Strong-coupling field theory. I. Variational approach toφ4theory

Abstract

Theoretical attempts to understand hadrons in terms of confined quark constituents lead naturally to the study of quantum field theory with methods that can be applied when strong interactions are present. In this paper nonperturbative, variational techniques are developed and applied to calculating the ground state and low-lying collective excitations ("kinks") of theories rendered finite on a discrete lattice. Particular application is made to a scalar theory with a self-coupling of the form $\ensuremath{\lambda}{({\ensuremath{\varphi}}^{2}\ensuremath{-}{f}^{2})}^{2}$ in two dimensions. Working in configuration space we reduce the theory to coupled Schr\"odinger problems and establish the conditions for the variational solution to exhibit a phase transition between ground states with $〈\ensuremath{\varphi}〉=0$ and those exhibiting a spontaneously broken symmetry such that $〈\ensuremath{\varphi}〉\ensuremath{\ne}0$. The phase transition is a second-order one in a simple trial state constructed in a single-site product basis. Low-lying excitations are constructed that are analogs of the classical "kink" solutions. The single-site basis is also generalized to form "blocks" of coupled lattice sites, and general properties of a block formalism are explored. The usual renormalization $\mathrm{limit}\mathrm{of}\mathrm{cutoff}\ensuremath{\rightarrow}\ensuremath{\infty}$, or lattice spacing\ensuremath{\rightarrow}0, is also studied as well as the relation of our approach to the conventional renormalization program.