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Geometry, Topology and Quantization

  • Pratul Bandyopadhyay

Part of the Mathematics and Its Applications book series (MAIA, volume 386)

Table of contents

  1. Front Matter
    Pages i-x
  2. Pratul Bandyopadhyay
    Pages 1-33
  3. Pratul Bandyopadhyay
    Pages 35-66
  4. Pratul Bandyopadhyay
    Pages 67-97
  5. Pratul Bandyopadhyay
    Pages 99-125
  6. Pratul Bandyopadhyay
    Pages 127-181
  7. Pratul Bandyopadhyay
    Pages 183-216
  8. Back Matter
    Pages 217-230

About this book

Introduction

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.

Keywords

Particle Physics differential geometry manifold mathematical physics quantum mechanics

Authors and affiliations

  • Pratul Bandyopadhyay
    • 1
  1. 1.Indian Statistical InstituteCalcuttaIndia

Bibliographic information

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