Quantum field theory in static external potentials and Hadamard states

TL;DR

This paper proves that the ground state solutions for Dirac and Klein-Gordon equations in static external potentials satisfy the Hadamard condition, with implications for quantum field theory in curved spacetime.

Contribution

It establishes the Hadamard property for ground states in static external potentials and extends results to Klein-Gordon fields using Krein space formalism.

Findings

Ground states in static potentials satisfy the Hadamard condition.

A Fourier support condition characterizes the Hadamard property.

Overcritical potentials admit no ground states, but many Hadamard states can be constructed.

Abstract

We prove that the ground state for the Dirac equation on Minkowski space in static, smooth external potentials satisfies the Hadamard condition. We show that it follows from a condition on the support of the Fourier transform of the corresponding positive frequency solution. Using a Krein space formalism, we establish an analogous result in the Klein-Gordon case for a wide class of smooth potentials. Finally, we investigate overcritical potentials, i.e. which admit no ground states. It turns out, that numerous Hadamard states can be constructed by mimicking the construction of ground states, but this leads to a naturally distinguished one only under more restrictive assumptions on the potentials.