Light-Front Quantisation as an Initial-Boundary Value Problem

TL;DR

This paper explores how additional boundary conditions in light-front quantisation ensure well-posedness and consistent quantization, particularly for massive scalar fields in 1+1 dimensions, supporting the validity of discretised light cone quantisation.

Contribution

It demonstrates that boundary conditions in the spatial variable, alongside fixed light cone time, make the initial-boundary value problem well posed in light-front quantisation.

Findings

Initial conditions plus boundary conditions yield a consistent commutator algebra.

Hamiltonian and Euler-Lagrange equations become equivalent under these conditions.

Supports the validity of discretised light cone quantisation.

Abstract

In the light front quantisation scheme initial conditions are usually provided on a single lightlike hyperplane. This, however, is insufficient to yield a unique solution of the field equations. We investigate under which additional conditions the problem of solving the field equations becomes well posed. The consequences for quantisation are studied within a Hamiltonian formulation by using the method of Faddeev and Jackiw for dealing with first-order Lagrangians. For the prototype field theory of massive scalar fields in 1+1 dimensions, we find that initial conditions for fixed light cone time {\sl and} boundary conditions in the spatial variable are sufficient to yield a consistent commutator algebra. Data on a second lightlike hyperplane are not necessary. Hamiltonian and Euler-Lagrange equations of motion become equivalent; the description of the dynamics remains canonical and simple. In this way we justify the approach of discretised light cone quantisation.