Point-Form Quantum Field Theory and Meson Form Factors

Recently we have reconsidered the quantization of relativistic field theories on a Lorentz-invariant surface of the form xμx μ = τ [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. Thus we speak of “point-form quantum field theory” (PFQFT). Old papers on PFQFT (see, e.g., [2, 3]) dealt mainly with the evolution of quantum fields in the parameter τ and made use of a Fock-space basis which is related to the generators of the Lorentz group. 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 xμx μ = τ leads to the same Fock-space representation of the Poincaré generators as equaltime quantization. Moreover, we have suggested a generalized interaction picture which leads to a manifestly Lorentz covariant expression for the scattering operator as path-ordered exponential of the interaction part of the 4-momentum operator (along arbitrary timelike paths). We furthermore showed that the perturbative expansion of the scattering operator, defined in such a way, is (order by order) equivalent to usual time-ordered perturbation theory. The nice feature that the operator formalism becomes manifestly Lorentz covariant if fields are quantized on the space-time hyperboloid xμx μ = τ was not our only motivation to study PFQFT. PFQFT serves also as a natural starting point for the construction of effective interactions, currents, etc., which can be applied to point-form quantum mechanics. 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. A procedure that resolves this problem has been proposed by Bakamjian and