States on timelike hypersurfaces in quantum field theory

TL;DR

This paper explores defining quantum states on timelike hypersurfaces in the Klein-Gordon theory, revealing a well-defined vacuum and particle states that behave smoothly under Euclidean rotations, expanding the understanding of quantum field states beyond spacelike surfaces.

Contribution

It demonstrates the existence of well-defined vacuum and particle states on timelike hypersurfaces, including their properties and the behavior of multi-particle wave functionals, which was previously unclear.

Findings

Vacuum wave functional exists on any hyperplane.

Particles on timelike hyperplanes have incoming and outgoing variants.

Multi-particle wave functionals resemble those on spacelike hypersurfaces.

Abstract

We investigate the possibility of defining states on timelike hypersurfaces in quantum field theory. To this end we consider hyperplanes in the real massive Klein-Gordon theory using the Schroedinger representation. We find a well defined vacuum wave functional, existing on any hyperplane, with the remarkable property that it changes smoothly even under Euclidean rotation through the light-cone. Moreover, particles on timelike hyperplanes exist and occur in two variants, incoming and outgoing, distinguished by the sign of the energy. Multi-particle wave functionals take a form similar to those on spacelike hypersurfaces. The role of unitarity and the inner product is discussed.