Diagrammatics for Wightman Correlators in General States: The Cases of the Simple Harmonic and Anharmonic Oscillators

TL;DR

This paper extends diagrammatic methods for computing Wightman correlators from vacuum states to general states in harmonic and anharmonic oscillators, introducing a Wick theorem and diagrammatics for these cases.

Contribution

It develops a Wick theorem and diagrammatic formalism for Wightman correlators in general states of harmonic and anharmonic oscillators, broadening existing vacuum-based techniques.

Findings

Wick theorem for general density matrices of harmonic oscillators

Diagrammatic formalism for anharmonic oscillator correlators

Extension of diagrammatic methods beyond vacuum states

Abstract

The machinery of computing vacuum expectation values of a time-ordered sequence of position operators of the simple harmonic oscillator is already well established. It rests on a Wick theorem, which enables one to decompose such a quantity in terms of products of \emph{pairwise contractions}, which are vacuum expectation values of a time-ordered sequence of position operators taken two at a time. This result naturally leads to a diagrammatic approach of computing such correlators, and is already well known in the form of Feynman diagrams. We generalise this setup to encompass expectation values of a general ordered sequence of position operators (Wightman sequences) in general density matrices of the simple harmonic oscillator. A Wick theorem is first developed for this situation and consequently a diagrammatics is laid down. Wightman correlators in general density matrices of the \emph{anharmonic} oscillator are also analysed and a diagrammatic formalism is developed for them too.