Validity of the Semiclassical Approximation in 1+1 Electrodynamics: Numerical Solutions to the Linear Response Equation

TL;DR

This paper investigates the validity of the semiclassical approximation in 1+1 electrodynamics by comparing approximate and numerical solutions to the linear response equation, focusing on a massive spin 1/2 field.

Contribution

It provides a numerical analysis of the linear response equation to test the robustness of the semiclassical approximation in a specific quantum field scenario.

Findings

Approximate solutions closely match numerical solutions at early times.

Significant growth in the two-point correlation function indicates potential breakdown of the approximation.

The method's robustness varies depending on the scale of particle production.

Abstract

From previous work arXiv:2010.09811, the semiclassical backreaction equation in 1+1 dimensions was solved and a criterion was implemented to assess the validity of the semiclassical approximation in this case. The criterion involves the behavior of solutions to the linear response equation which describes perturbations about solutions to the semiclassical backreaction equation. The linear response equation involves a time integral over a two-point correlation function for the current induced by the quantum field and it is expected that significant growth in this two-point function (and therefore in quantum fluctuations) will result in significant growth in solutions to the linear response equation. It was conjectured for early times that the difference of two nearby solutions to the semiclassical backreaction equation, with similar initial conditions, can act as an approximate solution to the linear response equation. A comparative analysis between the approximate and numerical solutions to the linear response equation, for the critical scale for particle production, will be presented for the case of a massive, quantized spin 1/2 field in order to determine how robust the approximation method is for representing its solutions.