overall, the paper provides a versatile toolkit for studying semiclassical limits, Ehrenfest relations, and potential observational implications in a background-independent quantum field theory framework.

Abstract

In this article we outline a rather general construction of diffeomorphism covariant coherent states for quantum gauge theories. By this we mean states $ψ_{(A,E)}$, labelled by a point (A,E) in the classical phase space, consisting of canonically conjugate pairs of connections A and electric fields E respectively, such that (a) they are eigenstates of a corresponding annihilation operator which is a generalization of A-iE smeared in a suitable way, (b) normal ordered polynomials of generalized annihilation and creation operators have the correct expectation value, (c) they saturate the Heisenberg uncertainty bound for the fluctuations of $\hat{A},\hat{E}$ and (d) they do not use any background structure for their definition, that is, they are diffeomorphism covariant. This is the first paper in a series of articles entitled ``Gauge Field Theory Coherent States (GCS)'' which aim at connecting non-perturbative quantum general relativity with the low energy physics of the standard model. In particular, coherent states enable us for the first time to take into account quantum metrics which are excited {\it everywhere} in an asymptotically flat spacetime manifold. The formalism introduced in this paper is immediately applicable also to lattice gauge theory in the presence of a (Minkowski) background structure on a possibly {\it infinite lattice}.

Gauge Field Theory Coherent States (GCS) : I. General Properties

TL;DR

This paper develops a general framework for constructing diffeomorphism-covariant gauge-theory coherent states by a complexifier-based transform, comparing three families of constructions and endorsing the heat-kernel complexifier (Option 2) for practical semiclassical analysis. It establishes a rigorous kinematical setup using holonomy-flux variables, spin-network kinematics, and a Segal-Bargmann–type representation, and outlines strategies to build gauge- and diffeomorphism-invariant coherent states. The work also presents a concrete 2+1 Euclidean gravity model to illustrate gauge-invariant coherent states and their semiclassical behavior, laying groundwork for connecting non-perturbative quantum gravity with low-energy Standard Model physics. Overall, the paper provides a versatile toolkit for studying semiclassical limits, Ehrenfest relations, and potential observational implications in a background-independent quantum field theory framework.

Abstract

In this article we outline a rather general construction of diffeomorphism covariant coherent states for quantum gauge theories. By this we mean states , labelled by a point (A,E) in the classical phase space, consisting of canonically conjugate pairs of connections A and electric fields E respectively, such that (a) they are eigenstates of a corresponding annihilation operator which is a generalization of A-iE smeared in a suitable way, (b) normal ordered polynomials of generalized annihilation and creation operators have the correct expectation value, (c) they saturate the Heisenberg uncertainty bound for the fluctuations of and (d) they do not use any background structure for their definition, that is, they are diffeomorphism covariant. This is the first paper in a series of articles entitled ``Gauge Field Theory Coherent States (GCS)'' which aim at connecting non-perturbative quantum general relativity with the low energy physics of the standard model. In particular, coherent states enable us for the first time to take into account quantum metrics which are excited {\it everywhere} in an asymptotically flat spacetime manifold. The formalism introduced in this paper is immediately applicable also to lattice gauge theory in the presence of a (Minkowski) background structure on a possibly {\it infinite lattice}.

Paper Structure

This paper contains 11 sections, 1 theorem, 91 equations.

Table of Contents

  1. Introduction
  2. Kinematical Structure of Diffeomorphism Invariant Quantum Gauge Theories
  3. Classical Theory
  4. Quantum Theory
  5. Coherent States from a Coherent State Transform
  6. Review of Known Results
  7. Option 1) : The Volume Operator as the Complexifier
  8. Option 2) : The Heat Kernel Complexifier
  9. Coherent States Directly for Gauge Invariant Quantities
  10. Diffeomorphism Invariant Coherent States
  11. Model for Gauge Invariant Coherent States : Euclidean 2+1 gravity

Key Result

Lemma 3.1

The complexification of the holonomy for compact and semisimple $G$ is given directly as a left polar decomposition, where the right unitary factor is the holonomy of the compact gauge group while the left positive definite hermitean factor is just the exponential of $-i p^e_j\tau_j/2$.

Theorems & Definitions (2)

  • Definition 2.1
  • Lemma 3.1