Renormalized Poincar\'e algebra for effective particles in quantum field theory

TL;DR

This paper develops a renormalized Poincaré algebra framework for effective particles in scalar quantum field theory, ensuring proper transformation properties and paving the way for observable calculations in QCD.

Contribution

It introduces a boost-invariant renormalization group approach to express Poincaré generators in terms of effective particle operators, verifying their algebra at various orders in coupling.

Findings

Generators satisfy Poincaré algebra at orders 1, g, g^2

One-particle states transform correctly under Poincaré transformations

Framework facilitates calculation of hadron observables in QCD

Abstract

Using an expansion in powers of an infinitesimally small coupling constant $g$, all generators of the Poincar\'e group in local scalar quantum field theory with interaction term $g \phi^3$ are expressed in terms of annihilation and creation operators $a_\lambda$ and $a^\dagger_\lambda$ that result from a boost-invariant renormalization group procedure for effective particles. The group parameter $\lambda$ is equal to the momentum-space width of form factors that appear in vertices of the effective-particle Hamiltonians, $H_\lambda$. It is verified for terms order 1, $g$, and $g^2$, that the calculated generators satisfy required commutation relations for arbitrary values of $\lambda$. One-particle eigenstates of $H_\lambda$ are shown to properly transform under all Poincar\'e 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.