Quantization on Space-Time Hyperboloids

TL;DR

This paper develops a point-form canonical quantization method for relativistic fields on a hyperboloid, demonstrating Lorentz invariance, equivalence to equal-time quantization, and formulating a generalized scattering theory.

Contribution

It introduces a novel point-form quantization approach on hyperboloids, establishing its equivalence to traditional methods and extending scattering theory for interacting fields.

Findings

Lorentz invariance is preserved in hyperboloid quantization.

Equivalence between hyperboloid and equal-time quantization is proven.

A generalized scattering operator consistent with Dyson expansion is formulated.

Abstract

We quantize a relativistic massive complex spin-0 field and a relativistic massive spin-1/2 field on a space-time hyperboloid. We call this procedure point-form canonical quantization. Lorentz invariance of the hyperboloid implies that the 4 generators for translations become dynamic and interaction dependent, whereas the 6 generators for Lorentz transformations remain kinematic and interaction free. We expand the fields in terms of usual plane waves and prove the equivalence to equal-time quantization by representing the Poincare generators in a momentum basis. We formulate a generalized scattering theory for interacting fields by considering evolution of the system generated by the interaction dependent four-momentum operator. Finally we expand our generalized scattering operator in powers of the interaction and show its equivalence to the Dyson expansion of usual time-ordered perturbation theory.