Point-form quantum field theory

TL;DR

This paper explores a covariant point-form quantization of relativistic quantum fields on a hyperboloid surface, maintaining Lorentz invariance and reproducing standard scattering results while offering a different perspective on the dynamics.

Contribution

It introduces a novel covariant point-form quantization method on hyperboloids, preserving Poincaré symmetry and reproducing standard perturbative scattering theory.

Findings

Fock-space representations match equal-time quantization for free fields

Perturbative S-matrix expansion is recovered in this framework

Interaction effects are incorporated through the 4-momentum operator

Abstract

We examine canonical quantization of relativistic field theories on the forward hyperboloid, a Lorentz-invariant surface of the form $x_\mu x^\mu = \tau^2$. This choice of quantization surface implies that all components of the 4-momentum operator are affected by interactions (if present), whereas rotation and boost generators remain interaction free -- a feature characteristic of Dirac's `` point-form\rq\rq of relativistic dynamics. Unlike previous attempts to quantize fields on space-time hyperboloids, we keep the usual plane-wave expansion of the field operators and consider evolution of the system generated by the 4-momentum operator. We verify that the Fock-space representations of the Poincar\'e generators for free scalar and spin-1/2 fields look the same as for equal-time quantization. Scattering is formulated for interacting fields in a covariant interaction picture and it is shown that the familiar perturbative expansion of the S-operator is recovered by our approach. An appendix analyzes special distributions, integrals over the forward hyperboloid, that are used repeatedly in the paper.