Point-form quantum field theory

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.Comment: 30 page

Electromagnetic meson form factor from a relativistic coupled-channels approach

“…It is, in particular, not possible to factorize the four-momentum operator of an interacting point-form quantum field theory as in Eq. (3) [18]. The vertex operatorsK andK † in Eq.…”

Lorentz-boost eigenmodes

“…Similar, but not the same, set of Lorentz-boost eigenmodes have been considered in the context of "point-form" quantum field theory [25][26][27][28]. The difference is that the previous solutions [25][26][27][28] were obtained to satisfy 1D Klein-Gordon equation, and these are initially defined only inside the forward light cone; extending these solutions to the whole Minkowski spacetime faces some difficulties. In contrast, the modes considered in this work are solutions of the 1D massless wave equation (the mass term is taken into account in the equation for the transverse wavefunction envelope), and our modes are immediately well-defined in the whole Minkowski spacetime.…”

Point-form quantum field theory and meson form factors

“…Such a choice for the basis and the "time parameter", however, gave rise to conceptual difficulties. To avoid these problems we have kept the usual momentum basis and considered evolution of the system as generated by the 4-momentum operator [1]. In this way we were able to show for free fields that quantization on the space-time hyperboloid x µ x µ = τ 2 leads to the same Fock-space representation of the Poincaré generators as equaltime quantization.…”

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Recently we have reconsidered the quantization of relativistic field theories on a Lorentz-invariant surface of the form x µ x µ = τ 2 [1]. With this choice of the quantization surface all components of the 4-momentum operator become interaction dependent, whereas the generators of Lorentz transformations stay free of interactions -a feature characteristic for Dirac's "point form" of relativistic dynamics.

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“…The main difficulty of finding a quantum mechanical realization of the Poincaré algebra, which describes a finite number of interacting particles, is caused by the fact that interaction terms in the Poincaré generators have to satisfy non-linear constraints, in general. 1 A procedure that resolves this problem has been proposed by Bakamjian and Thomas [4]. Its point-form version amounts to the assumption that the free 4-velocity operatorV µ free can be factored out of the (interacting) 4-momentum…”