A time operator and the time-of-arrival problem in quantum field theory

TL;DR

This paper constructs a time operator in quantum field theory using states localized on timelike hypersurfaces, addressing the long-standing time-of-arrival problem and demonstrating results consistent with causal particle propagation.

Contribution

It introduces a novel approach to defining localized states on timelike hypersurfaces, enabling a quantum time operator and advancing the understanding of the time-of-arrival problem.

Findings

Time-localized states are constructed on timelike hypersurfaces.

Results align with semiclassical causal propagation expectations.

Superluminal localization effects are exponentially suppressed.

Abstract

The Newton-Wigner states and operator are widely accepted to provide an adequate notion of spatial localization of a particle in quantum field theory on a spacelike hypersurface. Replacing the spacelike with a timelike hypersurface, we construct one-particle states of massive Klein-Gordon theory that are localized on the hypersurface in the temporal as well as two spatial directions. This addresses the longstanding problem of a "time operator" in quantum theory. It is made possible by recent advances in quantization on timelike hypersurfaces and the introduction of evanescent particles. As a first application of time-localized states, we consider the time-of-arrival problem. Our results are in accordance with semiclassical expectations of causal propagation of massless and massive particles. As in the Newton-Wigner case, localization is not perfect, but apparent superluminal propagation is exponentially suppressed.