Point Form Quantum Field Theory on Velocity Grids I: Bosonic Contractions

TL;DR

This paper explores a novel discretized velocity space approach to continuum quantum field theories, focusing on bosonic contractions and algebraic structures, to address issues with infinite mode creation and convergence.

Contribution

It introduces a method to derive boson algebra as a contraction limit of a finite-mode unitary algebra, avoiding infinite mode problems in quantum field theories.

Findings

Boson algebra arises as a contraction limit of a finite-mode algebra.

Finite mode Hamiltonians exhibit generic properties analyzed.

Convergence rate of boson contraction studied with simple models.

Abstract

In constrast to discretized space-time approximations to continuum quantum field theories, discretized velocity space approximations to continuum quantum field theories are investigated. A four-momentum operator is given in terms of bare fermion-antifermion-boson creation and annihilation operators with discrete indices. In continuum quantum field theories the fermion-antifermion creation and annihilation operators appear as bilinears in the four-momentum operator and generate a unitary algebra. When the number of modes range over only a finite number of values, the algebra is that associated with the Lie algebra of U(2N). By keeping N finite (but arbitrary) problems due to an infinite Lorentz volume and to the creation of infinite numbers of bare fermion-antifermion pairs are avoided. But even with a finite number of modes, it is still possible to create an infinite number of bare bosons. We show how the full boson algebra arises as the contraction limit of another unitary algebra that restricts the number of bare bosons in any mode to be finite. Generic properties of finite mode Hamiltonians are investigated, as are several simple models to see the rate of convergence of the boson contraction; the possibility of fine tuning the bare strong coupling constant is also briefly discussed.