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Coupled Proca theories: Green-hyperbolicity, quantization and applications to polarization measurement

Christopher J. Fewster, Christiane K. M. Klein

Abstract

The Proca field describes a massive relativistic spin-$1$ particle and was originally formulated in Minkowski spacetime. Here we consider a variety of generalizations in globally hyperbolic spacetimes, including couplings between a number of Proca fields via a mass-matrix, the charged Proca field with arbitrary magnetic moment in an arbitrary external electromagnetic field, and a Proca-Klein-Gordon theory with a spacetime-dependent bilinear coupling. The equations are analysed using a general auxiliary field method, introduced here, which provides practical criteria for showing that a given operator is (semi)-Green-hyperbolic. The method goes beyond what is achieved in existing analyses of deformed equations, which for example place restrictions on the magnetic moment and electromagnetic potential that can be coupled to a Proca field. The theories considered can be quantized following a common pattern and all the examples treated in this work admit Hadamard states on any globally hyperbolic spacetime. As an application, the Proca-Klein-Gordon system is used to develop a measurement scheme sensitive to the Proca polarization, using a Klein-Gordon field as the probe. For a suitable family of $n$-particle Proca states, the leading-order probe response accords with Malus' law, confirming that this system acts as a polarization-sensitive detector.

Coupled Proca theories: Green-hyperbolicity, quantization and applications to polarization measurement

Abstract

The Proca field describes a massive relativistic spin- particle and was originally formulated in Minkowski spacetime. Here we consider a variety of generalizations in globally hyperbolic spacetimes, including couplings between a number of Proca fields via a mass-matrix, the charged Proca field with arbitrary magnetic moment in an arbitrary external electromagnetic field, and a Proca-Klein-Gordon theory with a spacetime-dependent bilinear coupling. The equations are analysed using a general auxiliary field method, introduced here, which provides practical criteria for showing that a given operator is (semi)-Green-hyperbolic. The method goes beyond what is achieved in existing analyses of deformed equations, which for example place restrictions on the magnetic moment and electromagnetic potential that can be coupled to a Proca field. The theories considered can be quantized following a common pattern and all the examples treated in this work admit Hadamard states on any globally hyperbolic spacetime. As an application, the Proca-Klein-Gordon system is used to develop a measurement scheme sensitive to the Proca polarization, using a Klein-Gordon field as the probe. For a suitable family of -particle Proca states, the leading-order probe response accords with Malus' law, confirming that this system acts as a polarization-sensitive detector.

Paper Structure

This paper contains 28 sections, 9 theorems, 219 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Further literature and remarks
  3. Structure
  4. Notation
  5. The charged Proca equation
  6. Auxiliary field method
  7. Green-hyperbolic operators
  8. Warm-up: the neutral Proca field
  9. Abstract existence result
  10. Applications
  11. Proca multiplets with variable mass matrix
  12. Charged Proca field: general set-up and special case $\kappa=1$
  13. The charged Proca field for general $\kappa\in\mathbb{R}$
  14. An alternative treatment of the neutral Proca field
  15. Coupled Proca--scalar system
  16. ...and 13 more sections

Key Result

Lemma 3.1

Let $P:\Gamma^\infty(B_1)\to\Gamma^\infty(B_2)$, $Q$, $C$ and $D$ be differential operators so that $CP=QD$, and suppose that $P$ and $Q$ are semi-Green-hyperbolic. Then $DE_P^\pm = E_Q^\pm C$ on $\Gamma_0^\infty(B_2)$.

Figures (2)

  • Figure 1: Commutative diagram for Theorem \ref{['thm:abs']}.
  • Figure 2: The parametrization of $k_\mu$ viewed from above the $xy$-plane

Theorems & Definitions (19)

  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Theorem 3.5
  • proof
  • Remark 3.6
  • ...and 9 more