A Possible Way for Constructing Generators of the Poincaré Group in Quantum Field Theory

Abstract

Starting from the instant form of relativistic quantum dynamics for a system of interacting fields, where amongst the ten generators of the Poincaré group only the Hamiltonian and the boost operators carry interactions, we offer an algebraic method to satisfy the Poincaré 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 ≥ 1) 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.

Clothed particles in mesodynamics, quantum electrodynamics and other field models

“…By doing so, we find links 8) and so on. Moreover, since the operators α are assumed to meet the same commutation rules as the operators…”

“…Therefore one has to seek other ways to provide the relativistic invariance (RI) in Dirac sense (see, e.g., [8]). Further, by using the Fourier expansions…”

The Method of Unitary Clothing Transformations in Quantum Field Theory: Applications in the Theory of Nuclear Forces and Reactions

“…But we recall it as a special case of the microcausality requirement that is realized in local field models. Beyond such models, as shown in Appendix B of (Shebeko & Frolov , 2011), Eqs. (56) and (49) may be incompatible.…”

“…First, we consider an algebraic method (Shebeko & Frolov , 2011) to meet the Poincaré commutators for a wide class of field theoretic models (local and nonlocal ones taking into account their invariance with respect to space translations). In particular, this recursive method is appropriate for models with derivative couplings and spins ≥1 , typical of the meson theory of nuclear forces, where only some part of the interaction density in the Dirac picture has the property to be a Lorentz scalar.…”

“…2). One of them, proposed in (Shebeko & Frolov , 2011), has some touching points with the other developed in (Kita, 1966;1968) and essentially repeated many years after by Chandler (Chandler, 2003). In paper (Kita, 1968) the author considers three kinds of neutral spinless bosons and nonlocal interaction between them in a relativistic version of the Lee model with a cutoff in momentum space.…”