On two approaches to quantization in strong background fields

TL;DR

This paper compares two quantization methods in strong background fields, analyzing their differences, similarities, and conditions for equivalence, with implications for symmetry preservation in quantum field theory.

Contribution

It provides a detailed comparison of the Heisenberg and Schrödinger quantization approaches in strong background fields, identifying when they produce equivalent results.

Findings

Both methods can yield the same results under specific conditions.

The Heisenberg approach preserves symmetries better in certain cases.

The Schrödinger approach introduces an initial Cauchy surface, breaking some symmetries.

Abstract

There are two ways to quantize free (gaussian) theory in strong background fields. In one of them, which we refer to as the Heisenberg approach, the mode functions are defined once and for entire space-time. In this approach there is no any apparent presence of an initial Cauchy surface. Another method, which we refer to as the Schrodinger approach, assumes the presence of the initial Cauchy surface on which one defines a spatial basis of modes and only then considers the evolution of the creation and annihilation operators in time with the use of the (free) Hamiltonian. The first method usually used to respect the symmetries of the problem (e.g. isometry of the de Sitter space-time), while in the second method the isometry is apparently broken by the presence of the initial Cauchy surface. In this paper we compare the two methods of quantization and find conditions under which they give the same result.