General Wick's Theorem for bosonic and fermionic operators

TL;DR

This paper introduces a General Wick's Theorem that unifies the treatment of bosonic and fermionic operators, simplifying complex calculations in quantum field theory by relating different operator orderings.

Contribution

It proves a unified General Wick's Theorem applicable to both bosonic and fermionic operators, extending the classical Wick's theorem as a special case.

Findings

GWT applies equally to bosonic and fermionic operators

The theorem simplifies calculations involving operator orderings

Examples demonstrate reduced computational effort

Abstract

Wick's theorem provides a connection between time ordered products of bosonic or fermionic fields, and their normal ordered counterparts. We consider a generic pair of operator orderings and we prove, by induction, the theorem that relates them. We name this the General Wick's Theorem (GWT) because it carries Wick's theorem as special instance, when one applies the GWT to time and normal orderings. We establish the GWT both for bosonic and fermionic operators, i.e. operators that satisfy c-number commutation and anticommutation relations respectively. We remarkably show that the GWT is the same, independently from the type of operator involved. By means of a few examples, we show how the GWT helps treating demanding problems by reducing the amount of calculations required.