The semiclassical propagator in field theory

TL;DR

This work studies the semiclassical propagation of a scalar field in a slowly varying background with a spatially dependent mass. By formulating the Schwinger-Dyson equations on the Keldysh contour and performing a gradient (Wigner) expansion, the authors derive the zeroth-order propagator $G^{r,a}_0 = 1/(k^2 - m^2 \pm i \omega \Gamma)$ and the first nontrivial gradient corrections, yielding $G_2 = 1/z - \delta_2/z^3 - \delta_1/z^4$ with $\delta_2 = m^{2\prime\prime}/2$ and $\delta_1 = (m^{2\prime})^2/2 + k_z^2 m^{2\prime\prime}$. Spectral integrals and a boundary condition $G \to 1/z$ at large $|z|$ connect this to a semiclassical dispersion, whose on-shell momentum $k_{\rm sc}$ reproduces the WKB momentum $k_x$; a regulated integral representation also emerges for $G$. The analysis reveals that localization in position space induces quantum correlations in momentum space and that higher-order momentum derivatives are intrinsic to dynamical quantities, implying that plasma dynamics in slowly varying backgrounds cannot be captured by a single classical Boltzmann equation. The results have relevance for electroweak baryogenesis and related plasma systems, offering a controlled framework to include gradient corrections and derivative couplings in transport descriptions.

Abstract

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.

Figures (3)

• Figure 1: The integration contour for spectral integral in complex $\omega$-plane.
• Figure 2: The integration contours for spectral integral in the complex $z$-plane.
• Figure 3: Plotted is the quantity $\kappa \equiv k_{\rm sc}/k_0 - 1$ for three lowest order gradient approximations for $G$ along with the formal integral solution (solid line). Inset: deviation of $\kappa_n$ from the integral result for small $\delta/k_0^2$.