Quantum Measurement of Space-Time Events

TL;DR

This paper develops a relativistic quantum theory of phase-space events using complex Minkowski space, enabling covariant measurement and localization of space-time events with a new mathematical framework.

Contribution

It introduces a novel quantum measurement framework for space-time events based on the geometry of the future tube in complex Minkowski space.

Findings

Constructed a unitary representation of conformal transformations on the Hilbert space.

Formulated a covariant quantum measurement theory using positive operator valued measures.

Established a localization theorem for phase-space events related to the Compton wavelength.

Abstract

The phase space of a relativistic system can be identified with the future tube of complexified Minkowski space. As well as a complex structure and a symplectic structure, the future tube, seen as an eight-dimensional real manifold, is endowed with a natural positive-definite Riemannian metric that accommodates the underlying geometry of the indefinite Minkowski space metric, together with its symmetry group. A unitary representation of the 15-parameter group of conformal transformations can then be constructed that acts upon the Hilbert space of square-integrable holomorphic functions on the future tube. These structures are enough to allow one to put forward a quantum theory of phase-space events. In particular, a theory of quantum measurement can be formulated in a relativistic setting, based on the use of positive operator valued measures, for the detection of phase-space events, hence allowing one to assign probabilities to the outcomes of joint space-time and four-momentum measurements in a manifestly covariant framework. This leads to a localization theorem for phase-space events in relativistic quantum theory, determined by the associated Compton wavelength.