Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime

TL;DR

This paper extends Radzikowski's wavefront set spectrum condition characterization of Hadamard states from scalar to vector-valued quantum fields in curved spacetime, clarifying the structure and short-distance behavior of these states.

Contribution

It generalizes the wavefront set spectrum condition to vector fields and addresses a gap in the scalar case proof, enhancing the understanding of quantum fields in curved spacetime.

Findings

Wavefront set spectrum condition equivalent to Hadamard condition for vector fields.

Resolved a gap in the proof for scalar fields.

Short-distance limits of vector Hadamard states match flat-space vacuum states.

Abstract

Some years ago, Radzikowski has found a characterization of Hadamard states for scalar quantum fields on a four-dimensional globally hyperbolic spacetime in terms of a specific form of the wavefront set of their two-point functions (termed `wavefront set spectrum condition'), thereby initiating a major progress in the understanding of Hadamard states and the further development of quantum field theory in curved spacetime. In the present work, we extend this important result on the equivalence of the wavefront set spectrum condition with the Hadamard condition from scalar fields to vector fields (sections in a vector bundle) which are subject to a wave-equation and are quantized so as to fulfill the covariant canonical commutation relations, or which obey a Dirac equation and are quantized according to the covariant anti-commutation relations, in any globally hyperbolic spacetime having dimension three or higher. In proving this result, a gap which is present in the published proof for the scalar field case will be removed. Moreover we determine the short-distance scaling limits of Hadamard states for vector-bundle valued fields, finding them to coincide with the corresponding flat-space, massless vacuum states.