Geometry, Topology and Quantization

235 Pages·1996·18.31 MB·English

by  Pratul Bandyopadhyay (auth.)

Geometry, Topology and Quantization
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre/or Mathematics and Computer Science. Amsterdam. The Netherlands
Volume 386
Geometry, Topology
and Quantization
by
Pratul Bandyopadhyay
Indian Statisticallnstitute,
Calcutta, India
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress
ISBN 978-94-010-6282-4 ISBN 978-94-011-5426-0 (eBook)
DOI 10.1007/978-94-011-5426-0
Printed on acid-free paper
All Rights Reserved
© 1996 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1996
Softcover reprint ofthe hardcover 1st edition 1996
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Contents
Preface ix
1 Manifold and Differential Forms 1
1.1 Manifold............. 1
1.1.1 Topological Space. . . . 1
1.1.2 Differentiable Manifold . 2
1.1.3 Hausdorff and Metric Space 3
1.1.4 Tangent and Cotangent Space 4
1.1.5 Group Manifold . 6
1.2 Differential Forms . . . . 6
1.2.1 Definitions.... 6
1.2.2 Stokes' Theorem 9
1.3 Homology and Cohomology 10
1.3.1 Simplex, Simplicial Complex and Homology 10
1.3.2 de Rham Cohomology 13
1.4 Fibre Bundles . . . 16
1.4.1 Definitions... 16
1.4.2 G-Structure.. 19
1.4.3 Lie Derivative . 19
1.4.4 Connection, Curvature and Parallel Transport 21
1.4.5 Levi-Civita Connection . 24
1.4.6 Bianchi Identities 25
1.4.7 Holonomy Group 27
1.5 Characteristic Classes. . 28
1.5.1 Definitions.... 28
1.5.2 Pontryagin, Euler, Chern and Stiefel-Whitney Classes. 29
1.5.3 Global Invariants . . . . . . . . . . . . . . . . . . . . . 30
2 Spinor Structure and Twistor Geometry 35
2.1 Minkowski Space-Time. . . . . . . . . . . 35
2.1.1 Minkowski Vector Space . . . . . . 35
2.1.2 Lorentz and Poincare Transformation . 36
2.1.3 Poincare Transformation . . . . . 37
2.2 Spinors and Spin Structure. . . . . . . . 37
2.2.1 Spinor Space and Spinor Algebra 37
v
vi CONTENTS
2.2.2 Spinors and Tensors ... 41
2.2.3 Universal Covering Space. 42
2.2.4 Spinor Structure ..... 42
2.3 Conformal Spinors ........ 44
2.3.1 Conformal Transformations 44
2.3.2 Spinors in E( 4,2) Space. 44
2.4 Supersymmetryand Superspace . . 47
2.4.1 Supersymmetry Algebra . . 47
2.4.2 Conformal Spinors, Supersymmetry and Internal Symmetry. 49
2.4.3 Superspace ... 52
2.5 Twistor Geometry .......................... 53
2.5.1 Twistor Equation. . . . . . . . . . . . . . . . . . . . . . . 53
2.5.2 Twistor Geometry, Complexified Space-Time and Fermion
Number. . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5.3 Twistors and Cartan Semispinors . . . . . . . . . . . 61
2.5.4 Twistor Geometry, Spinor Structure and Super-space 62
3 Quantization 67
3.1 Geometric Quantization 67
3.1.1 The Quantum Condition 67
3.1.2 Prequantization..... 68
3.1.3 The Integrability Condition 69
3.1.4 Quantization ........ 70
3.2 Klauder Quantization. . . . . . . . 72
3.2.1 Quantization and Coordinate Independence 72
3.2.2 Symplectic Structure and Universal Magnetic Field 78
3.2.3 Landau Levels and Geometric Quantization 79
3.3 Stochastic Quantization ............... 81
3.3.1 Stochastic Quantization: Nelson's Approach 81
3.3.2 Stochastic Field Theory ........... 83
3.3.3 Stochastic Quantization: Parisi-Wu Approach 85
3.3.4 Stochastic Quantization and Supersymmetry . 86
3.3.5 Relativistic Generalization and Quantization of a Fermi Field 87
3.3.6 Stochastic Quantization in Minkowski Space-Time. . . . . .. 91
3.3.7 Stochastic Quantization in Minkowski Space-Time and Thermo
Field Dynamics . . . . . . . . . . . . . . . . . . . . . . . . .. 94
4 Quantization And Gauge Field 99
4.1 Equivalence of Stochastic, Klauder and Geometric Quantization 99
4.1.1 Stochastic Phase Space and Symplectic Structure 99
4.1.2 Role of Gauge Field. . . . . . . . . . . . . . . . . . 102
4.1.3 Equivalence of Different Quantization Procedures . 103
4.2 Gauge Theoretic Extension. . . . . . . . . . . . . . . . . . 105
4.2.1 Quantization of a Fermion and SL(2,C) Gauge Structure . 105
CONTENTS vii
4.2.2 Relativistic Quantum Particle as a Gauge Theoretic Extended
Body. . . . . . . . . . . . . . . . . . . . . . 108
4.2.3 SU(2) and U(l) Gauge Bundle .................. 109
4.3 Locality and Nonlocality in Quantum Mechanics . . . . . . . . . . . . 110
4.3.1 Nonrelativistic Quantum Mechanics and Sharp Point Limit .. 110
4.3.2 Localization of a Relativistic Quantum Particle . 112
4.3.3 Locality and Separability. . 115
4.4 Quantization and Berry Phase . . . . . . 116
4.4.1 The Geometric Phase. . . . . . . 116
4.4.2 Non-Abelian Geometric Phase . . 117
4.4.3 Non-adiabatic Generalization . . 119
4.4.4 Classical Limit of the Geometric Phase . 120
4.4.5 Topological Character of the Berry Phase. . 122
4.4.6 Quantization, Gauge Degrees of Freedom and Berry
Connection ....................... . 124
5 Fermions and Topology 127
5.1 Quantization of a Fermion, Nonlinear Sigma Model and Vortex Line. 127
5.1.1 Bosonization: Skyrme Model ................... 127
5.1.2 Gauge Theoretic Extension of a Fermion and Nonlinear Sigma
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.1.3 Boson-Fermion Transmutation. . . . . . . . . . . . . . . 131
5.1.4 Vortex line, Magnetic Flux and Fermion Quantization. . 133
5.2 Quantization and Anomaly. . . . . . . . . . . . . . . . . . . . . 137
5.2.1 Quantum Mechanical Symmetry Breaking and Anomaly . 137
5.2.2 Anomaly and Schwinger Term. . . . . . . . . . . 142
5.2.3 Path Integral Formalism and Chiral Anomaly . . . 144
5.2.4 Quantization of a Fermion and Chiral Anomaly . . 148
5.2.5 Quantization of a String and Conformal Anomaly . 153
5.3 Anomaly and Topology. . . . . . . . . . . 159
5.3.1 Topological Aspects of Anomaly. . . 159
5.3.2 Chiral Anomaly and Berry Phase . . 170
5.3.3 Berry Phase and Fermion Number. . 179
6 Topological Field Theory 183
6.1 General Aspects. . . . . . . . . . . . . . . . . . . 183
6.1.1 Definitions................. . 183
6.1.2 Topological Field theory: Witten Type. . 185
6.1.3 Topological Field Theory: Schwarz Type. . 190
6.2 Quantization, Supersymmetry and Topological Field Theory . 197
6.2.1 Topological Field Theory and Supersymmetry . . . . . 197
6.2.2 Supersymmetric Sigma Model . . . . . . . . . . . . . . 200
6.2.3 Quantization, Supersymmetry and Topological Field Theory . 202
6.3 Geometry and Topological Field Theory .......... . 205
6.3.1 Donaldson Invariants and Topological Field Theory ...... 205
viii CONTENTS
6.3.2 Geometry of Topological Gauge Theory. . . . . . . . . . . . . 208
6.3.3 Quantization, Topological Action, and Topological Field The-
ory in Different Dimensions ................... 211
References 217
Index 229
Preface
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. The role of gauge field
emphasized in various quantization procedures has its implication in the topological
feature associated with a quantum particle manifested through the Berry phase.
The topological feature associated with the quantization of a Fermi field is revealed
through the topological origin of fermion number and chiral anomaly which again
is related to the Berry phase. This monograph deals with these features associated
with the quantization procedure.
In chapters 1 and 2, preliminary mathematical formulations related to differential
forms, spinor structure and twist or geometry have been discussed. In chapter 3,
we have discussed various quantization procedures and in chapter 4 the role of
gauge field and its various implications have been discussed. Chapter 5 deals with
the topological features associated with fermions manifested through the origin of
fermion number and chiral anomaly. In chapter 6, we have discussed some aspects of
topological field theory emphasizing the relevance of the topological term associated
with the quantization of a Fermi field in such theories.
I would like to' ~lank my colleagues and students who have helped me in var
ious phases of preparing the manuscript. I would specifically mention K.Hajra,
B.Basu, G.Goswami, D.Banerjee, L.Mullick, B.B.Chaudhuri, S.Parui and P.P.Basu
ix
x Preface
who helped me immensely in numerous ways. I express my gratitude to B.Roy,
D.Chaudhuri, T.Pal, D.C.Dalal, G.C.Layek and S.Mukherjee for helping me in
preparing the LATeX version of the manuscript.
Calcutta, India
June 1996 Pratul Bandyopadhyay