Normal Ordering Normal Modes

TL;DR

This paper derives the transformation (Wick map) between normal ordering schemes in quantum field theory soliton sectors, providing explicit formulas and recursion relations for scalar fields in 1+1 dimensions.

Contribution

It introduces a method to relate normal ordering in soliton normal modes to plane wave normal ordering, with explicit solutions and recursion formulas for scalar fields.

Findings

Derived the Wick map between different normal orderings.

Provided explicit solutions for products of fields at the same point.

Established a recursion formula for $j$-point functions.

Abstract

In a soliton sector of a quantum field theory, it is often convenient to expand the quantum fields in terms of normal modes. Normal mode creation and annihilation operators can be normal ordered, and their normal ordered products have vanishing expectation values in the one-loop soliton ground state. The Hamiltonian of the theory, however, is usually normal ordered in the basis of operators which create plane waves. In this paper we find the Wick map between the two normal orderings. For concreteness, we restrict our attention to Schrodinger picture scalar fields in 1+1 dimensions, although we expect that our results readily generalize beyond this case. We find that plane wave ordered $n$-point functions of fields are sums of terms which factorize into $j$-point functions of zero modes, breather and continuum normal modes. We find a recursion formula in $j$ and, for products of fields at the same point, we solve the recursion formula at all $j$.