Point-form quantum field theory and meson form factors

Abstract

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. 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 x µ x µ = τ 2 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 x µ x µ = τ 2 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. 1 A procedure that resolves this problem has been proposed by Bakamjian and * E-mail address: wolfgang.schweiger@uni-graz.at 1 These constraints are automatically satisfied if a local interacting field theory is quantized.

Electromagneticρ-meson form factors in point-form relativistic quantum mechanics

“…For pseudoscalar mesons such as the pion, the current derived along the same lines is conserved and can be parametrized by 1 physical and 1 spurious form fac-tor [50]. The structure of this current reveals an interesting correspondence to the covariant light-front approach [19,20].…”

“…These are finite expressions and independent of k. It has been shown that they are identical to the corresponding ones obtained from the current matrix elements in Eqs. (50), (51) and (52). The physical current that has all required properties is then…”

Survey ofJ=0, 1 mesons in a Bethe-Salpeter approach

“…On the theoretical side the key challenge is to understand mesons (and hadrons in general) as bound states of QCD's elementary degrees of freedom, quarks and gluons. The various approaches used to provide direct or indirect insight regarding this problem are (relativistic) quark models (e. g. [1,2,3,4,5,6,7,8,9] and references therein), reductions of the Bethe-Salpeter equation (e. g. [10,11,12,13] and references therein), lattice QCD (e. g. [14,15,16,17,18,19] and references therein), effective field theories (e. g. [20] and references therein), and the Dyson-Schwinger approach used herein. On the experimental side, present-day challenges can be exemplified by the recent measurement of the pseudoscalar ground-state mass in the bottomonium system [21].…”

An estimation of non-valence contributions to form factors of heavy-light mesons

“…The point-form [1] of relativistic quantum mechanics has been successfully employed to calculate hadron form factors within the framework of constituent quark models [2][3][4][5][6][7][8][9][10][11][12]. In this formalism the physical process in which a particular form factor is measuered is described in a Poincaré invariant way by means of the Bakamjian-Thomas construction [13].…”