Canonical Quantization of Constrained Systems and Coadjoint Orbits of Diff

It is shown that Dirac's treatment of constrained Hamiltonian systems and Schwinger's action principle quantization lead to identical commutation relations. An explicit relation between the Lagrange multipliers in the action principle approach and the additional terms in the Dirac bracket is derived. The equivalence of the two methods is demonstrated in the case of the non-linear sigma model. Dirac's method is extended to superspace and this extension is applied to the chiral superfield. The Dirac brackets of the massive interacting chiral superfield are derived and shown to give the correct commutation relations for the component fields. The Hamiltonian of the theory is given and the Hamiltonian equations of motion are computed. They agree with the component field results. An infinite sequence of differential operators which are covariant under the coadjoint action of Diff(S ^1) and analogues to Hill's operator is constructed. They map conformal fields of negative integer and half-integer weight to their dual space. Some properties of these operators are derived and possible applications are discussed. The Korteweg-de Vries equation is formulated as a coadjoint orbit of Diff(S^1).