Normal Ordering Normal Modes

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 $j$.

arXiv comment26 pages, no figures

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• Excited Kinks as Quantum States
• The Two-Loop ${\varphi }^4$ Kink Mass
• A Reduced Inner Product for Kink States
• Moving Kinks and Their Wave Packets
• An Alternative to Collective Coordinates
• Kink Form Factors
• Removing Tadpoles in a Soliton Sector