Global and microlocal aspects of Dirac operators: Propagators and Hadamard states

Matteo Capoferri; Simone Murro

doi:10.1002/mana.12032

Global and microlocal aspects of Dirac operators: Propagators and Hadamard states

Matteo Capoferri, Simone Murro

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Abstract

We propose a geometric approach to construct the Cauchy evolution operator for the Lorentzian Dirac operator on Cauchy-compact globally hyperbolic 4-manifolds. We realize the Cauchy evolution operator as the sum of two invariantly defined oscillatory integrals—the positive and negative Dirac propagators—global in space and in time, with distinguished complex-valued geometric phase functions. As applications, we relate the Cauchy evolution operators with the Feynman propagator and construct Cauchy surfaces covariances of quasifree Hadamard states.

Original language	English
Pages (from-to)	2942-2974
Number of pages	33
Journal	Mathematische Nachrichten
Volume	298
Issue number	9
Early online date	27 Jun 2025
DOIs	https://doi.org/10.1002/mana.12032
Publication status	Published - Sept 2025

Keywords

Cauchy evolution operator
Feynman propagator
Fourier integral operators
global propagators
globally hyperbolic manifolds
Hadamard states
Lorentzian Dirac operator
pseudodifferential projections

ASJC Scopus subject areas

General Mathematics

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abstract = "We propose a geometric approach to construct the Cauchy evolution operator for the Lorentzian Dirac operator on Cauchy-compact globally hyperbolic 4-manifolds. We realize the Cauchy evolution operator as the sum of two invariantly defined oscillatory integrals—the positive and negative Dirac propagators—global in space and in time, with distinguished complex-valued geometric phase functions. As applications, we relate the Cauchy evolution operators with the Feynman propagator and construct Cauchy surfaces covariances of quasifree Hadamard states.",

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AU - Capoferri, Matteo

AU - Murro, Simone

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N2 - We propose a geometric approach to construct the Cauchy evolution operator for the Lorentzian Dirac operator on Cauchy-compact globally hyperbolic 4-manifolds. We realize the Cauchy evolution operator as the sum of two invariantly defined oscillatory integrals—the positive and negative Dirac propagators—global in space and in time, with distinguished complex-valued geometric phase functions. As applications, we relate the Cauchy evolution operators with the Feynman propagator and construct Cauchy surfaces covariances of quasifree Hadamard states.

AB - We propose a geometric approach to construct the Cauchy evolution operator for the Lorentzian Dirac operator on Cauchy-compact globally hyperbolic 4-manifolds. We realize the Cauchy evolution operator as the sum of two invariantly defined oscillatory integrals—the positive and negative Dirac propagators—global in space and in time, with distinguished complex-valued geometric phase functions. As applications, we relate the Cauchy evolution operators with the Feynman propagator and construct Cauchy surfaces covariances of quasifree Hadamard states.

KW - Cauchy evolution operator

KW - Feynman propagator

KW - Fourier integral operators

KW - global propagators

KW - globally hyperbolic manifolds

KW - Hadamard states

KW - Lorentzian Dirac operator

KW - pseudodifferential projections

UR - https://www.scopus.com/pages/publications/105009327486

U2 - 10.1002/mana.12032

DO - 10.1002/mana.12032

M3 - Article

SN - 0025-584X

VL - 298

SP - 2942

EP - 2974

JO - Mathematische Nachrichten

JF - Mathematische Nachrichten

IS - 9

ER -

Global and microlocal aspects of Dirac operators: Propagators and Hadamard states

Abstract

Keywords

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