Geometry, Topology and Quantization, 1996 Mathematics and Its Applications Series, Vol. 386

This is a monograph on geometrical and topological features which arise in various quantization procedures. Quantization schemes consider the feasibility of arriving at a quantum system from a classical one and these involve three major procedures viz. i) geometric quantization, ii) Klauder quantization, and iii) stochastic quanti­ zation. In geometric quantization we have to incorporate a hermitian line bundle to effectively generate the quantum Hamiltonian operator from a classical Hamil­ tonian. Klauder quantization also takes into account the role of the connection one-form along with coordinate independence. In stochastic quantization as pro­ posed by Nelson, Schrodinger equation is derived from Brownian motion processes; however, we have difficulty in its relativistic generalization. It has been pointed out by several authors that this may be circumvented by formulating a new geometry where Brownian motion proceses are considered in external as well as in internal space and, when the complexified space-time is considered, the usual path integral formulation is achieved. When this internal space variable is considered as a direc­ tion vector introducing an anisotropy in the internal space, we have the quantization of a Fermi field. This helps us to formulate a stochastic phase space formalism when the internal extension can be treated as a gauge theoretic extension. This suggests that massive fermions may be considered as Skyrme solitons. The nonrelativistic quantum mechanics is achieved in the sharp point limit.

1 Manifold and Differential Forms.- 2 Spinor Structure and Twistor Geometry.- 3 Quantization.- 4 Quantization And Gauge Field.- 5 Fermions and Topology.- 6 Topological Field Theory.- References.

This monograph deals with the geometrical and topological aspects associated with the quantization procedure, and it is shown how these features are manifested in anomaly and Berry phase. This book is unique in its emphasis on the topological aspects of a fermion which arise as a consequence of the quantization procedure. Also, an overview of quantization procedures is presented, tracing the equivalence of these methods by noting that the gauge field plays a significant role in all these procedures, as it contains the ingredients of topological features.