On the construction of Hadamard states from Feynman propagators
Christopher J. Fewster, Alexander Strohmaier
TL;DR
The paper addresses constructing Hadamard states from Feynman propagators for bundle-valued quantum fields on globally hyperbolic spacetimes. It shows, using a microlocal construction plus a simple domination lemma, that a Feynman propagator satisfying the first positivity condition can be adjusted by a smoothing kernel to satisfy both positivity conditions, thereby yielding Hadamard states for complex, Hermitian, Dirac-type, and Majorana-type theories. The approach unifies bosonic and fermionic cases by working through normally hyperbolic and Dirac-type operators, and provides explicit formulations of the two-point functions in terms of the Feynman propagator and Pauli–Jordan distributions. This delivers a general toolbox that converts microlocal splittings into Hadamard states across a broad class of bundle-valued fields, with clear implications for gauge-invariant quasifree states and their algebraic realization.
Abstract
The Wightman two-point function of any Hadamard state of a linear quantum field theory determines a corresponding Feynman propagator. Conversely, however, a Feynman propagator determines a state only if certain positivity conditions are fulfilled. We point out that the construction of a state from a Feynman propagator involves a slightly subtle point that we address and resolve. Starting from a recent generalisation of the Duistermaat-Hörmander theory of distinguished parametrices to normally hyperbolic and Dirac-type operators acting on sections of hermitian vector bundles, we complete this work by showing how Feynman propagators can be chosen so as to define Hadamard states. The theories considered are: the complex bosonic field governed by a normally hyperbolic operator; the corresponding hermitian theory if the operator commutes with a complex conjugation; the Dirac fermionic theory governed by a Dirac-type operator, and the corresponding Majorana theory in the case where the operator commutes with a skew complex conjugation. The additional key ingredients that we supply are simple domination properties of self-adjoint smooth kernels.