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Polarisation sets of Green operators for normally hyperbolic equations

Christopher J. Fewster

TL;DR

This work addresses the microlocal structure of normally hyperbolic operators on vector bundles over globally hyperbolic spacetimes by computing the polarisation sets of their advanced and retarded Green kernels and their difference. The authors establish a general formula expressing the polarisation sets in terms of parallel transport along null geodesics with respect to the Weitzenböck connection, yielding precise fibre data that complements the standard wavefront-set description. They apply these results to the Proca field, resolving a gap in Moretti–Murro–Volpe and clarifying the Hadamard-state structure for massive spin-1 fields, while also providing a detailed, general framework for the polarisation of Green operators. The findings enable direct calculation of wavefront sets for related operators and offer a robust tool for quantum field theory in curved spacetimes, with implications for Hadamard states and interacting field theories. The work thus strengthens the bridge between microlocal analysis and quantum field theory on curved backgrounds, and opens questions about richer stratifications of polarisation that reveal propagating physical degrees of freedom beyond constraints.

Abstract

The polarisation set of a vector-valued distribution generalises the wavefront set and captures fibre-directional information about its singularities in addition to their phase space description. Motivated by problems in quantum field theory on curved spacetimes, we consider normally hyperbolic operators on vector bundles over globally hyperbolic spacetimes, and compute the polarisation sets of the kernel distributions for their advanced and retarded Green operators and the difference thereof. This permits the computation of related polarisation and wavefront sets for operators whose solution theory is related to the normally hyperbolic case. As a particular example, we consider the Proca equation that describes massive relativistic spin-1 particles, identifying and closing a gap in a recent paper on that subject.

Polarisation sets of Green operators for normally hyperbolic equations

TL;DR

This work addresses the microlocal structure of normally hyperbolic operators on vector bundles over globally hyperbolic spacetimes by computing the polarisation sets of their advanced and retarded Green kernels and their difference. The authors establish a general formula expressing the polarisation sets in terms of parallel transport along null geodesics with respect to the Weitzenböck connection, yielding precise fibre data that complements the standard wavefront-set description. They apply these results to the Proca field, resolving a gap in Moretti–Murro–Volpe and clarifying the Hadamard-state structure for massive spin-1 fields, while also providing a detailed, general framework for the polarisation of Green operators. The findings enable direct calculation of wavefront sets for related operators and offer a robust tool for quantum field theory in curved spacetimes, with implications for Hadamard states and interacting field theories. The work thus strengthens the bridge between microlocal analysis and quantum field theory on curved backgrounds, and opens questions about richer stratifications of polarisation that reveal propagating physical degrees of freedom beyond constraints.

Abstract

The polarisation set of a vector-valued distribution generalises the wavefront set and captures fibre-directional information about its singularities in addition to their phase space description. Motivated by problems in quantum field theory on curved spacetimes, we consider normally hyperbolic operators on vector bundles over globally hyperbolic spacetimes, and compute the polarisation sets of the kernel distributions for their advanced and retarded Green operators and the difference thereof. This permits the computation of related polarisation and wavefront sets for operators whose solution theory is related to the normally hyperbolic case. As a particular example, we consider the Proca equation that describes massive relativistic spin-1 particles, identifying and closing a gap in a recent paper on that subject.

Paper Structure

This paper contains 29 sections, 15 theorems, 131 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background on distributions and pseudodifferential operators
  3. General conventions
  4. Manifolds and bundles
  5. Distributions and kernels
  6. Pseudodifferential operators
  7. Subprincipal and refined principal symbols
  8. The formal dual
  9. The polarisation set
  10. Definition and basic properties
  11. Examples
  12. Bundle pullbacks
  13. Tensor pullbacks
  14. Propagation of polarisation for systems of real principal type
  15. Polarisation sets of Green operators
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $B$ be a finite-rank complex vector bundle over a globally hyperbolic spacetime $(M,g)$. Let $P$ be a normally hyperbolic operator on $\Gamma^\infty(B\otimes\Omega^{1/2})$, and $\mathcal{R}_\textnormal{pol}^\pm$, $\mathcal{R}_\textnormal{pol}$ be as defined in eq:Rcpolsets. Then:

Figures (1)

  • Figure 1: (a) The points $x$, $x'$, $y$, $y'$ and $x"$ along a null geodesic and region $N$ (shaded) and subregions $N^\pm$ appearing in step $4$ of the proof of Theorem \ref{['thm:WFpolEPpmandEP']}. The unlabelled dotted lines are the Cauchy surfaces $\Sigma^\pm$. One has $(y,l;y',-l')\in \mathop{\mathrm{WF}}\nolimits(E_{\tilde{P}}^+)$ at the outset. (b) Propagation of singularities for $\tilde{P}\otimes 1$ is used along the geodesic between $y$ and $x$ without meeting $y'$ to infer that $(x,k;y',-l')\in \mathop{\mathrm{WF}}\nolimits(E_{\tilde{P}}^+)$. (c) Propagation of singularities for $1\otimes\leftidx{^{\star}}{\tilde{P}}{}$ is used along the geodesic between $y'$ and $x"$ without meeting $x$ to infer that $(x,k;x",-k")\in\mathop{\mathrm{WF}}\nolimits(E_{\tilde{P}}^+)\cap\mathop{\mathrm{WF}}\nolimits(E_{P}^+)$. (d) Finally, propagation of singularities for $1\otimes\leftidx{^{\star}}{P}{}$ is used along the geodesic between $x"$ and $x'$ without meeting $x$ to infer that $(x,k;x',-k')\in\mathop{\mathrm{WF}}\nolimits(E_P^+)$.

Theorems & Definitions (30)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 20 more