Similarity Renormalization Group Approach to Boost Invariant Hamiltonian Dynamics

TL;DR

This paper develops a method to derive boost invariant Hamiltonians in quantum field theory using similarity renormalization group techniques, enabling consistent descriptions of relativistic systems across different frames.

Contribution

It introduces a novel approach to construct boost invariant effective Hamiltonians with renormalization group equations, applicable to QED, Yukawa theory, and QCD, including confinement effects.

Findings

Derived boost invariant Hamiltonians for Yukawa, QED, and QCD.

Regularized relativistic ultraviolet singularities with form factors.

Found a confining quark-antiquark interaction in QCD.

Abstract

We outline a method of deriving boost invariant hamiltonians for effective particles in quantum field theory. The hamiltonians are defined and calculated using creation and annihilation operators in light-front dynamics. The renormalization group equations are written for a sequence of unitary transformations which gradually transform the bare canonical creation and annihilation operators of a local theory to the creation and annihilation operators of effective particles in an effective theory with the same dynamical content but a finite range of energy transfers due to form factors in the interaction vertices. The boost invariant effective hamiltonians can be used to describe the constituent dynamics in relativistically moving systems including the rest and the infinite momentum frame. The general equations are illustrated in perturbation theory by second-order calculations of self-energy and two-particle interaction terms in Yukawa theory, QED and QCD. In Yukawa theory, one obtains the generalized Yukawa potential including its full off-energy-shell extension and form factors in the vertices. In QED, the effective hamiltonian eigenvalue problem converges for small coupling constants to the Schr\"odinger equation but the typical relativistic ultraviolet singularities at short distances between constituents are regularized by the similarity form factors. In the second-order QCD effective hamiltonian one obtains a boost invariant logarithmically confining quark-anti-quark interaction term which may remain uncanceled in the non-abelian dynamics of effective quarks and gluons.