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.

The $ϕ^4$ Kink Mass at Two Loops

“…12) where we have changed normal ordering in terms of plane wave creation operators :: a to normal ordering in terms of normal mode creation operators :: b using the Wick's theorem of Ref. [19].…”

Moving Kinks and Their Wave Packets

“…The plane wave normal ordering of these terms is easily converted to normal mode normal ordering using the Wick's theorem of Ref. [26]. They therefore simply add terms to the left hand side of (3.38) that are cubic and higher in φ 0 and B † .…”

Evidence for the unbinding of the 𝜙4 kink’s shape mode

“…(A.3) is normal mode normal ordered using a Wick's theorem from ref. [28] : φ j (x) : a = The vertices in our diagrammatic approach correspond to the normal mode normal ordered interaction terms. Therefore the : φ 3 : a term in H I leads to two vertices.…”