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

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.

27 pages, 8 figures; v2: minor modifications

References in corpus (14)

• Time-of-arrival in quantum mechanics
• A "general boundary" formulation for quantum mechanics and quantum gravity
• General boundary quantum field theory: Timelike hypersurfaces in Klein-Gordon theory
• Spatially asymptotic S-matrix from general boundary formulation
• States on timelike hypersurfaces in quantum field theory
• S-matrix at spatial infinity
• Time of arrival and localization of relativistic particles
• A local and operational framework for the foundations of physics
• A positive formalism for quantum theory in the general boundary formulation
• Locality and General Vacua in Quantum Field Theory
• The vacuum as a Lagrangian subspace
• Evanescent Particles
• Scattering of Evanescent Particles
• Interaction of evanescent particles with an Unruh-DeWitt detector