Time Evolution in the External Field: the Unitarity Paradox

TL;DR

This paper investigates the apparent contradiction between unitarity in quantum field theory and non-unitary evolution in external fields, demonstrating that non-unitarity in approximations does not violate the unitarity of the exact theory.

Contribution

It provides a detailed analysis of the unitarity paradox using a simple quantum mechanical model analogous to large-N field theory, reconciling approximate non-unitary evolution with exact unitarity.

Findings

Exact quantum model maintains unitarity despite non-unitary approximate evolution.

Non-unitarity in leading-order approximations does not imply fundamental non-unitarity.

Constructive field theory methods successfully construct the Hilbert space and evolution operator.

Abstract

One of the axioms of quantum field theory is the property of unitarity of the evolution operator. However, if one considers the quantum electrodynamics in the external field in the leading order of perturbation theory, one will find that the evolution transformation is a non-unitary canonical transformation of creation and annihilation operators. This observation was one of the arguments for the hypothesis that one should choose different representations of the canonical commutation relations at different moments of time in the exact quantum field theory. In this paper the contradiction is analyzed for the case of a simple quantum mechanical model being an analog of the leading order of the large-N field theory. On the one hand, this model is renormalized with the help of the constructive field theory methods; the Hilbert space and unitary evolution operator are constructed. On the other hand, the leading order of the evolution transformation in the strong external field is shown to be non-unitary. Thus, unitarity of evolution in the exact theory is not in contradiction with non-unitarity of the approximate theory.