A possible way for constructing generators of the Poincare group in quantum field theory

TL;DR

This paper presents an algebraic method to construct Poincare group generators in relativistic quantum field theory without relying on Lagrangian formalism, applicable to local and nonlocal models including meson exchange theories.

Contribution

It introduces a recursive algebraic approach to build Poincare generators directly from the interaction density, avoiding Lagrangian formalism and applicable to complex field models.

Findings

Derived explicit expressions for Poincare generators in clothed-particle representation.

Established relations between mass renormalization and covariant integrals with cutoffs.

Demonstrated applicability to local and nonlocal meson-nucleon interaction models.

Abstract

Starting from the instant form of relativistic quantum dynamics for a system of interacting fields, where amongst the ten generators of the Poincare group only the Hamiltonian and the boost operators carry interactions, we offer an algebraic method to satisfy the Poincare commutators. We do not need to employ the Lagrangian formalism for local fields with the Noether representation of the generators. Our approach is based on an opportunity to separate in the primary interaction density a part which is the Lorentz scalar. It makes possible apply the recursive relations obtained in this work to construct the boosts in case of both local field models (for instance with derivative couplings and spins $\geq1$) and their nonlocal extensions. Such models are typical of the meson theory of nuclear forces, where one has to take into account vector meson exchanges and introduce meson-nucleon vertices with cutoffs in momentum space. Considerable attention is paid to finding analytic expressions for the generators in the clothed-particle representation, in which the so-called bad terms are simultaneously removed from the Hamiltonian and the boosts. Moreover, the mass renormalization terms introduced in the Hamiltonian at the very beginning turn out to be related to certain covariant integrals that are convergent in the field models with appropriate cutoff factors.