The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes
R. Brunetti, K. Fredenhagen, M. Koehler
TL;DR
The authors formulate the microlocal spectrum condition ($\mu$SC) as a wave front set constraint on all $n$-point distributions to extend the Minkowski spectrum condition to curved spacetimes. Building on Radzikowski's Hadamard-state analysis and Hörmander's microlocal tools, they prove that Hadamard states satisfy $\mu$SC and construct Wick polynomials (including the energy-momentum tensor) as operator-valued distributions that also satisfy $\mu$SC. This yields a rigorous, local, and covariant framework for defining products of fields and for perturbative constructions on globally hyperbolic manifolds. The results bridge Hadamard/QFT on curved spacetimes with Wightman-field concepts and set the stage for causal perturbation theory in curved backgrounds, potentially clarifying distinctions between free and interacting theories through stronger forms of $\mu$SC.
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'' ($μ$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.