Renormalized Poincaré algebra for effective particles in quantum field theory

Abstract

Using an expansion in powers of an infinitesimally small coupling constant $g$, all generators of the Poincaré group in local scalar quantum field theory with interaction term $g ϕ^3$ are expressed in terms of annihilation and creation operators $a_λ$ and $a^\dagger_λ$ that result from a boost-invariant renormalization group procedure for effective particles. The group parameter $λ$ is equal to the momentum-space width of form factors that appear in vertices of the effective-particle Hamiltonians, $H_λ$. It is verified for terms order 1, $g$, and $g^2$, that the calculated generators satisfy required commutation relations for arbitrary values of $λ$. One-particle eigenstates of $H_λ$ are shown to properly transform under all Poincaré transformations. The transformations are obtained by exponentiating the calculated algebra. From a phenomenological point of view, this study is a prerequisite to construction of observables such as spin and angular momentum of hadrons in quantum chromodynamics.