The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes

TL;DR

This paper introduces a microlocal spectrum condition for quantum fields on curved spacetimes, extending the Hadamard condition, and constructs Wick polynomials that satisfy this new condition, ensuring well-defined quantum observables.

Contribution

It defines a microlocal spectrum condition for quantum fields on curved spacetimes and constructs Wick polynomials satisfying this condition, advancing the mathematical foundation of quantum field theory in curved backgrounds.

Findings

The microlocal spectrum condition generalizes the spectrum condition to curved spacetimes.

Wick polynomials including the energy-momentum tensor satisfy the microlocal spectrum condition.

The construction provides a rigorous framework for quantum fields on curved spacetimes.

Abstract

Quantum fields propagating on a curved spacetime are investigated in terms of microlocal analysis. We discuss a condition on the wave front set for the corresponding n-point distributions, called ``microlocal spectrum condition'' ($\mu$SC). On Minkowski space, this condition is satisfied as a consequence of the usual spectrum condition. Based on Radzikowski's determination of the wave front set of the two-point function of a free scalar field, satisfying the Hadamard condition in the Kay and Wald sense, we construct in the second part of this paper all Wick polynomials including the energy-momentum tensor for this field as operator valued distributions on the manifold and prove that they satisfy our microlocal spectrum condition.