Point-form quantum field theory

We examine canonical quantization of relativistic field theories on the forward hyperboloid, a Lorentz-invariant surface of the form x_μx^μ = τ². 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” 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 Poincaré 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.