Some non-perturbative semi-classical methods in quantum field theory (a pedagogical review)

Abstract

A pedagogical introduction is given to non-perturbative semi-classical methods for finding solutions to quantum field theories. Both the weak coupling method based on a time-independent classical solution, and the WKB method based on all periodic orbits are developed in detail, proceeding ffrom elementary quantum mechanics to field theory in stages. Both methods are then illustrated in model field theories. The [λø⁴]₂ theory to which the weak coupling method is applied yields a new family of "kink" states whose properties are discussed. The WKB method is illustrated by quantizing "soliton" and "doublet" solutions of the two-dimensional sine-Gordon theory. The results are compared to consequences of Coleman's equivalence proof relating this system to the massive Thirring model. The speculation that solitons may be fermions is discussed, along with indications that the WKB method may ne yielding exact mass ratios for this theory. A final section is devoted to solutions of more realistic four-dimensional models containing fermions, internal symmetry etc. Some quark-confinement models and vortex type solutions come under this category. General remarks are made on this family of solutions, and illustrated using 't Hooft's monopole solution.