Geometry, Topology and QuantizationThis 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. |
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Common terms and phrases
Abelian adiabatic algebra anticommuting associated Bandyopadhyay Berry phase bosonic Brownian motion BRST Chern-Simons chiral anomaly classes classical cohomology commutator complex conformal spinors connection coordinates corresponding covariant curvature define denotes dimension dimensional Dirac direction vector equation Euclidean fermion number fibre formulation gauge field gauge fixing gauge invariant gauge potential gauge theory gauge transformation geometric quantization geometrical given gives rise Hamiltonian Hilbert space holonomy implies internal helicity internal space Lagrangian line bundle loop manifold mapping matrix metric Minkowski space momentum nonlinear nontrivial noted one-form operator orientation parameter particle path integral phase space Phys Pontryagin principal bundle quantization procedure quantum mechanics relation relativistic representation scalar sharp point limit spin spinor spinorial variable stochastic quantization superspace supersymmetry tangent tensor topological action topological field theory topological invariant twistor two-form vortex line Wess-Zumino term winding number Witten write zero
References to this book
5000 Jahre Geometrie: Geschichte, Kulturen, Menschen Christoph J. Scriba,Peter Schreiber No preview available - 2004 |