Commun. math. Phys. 60, 37--72 (1978) Communications in Mathematical Physics © by Springer-Verlag 1978 Soliton Mass and Surface Tension in the (ilckl4)2Quantum Field Model J. B611issard I*, J. Fr/Shlich 2.*, and B. Gidas 3.** 1 Universit6 de Provence, F-Marseille, France z Department of Mathematics, Princeton University, Princeton, New Jersey 08540, USA 3 Department of Mathematics, Rockefeller University, New York, New York 10021, USA Abstract. The spectrum of the mass operator on the soliton sectors of the anisotropic (2lq~14)2--and the (2q~4)2~quantum field models in the two pha region is analyzed. It is proven that, for small enough ,~. >0, the mass gap m~(2 on the soliton sector is positive, and m~()0=0(2-1). This involves estimating m,(2) from below by a quantity z(2) analogous to the surface tension in th statistical mechanics of two dimensional, classical spin systems and then estimating z(2) by methods of Euclidean field theory. In principle, our methods apply to any two dimensional quantum field modelwith a spontaneously broken, internal symmetry group. 1. Introduction: Main Subject, Models, Main Results 1.1 During the past few years the quantization of nonlinear waves (solitary solutions of nonlinear,classical field equations)has attracted a lot of interest and has been studied from various---more and less rigorous--points of view; see [1-6] and references given there, and [7-101 for a mathematically rigorous analysis. From these efforts emerged the (heuristic) picture that the homotopy classes finite energy solutions to some classical, nonlinear field equation are, for small enough h (oc Planck's constant), in a one-one correspondence with non-trivial, charged superselection (soliton) sectors of the relativistic quantum field theory formally determined by the same nonlinear field equation. It is felt that this picture might be a key to understanding some of the conservation laws and some of th (hadronic) extended particles observed in elementary particle physics. So far, however, many workers in the field have concentrated on the anMysi of quantum field models (or quantum spin systems [11]) in two space-time * Postal address: Centre de Physique Th6orique, Centre National de la Recherche S 31, chemin J. Aiguier, F-13274 Marseille, France ** A Sloan Foundation Fellow; Research supported in part by the U.S. National Scie Foundation under Grant No. MPS 75-11864. Present address: I.H.t~,S., F-91440 Bures-sur-Y France *** Supported in part by the National Science Foundation under Grant No. PHY 76-1 O010-3616/78/0060/0037/$07.20