Quantum mechanics on Riemannian manifold in Schwinger's quantization approach I

Abstract

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.

Hamiltonian and Lagrangian BRST Quantization in Riemann Manifold

“…In case of the system described in section IV, the convenient choice is Φk calculated in eqn [45][46][47][48].…”

“…Here Φ are the modified constraints in eqn [45][46][47][48] and H is the modified Hamiltonian in eqn (53). Now, this action satisfies the classical master equation…”

Hamiltonian and Lagrangian BRST Quantization in Riemann Manifold II

“…Our choice of Poisson bracket between the fields will be same as one taken for the general case. Hence, the matrix ω abij will have the form of Equation (48).…”

Topological Decompositions of the Pauli Group and their Influence on Dynamical Systems

“…Finding a Riemannian structure, in connection with a group, has a relevant meaning in several models of quantum mechanics. For instance, Chepilko and Romanenko [12] produced a series of contributions, illustrating how certain processes of quantization and some sophisticated variational principles may be easily understood in presence of Riemannian manifolds and groups. In this perspective one can also look at [30], which shows again the strong simplification of the structure of the hamiltonian in presence of models where we have both a Riemannian manifold and a group of symmetries.…”