The semiclassical propagator in field theory

We consider scalar field theory in a changing background field. As an example we study the simple case of a spatially varying mass for which we construct the semiclassical approximation to the propagator. The semiclassical dispersion relation is obtained by consideration of spectral integrals and agrees with the WKB result. Further we find that, as a consequence of localization, the semiclassical approximation necessarily contains quantum correlations in momentum space. 1. The semiclassical (WKB) method has been developed to describe motion of a quantum particle in a slowly varying background. The method is a systematic expansion of the wave function in powers of the Planck constant, or equivalently, in gradients of the background [1]. In field theory, on the other hand, one is usually concerned with coupling constant expansions, and applicability of the semiclassical method has appeared more limited. Nevertheless, there are many cases where the semiclassical method can be useful also in field theory. For example, it may be adequate to describe a part of the system by a classical field F to which some other quantum field φ is weakly coupled. Even though variation of F can ultimately be traced to some coupling constant which characterizes its dynamics, the full quantum description of F might not be required. In such cases the coupling constant expansion for φ, and the expansion in gradients of F can be regarded as independent. To be more precise, the expansion in gradients of F is adequate when the persistence scale of space-time correlations of F are large in comparison to the scale (the Compton wavelength) that characterizes the propagating species φ. Here we consider the propagator of a scalar field interacting with an external background field. Specifically we consider a field φ with the lagrangian L = (∂μφ)†(∂φ)−mφφ+ Lint , (1) where m=m(t, ~x) is a space-time varying mass and Lint contains interactions. Physically such a mass term can be induced by a spatially varying scalar field condensate during a first order phase transition. More concretely, our goal is to describe plasma dynamics at the electroweak phase transition, which at the moment offers the most attractive explanation of the matter-antimatter asymmetry of the Universe [2], soon to be tested by accelerator experiments. The scale at which the Higgs condensate varies is determined by the coupling of the Higgs field to other species in the plasma and by the dynamics of the phase boundary during the phase transition, while a typical Compton wave length of a particle in the plasma is given by the temperature, and hence these two scales in principle can be, and in fact are, different. Other applications of the semiclassical method abound, e.g. motion of electrons in the background of spatially varying potentials, etc. In this letter we develop an approximation for the propagator in powers of gradients,