Decoherent histories analysis of the relativistic particle

TL;DR

This paper applies the decoherent histories approach to analyze the probability of a relativistic particle crossing a spacelike surface, comparing different quantization methods and their implications for decoherence and measurement.

Contribution

It introduces a decoherent histories framework for relativistic particles, analyzing crossing probabilities with path integrals and comparing Klein-Gordon and Newton-Wigner quantizations.

Findings

Decoherence is approximate in Klein-Gordon quantization due to path integral paths going backwards in time.

Different quantization methods yield different crossing probabilities.

The approach clarifies the role of multiple crossings and the notion of measurement in relativistic quantum theory.

Abstract

The Klein-Gordon equation is a useful test arena for quantum cosmological models described by the Wheeler-DeWitt equation. We use the decoherent histories approach to quantum theory to obtain the probability that a free relativistic particle crosses a section of spacelike surface. The decoherence functional is constructed using path integral methods with initial states attached using the (positive definite) ``induced'' inner product between solutions to the constraint equation. The notion of crossing a spacelike surface requires some attention, given that the paths in the path integral may cross such a surface many times, but we show that first and last crossings are in essence the only useful possibilities. Different possible results for the probabilities are obtained, depending on how the relativistic particle is quantized (using the Klein-Gordon equation, or its square root, with the associated Newton-Wigner states). In the Klein-Gordon quantization, the decoherence is only approximate, due to the fact that the paths in the path integral may go backwards and forwards in time. We compare with the results obtained using operators which commute with the constraint (the ``evolving constants'' method).