Unitary evolution of free massless fields in de Sitter space-time

TL;DR

This paper investigates the quantum evolution of a massless scalar field in de Sitter space-time, demonstrating that the classical evolution cannot be implemented as a unitary transformation in the standard quantum framework.

Contribution

It shows that the symplectic map for the field's evolution cannot be unitarily realized in any Fock space with SO(4) symmetry, and that conformal redefinitions do not yield unitary dynamics.

Findings

Classical evolution represented by a canonical transformation.

No unitary implementation of the symplectic map in Fock space.

Conformal redefinitions do not produce unitary quantum dynamics.

Abstract

We consider the quantum dynamics of a massless scalar field in de Sitter space-time. The classical evolution is represented by a canonical transformation on the phase space for the field theory. By studying the corresponding Bogoliubov transformations, we show that the symplectic map that encodes the evolution between two instants of time cannot be unitarily implemented on any Fock space built from a SO(4)-symmetric complex structure. We will show also that, in contrast with some effectively lower dimensional examples arising from Quantum General Relativity such as Gowdy models, it is impossible to find a time dependent conformal redefinition of the massless scalar field leading to a quantum unitary dynamics.