The relation between instant and light-front formulations of quantum field theory

TL;DR

This paper establishes the non-perturbative equivalence between light-front and instant-form quantum field theories, showing they can share the same vacuum and one-particle states without relying on quantization or perturbation.

Contribution

It constructs equivalent representations of quantum field theories with both kinematic subgroups, addressing the inequivalence of free and interacting representations non-perturbatively.

Findings

Light-front and instant-form formulations can be made equivalent non-perturbatively.

Both formulations can share the same vacuum and one-particle states.

The approach avoids issues related to renormalization and vacuum structure in perturbative treatments.

Abstract

The scattering equivalence of quantum field theories formulated with light-front and instant-form kinematic subgroups is established using non-perturbative methods. The difficulty with field theoretic formulations of Dirac's forms of dynamics is that the free and interacting unitary representations of the Poincar\'e group are defined on inequivalent representations of the Hilbert space, which means that the concept of kinematic transformations must be modified on the Hilbert space of the field theory. This work addresses this problem by assuming the existence of a field theory with the expected properties and constructs equivalent representations with instant and front form kinematic subgroups. In this construction both the light-front and instant-form formulations share the same vacuum and one-particle states. The free field Fock space plays no role. There is no "quantization" of a classical theory. The property that survives from the perturbative approach is the notion of a kinematic subgroup, which means kinematic Poincar\'e transformations can be trivially implemented by acting on suitable basis vectors. This non-perturbative approach avoids dealing with issues that arise in perturbative treatments where is it necessary to have a consistent treatment of renormalization, rotational covariance, and the structure of the light-front vacuum. While addressing these issues in a computational framework is important for applications, this work may provide some insight into the nature of the expected resolution and identifies the origin of some of differences between the perturbative and non-perturbative approaches.