Singularities of Dirac-Coulomb propagators

TL;DR

This paper analyzes the singularities of Dirac-Coulomb propagators and N-point functions, establishing conditions under which certain states are Hadamard and the charge density is well-defined, even near the Coulomb singularity.

Contribution

It extends the diffractive propagation of singularities theorem to time-dependent Coulomb potentials, demonstrating the Hadamard property of in/out states and the well-definedness of charge density.

Findings

In and out Dirac-Coulomb vacua are Hadamard states for r ≠ 0.

Relative charge density between Hadamard states is locally integrable near r=0.

Results generalize previous theorems to include t-dependent potentials.

Abstract

In this paper we study singularities of propagators and $N$-point functions for Dirac fields in a Coulomb potential, possibly with a $t$-dependent smooth part for $|t|<T<\infty$. We show that the in and out Dirac-Coulomb vacua are Hadamard states for $r\neq 0$. Furthermore, we prove that the relative charge density of any two Hadamard states is well-defined as a locally integrable function including near $r=0$. The results are based on a diffractive propagation of singularities theorem for the Dirac-Coulomb system previously obtained by the first and third authors, generalized here to the case of $t$-dependent potentials.