The interaction representation in the quantum theory of fields

Abstract

The interaction representation has recently been introduced into the quantum theory of fields by Tomonaga and Schwinger. Applications of the theory to interacting meson-photon fields have led to apparent difficulties in determining invariant interaction Hamiltonians. Another troublesome feature is the necessity of verifying the integrability conditions of the so-called generalized Schrödinger equation. In the present paper the theory of the interaction representation is presented from a different point of view. It is shown that if two field operators with the same transformation character satisfy two different field equations, there is a unique unitary transformation connecting the field variables on any space-like surface given such a correspondence on one given space-like surface. A differential equation for determining this unique unitary transformation is found which is the analogue of Tomonaga’s generalized Schrödinger equation. This gives directly and simply an invariant interaction Hamiltonian and renders unnecessary the explicit verification of the integrability of the Schrödinger equation, since this is known to have a unique solution. To illustrate the simplification introduced by the present theory, the interaction Hamiltonian for the interacting scalar meson-photon fields is calculated. The result is the same as that obtained by Kanesawa & Tomonaga, but it is obtained by a straightforward calculation without the need to add terms to make the Hamiltonian an invariant.