Variational Wave Functionals in Quantum Field Theory

TL;DR

This paper develops variational methods for quantum field theory, using superpositions of Gaussians to approximate wave functionals and applying them to scalar and fermion fields in 1+1 dimensions, with potential for extension to higher dimensions.

Contribution

It introduces a variational approach with superpositions of Gaussians for quantum field wave functionals, including a scheme for refinement and application to scalar and fermion theories.

Findings

Numerical approximation for the effective potential using five variational parameters.

Formulation of a variational principle for fermion vacuum energy.

Discussion of extension challenges to higher dimensions.

Abstract

Variational (Rayleigh-Ritz) methods are applied to local quantum field theory. For scalar theories the wave functional is parametrized in the form of a superposition of Gaussians and the expectation value of the Hamiltonian is expressed in a form that can be minimized numerically. A scheme of successive refinements of the superposition is proposed that may converge to the exact functional. As an illustration, a simple numerical approximation for the effective potential is worked out based on minimization with respect to five variational parameters. A variational principle is formulated for the fermion vacuum energy as a functional of the scalar fields to which the fermions are coupled. The discussion in this paper is given for scalar and fermion interactions in 1+1 dimensions. The extension to higher dimensions encounters a more involved structure of ultraviolet divergences and is deferred to future work.