Initial Time Singularities in Non-Equilibrium Evolution of Condensates and Their Resolution in the Linearized Approximation

TL;DR

This paper addresses initial time singularities in the non-equilibrium evolution of scalar condensates in a Yukawa theory, proposing a method using Bogoliubov transformations to achieve singularity-free evolution in linearized approximations.

Contribution

It introduces a consistent approach to remove initial time singularities through initial state specification and Bogoliubov transformations in a renormalizable field theory.

Findings

Initial time singularities are enhanced in the Yukawa theory.

A Bogoliubov transformation can remove initial time singularities.

Analytical solution for inhomogeneous condensate evolution is provided.

Abstract

The real time non-equilibrium evolution of condensates in field theory requires an initial value problem specifying an initial quantum state or density matrix. Arbitrary specifications of the initial quantum state (pure or mixed) results in initial time singularities which are not removed by the usual renormalization counterterms. We study the initial time singularities in the linearized equation of motion for the scalar condensate in a renormalizable Yukawa theory in 3+1 dimensions. In this renormalizable theory the initial time singularities are enhanced. We present a consistent method for removing these initial time singularities by specifying initial states where the distribution of high energy quanta is determined by the initial conditions and the interaction effects. This is done through a Bogoliubov transformation which is consistently obtained in a perturbative expansion.The usual renormalization counterterms and the proper choice of the Bogoliubov coefficients lead to a singularity free evolution equation. We establish the relationship between the evolution equations in the linearized approximation and linear response theory. It is found that only a very specific form of the external source for linear response leads to a real time evolution equation which is singularity free. We focus on the evolution of spatially inhomogeneous scalar condensates by implementing the initial state preparation via a Bogoliubov transformation up to one-loop. As a concrete application, the evolution equation for an inhomogenous condensate is solved analytically and the results are carefully analyzed. Symmetry breaking by initial quantum states is discussed.