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Abstract. In this paper we extend Schwinger's quantization approach to the case of a supermanifold considered as a coset space of the Poincare group by the Lorentz group. In terms of coordinates parametrizing a supermanifold, quantum mechanics for a superparticle is constructed. As in models related to the usual Riemannian manifold, the key role in analyzes is played by Killing vectors. The main feature of quantum theory on the supermanifold consists of the fact that the spatial coordinates are not commute with each other and therefore are represented on wave functions by integral operators.

Abstract. In this paper we extend Schwinger's quantization approach to the case of a supermanifold considered as a coset space of the Poincare group by the Lorentz group. In terms of coordinates parametrizing a supermanifold, quantum mechanics for a superparticle is constructed. As in models related to the usual Riemannian manifold, the key role in analyzes is played by Killing vectors. The main feature of quantum theory on the supermanifold consists of the fact that the spatial coordinates are not commute with each other and therefore are represented on wave functions by integral operators.

The nonlinear o model Lagrangian for a rotating and vibrating quantum soliton expressed in terms of collective coordinates is shown to possess a symmetry under scale transformation of the chiral field. By utilizing this symmetry, an integro-differential equation determining the chiral angle is obtained. A consistency condition between this equation and the Schrodinger equation for the quantum soliton is discussed, and a relation between the total soliton energy and that related to rotation and vibration is obtained without solving the Schrodinger equation. In the limiting case of only a vibrating or a rotating soliton, the integro-differential equation is reduced to a differential one, and the chiral angle becomes independent of eigensolutions of the relevant Schrodinger equation. The effects of chiral-symmetry breaking due to the pion mass are also examined.