GRADUATE STUDENT SERIES IN PHYSICS
Series Editor:
Professor Douglas F Brewer, MA, DPhil
Emeritus Professor of Experimental Physics, University of Sussex

GEOMETRY, TOPOLOGY
AND PHYSICS
SECOND EDITION
MIKIO NAKAHARA
Department of Physics
Kinki University, Osaka, Japan

INSTITUTE OF PHYSICS PUBLISHING
Bristol and Philadelphia


­c IOP Publishing Ltd 2003
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ISBN 0 7503 0606 8
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Dedicated to my family

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CONTENTS

Preface to the First Edition
Preface to the Second Edition
How to Read this Book
Notation and Conventions
1

Quantum Physics
1.1 Analytical mechanics
1.1.1 Newtonian mechanics
1.1.2 Lagrangian formalism

1.1.3 Hamiltonian formalism
1.2 Canonical quantization
1.2.1 Hilbert space, bras and kets
1.2.2 Axioms of canonical quantization
1.2.3 Heisenberg equation, Heisenberg picture and Schrăodinger
picture
1.2.4 Wavefunction
1.2.5 Harmonic oscillator
1.3 Path integral quantization of a Bose particle
1.3.1 Path integral quantization
1.3.2 Imaginary time and partition function
1.3.3 Time-ordered product and generating functional
1.4 Harmonic oscillator
1.4.1 Transition amplitude
1.4.2 Partition function
1.5 Path integral quantization of a Fermi particle
1.5.1 Fermionic harmonic oscillator
1.5.2 Calculus of Grassmann numbers
1.5.3 Differentiation
1.5.4 Integration
1.5.5 Delta-function
1.5.6 Gaussian integral
1.5.7 Functional derivative
1.5.8 Complex conjugation
1.5.9 Coherent states and completeness relation

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1.5.10 Partition function of a fermionic oscillator

Quantization of a scalar field
1.6.1 Free scalar field
1.6.2 Interacting scalar field
1.7 Quantization of a Dirac field
1.8 Gauge theories
1.8.1 Abelian gauge theories
1.8.2 Non-Abelian gauge theories
1.8.3 Higgs fields
1.9 Magnetic monopoles
1.9.1 Dirac monopole
1.9.2 The Wu–Yang monopole
1.9.3 Charge quantization
1.10 Instantons
1.10.1 Introduction
1.10.2 The (anti-)self-dual solution
Problems
1.6

2

Mathematical Preliminaries
2.1 Maps
2.1.1 Definitions
2.1.2 Equivalence relation and equivalence class
2.2 Vector spaces
2.2.1 Vectors and vector spaces
2.2.2 Linear maps, images and kernels
2.2.3 Dual vector space
2.2.4 Inner product and adjoint
2.2.5 Tensors

2.3 Topological spaces
2.3.1 Definitions
2.3.2 Continuous maps
2.3.3 Neighbourhoods and Hausdorff spaces
2.3.4 Closed set
2.3.5 Compactness
2.3.6 Connectedness
2.4 Homeomorphisms and topological invariants
2.4.1 Homeomorphisms
2.4.2 Topological invariants
2.4.3 Homotopy type
2.4.4 Euler characteristic: an example
Problems

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3

Homology Groups
3.1 Abelian groups
3.1.1 Elementary group theory
3.1.2 Finitely generated Abelian groups and free Abelian groups
3.1.3 Cyclic groups
3.2 Simplexes and simplicial complexes
3.2.1 Simplexes
3.2.2 Simplicial complexes and polyhedra
3.3 Homology groups of simplicial complexes
3.3.1 Oriented simplexes
3.3.2 Chain group, cycle group and boundary group

3.3.3 Homology groups
3.3.4 Computation of H0(K )
3.3.5 More homology computations
3.4 General properties of homology groups
3.4.1 Connectedness and homology groups
3.4.2 Structure of homology groups
3.4.3 Betti numbers and the Euler–Poincar´e theorem
Problems

4

Homotopy Groups
4.1 Fundamental groups
4.1.1 Basic ideas
4.1.2 Paths and loops
4.1.3 Homotopy
4.1.4 Fundamental groups
4.2 General properties of fundamental groups
4.2.1 Arcwise connectedness and fundamental groups
4.2.2 Homotopic invariance of fundamental groups
4.3 Examples of fundamental groups
4.3.1 Fundamental group of torus
4.4 Fundamental groups of polyhedra
4.4.1 Free groups and relations
4.4.2 Calculating fundamental groups of polyhedra
4.4.3 Relations between H1(K ) and π1 (|K |)
4.5 Higher homotopy groups
4.5.1 Definitions
4.6 General properties of higher homotopy groups
4.6.1 Abelian nature of higher homotopy groups

4.6.2 Arcwise connectedness and higher homotopy groups
4.6.3 Homotopy invariance of higher homotopy groups
4.6.4 Higher homotopy groups of a product space
4.6.5 Universal covering spaces and higher homotopy groups
4.7 Examples of higher homotopy groups

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4.8

Orders in condensed matter systems
4.8.1 Order parameter
4.8.2 Superfluid 4 He and superconductors
4.8.3 General consideration
4.9 Defects in nematic liquid crystals
4.9.1 Order parameter of nematic liquid crystals
4.9.2 Line defects in nematic liquid crystals
4.9.3 Point defects in nematic liquid crystals
4.9.4 Higher dimensional texture
4.10 Textures in superfluid 3 He-A
4.10.1 Superfluid 3 He-A
4.10.2 Line defects and non-singular vortices in 3 He-A
4.10.3 Shankar monopole in 3 He-A
Problems

5

Manifolds
5.1 Manifolds

5.1.1 Heuristic introduction
5.1.2 Definitions
5.1.3 Examples
5.2 The calculus on manifolds
5.2.1 Differentiable maps
5.2.2 Vectors
5.2.3 One-forms
5.2.4 Tensors
5.2.5 Tensor fields
5.2.6 Induced maps
5.2.7 Submanifolds
5.3 Flows and Lie derivatives
5.3.1 One-parameter group of transformations
5.3.2 Lie derivatives
5.4 Differential forms
5.4.1 Definitions
5.4.2 Exterior derivatives
5.4.3 Interior product and Lie derivative of forms
5.5 Integration of differential forms
5.5.1 Orientation
5.5.2 Integration of forms
5.6 Lie groups and Lie algebras
5.6.1 Lie groups
5.6.2 Lie algebras
5.6.3 The one-parameter subgroup
5.6.4 Frames and structure equation
5.7 The action of Lie groups on manifolds

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5.7.1 Definitions
5.7.2 Orbits and isotropy groups
5.7.3 Induced vector fields
5.7.4 The adjoint representation
Problems
6

de Rham Cohomology Groups
6.1 Stokes’ theorem
6.1.1 Preliminary consideration
6.1.2 Stokes’ theorem
6.2 de Rham cohomology groups
6.2.1 Definitions
6.2.2 Duality of Hr (M) and H r (M); de Rham’s theorem
6.3 Poincar´e’s lemma
6.4 Structure of de Rham cohomology groups
6.4.1 Poincar´e duality
6.4.2 Cohomology rings
6.4.3 The Kăunneth formula
6.4.4 Pullback of de Rham cohomology groups
6.4.5 Homotopy and H 1(M)

7

Riemannian Geometry
7.1 Riemannian manifolds and pseudo-Riemannian manifolds
7.1.1 Metric tensors
7.1.2 Induced metric
7.2 Parallel transport, connection and covariant derivative

7.2.1 Heuristic introduction
7.2.2 Affine connections
7.2.3 Parallel transport and geodesics
7.2.4 The covariant derivative of tensor fields
7.2.5 The transformation properties of connection coefficients
7.2.6 The metric connection
7.3 Curvature and torsion
7.3.1 Definitions
7.3.2 Geometrical meaning of the Riemann tensor and the
torsion tensor
7.3.3 The Ricci tensor and the scalar curvature
7.4 Levi-Civita connections
7.4.1 The fundamental theorem
7.4.2 The Levi-Civita connection in the classical geometry of
surfaces
7.4.3 Geodesics
7.4.4 The normal coordinate system
7.4.5 Riemann curvature tensor with Levi-Civita connection
7.5 Holonomy

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7.6

Isometries and conformal transformations
7.6.1 Isometries
7.6.2 Conformal transformations
7.7 Killing vector fields and conformal Killing vector fields
7.7.1 Killing vector fields

7.7.2 Conformal Killing vector fields
7.8 Non-coordinate bases
7.8.1 Definitions
7.8.2 Cartan’s structure equations
7.8.3 The local frame
7.8.4 The Levi-Civita connection in a non-coordinate basis
7.9 Differential forms and Hodge theory
7.9.1 Invariant volume elements
7.9.2 Duality transformations (Hodge star)
7.9.3 Inner products of r -forms
7.9.4 Adjoints of exterior derivatives
7.9.5 The Laplacian, harmonic forms and the Hodge
decomposition theorem
7.9.6 Harmonic forms and de Rham cohomology groups
7.10 Aspects of general relativity
7.10.1 Introduction to general relativity
7.10.2 Einstein–Hilbert action
7.10.3 Spinors in curved spacetime
7.11 Bosonic string theory
7.11.1 The string action
7.11.2 Symmetries of the Polyakov strings
Problems
8

Complex Manifolds
8.1 Complex manifolds
8.1.1 Definitions
8.1.2 Examples
8.2 Calculus on complex manifolds
8.2.1 Holomorphic maps

8.2.2 Complexifications
8.2.3 Almost complex structure
8.3 Complex differential forms
8.3.1 Complexification of real differential forms
8.3.2 Differential forms on complex manifolds
8.3.3 Dolbeault operators
8.4 Hermitian manifolds and Hermitian differential geometry
8.4.1 The Hermitian metric
8.4.2 Kăahler form
8.4.3 Covariant derivatives

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8.5

8.6

8.7
8.8

9

8.4.4 Torsion and curvature
Kăahler manifolds and Kăahler differential geometry
8.5.1 Definitions
8.5.2 Kăahler geometry
8.5.3 The holonomy group of Kăahler manifolds
Harmonic forms and ∂-cohomology groups
†

8.6.1 The adjoint operators ∂ † and ∂
8.6.2 Laplacians and the Hodge theorem
8.6.3 Laplacians on a Kăahler manifold
8.6.4 The Hodge numbers of Kăahler manifolds
Almost complex manifolds
8.7.1 Definitions
Orbifolds
8.8.1 One-dimensional examples
8.8.2 Three-dimensional examples

Fibre Bundles
9.1 Tangent bundles
9.2 Fibre bundles
9.2.1 Definitions
9.2.2 Reconstruction of fibre bundles
9.2.3 Bundle maps
9.2.4 Equivalent bundles
9.2.5 Pullback bundles
9.2.6 Homotopy axiom
9.3 Vector bundles
9.3.1 Definitions and examples
9.3.2 Frames
9.3.3 Cotangent bundles and dual bundles
9.3.4 Sections of vector bundles
9.3.5 The product bundle and Whitney sum bundle
9.3.6 Tensor product bundles
9.4 Principal bundles
9.4.1 Definitions
9.4.2 Associated bundles
9.4.3 Triviality of bundles

Problems

10 Connections on Fibre Bundles
10.1 Connections on principal bundles
10.1.1 Definitions
10.1.2 The connection one-form
10.1.3 The local connection form and gauge potential
10.1.4 Horizontal lift and parallel transport
10.2 Holonomy

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337


10.2.1 Definitions
10.3 Curvature
10.3.1 Covariant derivatives in principal bundles
10.3.2 Curvature
10.3.3 Geometrical meaning of the curvature and the Ambrose–
Singer theorem
10.3.4 Local form of the curvature
10.3.5 The Bianchi identity
10.4 The covariant derivative on associated vector bundles
10.4.1 The covariant derivative on associated bundles
10.4.2 A local expression for the covariant derivative
10.4.3 Curvature rederived
10.4.4 A connection which preserves the inner product
10.4.5 Holomorphic vector bundles and Hermitian inner
products

10.5 Gauge theories
10.5.1 U(1) gauge theory
10.5.2 The Dirac magnetic monopole
10.5.3 The Aharonov–Bohm effect
10.5.4 Yang–Mills theory
10.5.5 Instantons
10.6 Berry’s phase
10.6.1 Derivation of Berry’s phase
10.6.2 Berry’s phase, Berry’s connection and Berry’s curvature
Problems
11 Characteristic Classes
11.1 Invariant polynomials and the Chern–Weil homomorphism
11.1.1 Invariant polynomials
11.2 Chern classes
11.2.1 Definitions
11.2.2 Properties of Chern classes
11.2.3 Splitting principle
11.2.4 Universal bundles and classifying spaces
11.3 Chern characters
11.3.1 Definitions
11.3.2 Properties of the Chern characters
11.3.3 Todd classes
11.4 Pontrjagin and Euler classes
11.4.1 Pontrjagin classes
11.4.2 Euler classes
ˆ
11.4.3 Hirzebruch L-polynomial and A-genus
11.5 Chern–Simons forms
11.5.1 Definition


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11.5.2 The Chern–Simons form of the Chern character
11.5.3 Cartan’s homotopy operator and applications
11.6 Stiefel–Whitney classes
11.6.1 Spin bundles
ˇ
11.6.2 Cech
cohomology groups
11.6.3 Stiefel–Whitney classes
12 Index Theorems
12.1 Elliptic operators and Fredholm operators
12.1.1 Elliptic operators
12.1.2 Fredholm operators
12.1.3 Elliptic complexes
12.2 The Atiyah–Singer index theorem
12.2.1 Statement of the theorem
12.3 The de Rham complex
12.4 The Dolbeault complex
12.4.1 The twisted Dolbeault complex and the Hirzebruch–
Riemann–Roch theorem
12.5 The signature complex
12.5.1 The Hirzebruch signature
12.5.2 The signature complex and the Hirzebruch signature
theorem
12.6 Spin complexes
12.6.1 Dirac operator
12.6.2 Twisted spin complexes
12.7 The heat kernel and generalized ζ -functions

12.7.1 The heat kernel and index theorem
12.7.2 Spectral ζ -functions
12.8 The Atiyah–Patodi–Singer index theorem
12.8.1 η-invariant and spectral flow
12.8.2 The Atiyah–Patodi–Singer (APS) index theorem
12.9 Supersymmetric quantum mechanics
12.9.1 Clifford algebra and fermions
12.9.2 Supersymmetric quantum mechanics in flat space
12.9.3 Supersymmetric quantum mechanics in a general
manifold
12.10 Supersymmetric proof of index theorem
12.10.1 The index
12.10.2 Path integral and index theorem
Problems

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13 Anomalies in Gauge Field Theories
13.1 Introduction
13.2 Abelian anomalies
13.2.1 Fujikawa’s method
13.3 Non-Abelian anomalies
13.4 The Wess–Zumino consistency conditions
13.4.1 The Becchi–Rouet–Stora operator and the Faddeev–
Popov ghost
13.4.2 The BRS operator, FP ghost and moduli space
13.4.3 The Wess–Zumino conditions
13.4.4 Descent equations and solutions of WZ conditions
13.5 Abelian anomalies versus non-Abelian anomalies

13.5.1 m dimensions versus m + 2 dimensions
13.6 The parity anomaly in odd-dimensional spaces
13.6.1 The parity anomaly
13.6.2 The dimensional ladder: 4–3–2
14 Bosonic String Theory
14.1 Differential geometry on Riemann surfaces
14.1.1 Metric and complex structure
14.1.2 Vectors, forms and tensors
14.1.3 Covariant derivatives
14.1.4 The Riemann–Roch theorem
14.2 Quantum theory of bosonic strings
14.2.1 Vacuum amplitude of Polyakov strings
14.2.2 Measures of integration
14.2.3 Complex tensor calculus and string measure
14.2.4 Moduli spaces of Riemann surfaces
14.3 One-loop amplitudes
14.3.1 Moduli spaces, CKV, Beltrami and quadratic differentials
14.3.2 The evaluation of determinants
References

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PREFACE TO THE FIRST EDITION

This book is a considerable expansion of lectures I gave at the School of
Mathematical and Physical Sciences, University of Sussex during the winter
term of 1986. The audience included postgraduate students and faculty members
working in particle physics, condensed matter physics and general relativity. The
lectures were quite informal and I have tried to keep this informality as much as

possible in this book. The proof of a theorem is given only when it is instructive
and not very technical; otherwise examples will make the theorem plausible.
Many figures will help the reader to obtain concrete images of the subjects.
In spite of the extensive use of the concepts of topology, differential geometry and other areas of contemporary mathematics in recent developments in
theoretical physics, it is rather difficult to find a self-contained book that is easily
accessible to postgraduate students in physics. This book is meant to fill the gap
between highly advanced books or research papers and the many excellent introductory books. As a reader, I imagined a first-year postgraduate student in theoretical physics who has some familiarity with quantum field theory and relativity.
In this book, the reader will find many examples from physics, in which topological and geometrical notions are very important. These examples are eclectic
collections from particle physics, general relativity and condensed matter physics.
Readers should feel free to skip examples that are out of their direct concern.
However, I believe these examples should be the theoretical minima to students
in theoretical physics. Mathematicians who are interested in the application of
their discipline to theoretical physics will also find this book interesting.
The book is largely divided into four parts. Chapters 1 and 2 deal with the
preliminary concepts in physics and mathematics, respectively. In chapter 1,
a brief summary of the physics treated in this book is given. The subjects
covered are path integrals, gauge theories (including monopoles and instantons),
defects in condensed matter physics, general relativity, Berry’s phase in quantum
mechanics and strings. Most of the subjects are subsequently explained in detail
from the topological and geometrical viewpoints. Chapter 2 supplements the
undergraduate mathematics that the average physicist has studied. If readers are
quite familiar with sets, maps and general topology, they may skip this chapter
and proceed to the next.
Chapters 3 to 8 are devoted to the basics of algebraic topology and
differential geometry. In chapters 3 and 4, the idea of the classification of spaces
with homology groups and homotopy groups is introduced. In chapter 5, we

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define a manifold, which is one of the central concepts in modern theoretical
physics. Differential forms defined there play very important roles throughout this
book. Differential forms allow us to define the dual of the homology group called
the de Rham cohomology group in chapter 6. Chapter 7 deals with a manifold
endowed with a metric. With the metric, we may define such geometrical
concepts as connection, covariant derivative, curvature, torsion and many more.
In chapter 8, a complex manifold is defined as a special manifold on which there
exists a natural complex structure.
Chapters 9 to 12 are devoted to the unification of topology and geometry.
In chapter 9, we define a fibre bundle and show that this is a natural setting
for many physical phenomena. The connection defined in chapter 7 is naturally
generalized to that on fibre bundles in chapter 10. Characteristic classes defined
in chapter 11 enable us to classify fibre bundles using various cohomology
classes. Characteristic classes are particularly important in the Atiyah–Singer
index theorem in chapter 12. We do not prove this, one of the most important
theorems in contemporary mathematics, but simply write down the special forms
of the theorem so that we may use them in practical applications in physics.
Chapters 13 and 14 are devoted to the most fascinating applications of
topology and geometry in contemporary physics. In chapter 13, we apply the
theory of fibre bundles, characteristic classes and index theorems to the study of
anomalies in gauge theories. In chapter 14, Polyakov’s bosonic string theory is
analysed from the geometrical point of view. We give an explicit computation of
the one-loop amplitude.
I would like to express deep gratitude to my teachers, friends and students.
Special thanks are due to Tetsuya Asai, David Bailin, Hiroshi Khono, David
Lancaster, Shigeki Matsutani, Hiroyuki Nagashima, David Pattarini, Felix E A
Pirani, Kenichi Tamano, David Waxman and David Wong. The basic concepts
in chapter 5 owe very much to the lectures by F E A Pirani at King’s College,
University of London. The evaluation of the string Laplacian in chapter 14 using
the Eisenstein series and the Kronecker limiting formula was suggested by T Asai.

I would like to thank Euan Squires, David Bailin and Hiroshi Khono for useful
comments and suggestions. David Bailin suggested that I should write this book.
He also advised Professor Douglas F Brewer to include this book in his series. I
would like to thank the Science and Engineering Research Council of the United
Kingdom, which made my stay at Sussex possible. It is a pity that I have no
secretary to thank for the beautiful typing. Word processing has been carried out
by myself on two NEC PC9801 computers. Jim A Revill of Adam Hilger helped
me in many ways while preparing the manuscript. His indulgence over my failure
to meet deadlines is also acknowledged. Many musicians have filled my office
with beautiful music during the preparation of the manuscript: I am grateful to
J S Bach, Ryuichi Sakamoto, Ravi Shankar and Erik Satie.
Mikio Nakahara
Shizuoka, February 1989

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PREFACE TO THE SECOND EDITION

The first edition of the present book was published in 1990. There has been
incredible progress in geometry and topology applied to theoretical physics and
vice versa since then. The boundaries among these disciplines are quite obscure
these days.
I found it impossible to take all the progress into these fields in this second
edition and decided to make the revision minimal. Besides correcting typos, errors
and miscellaneous small additions, I added the proof of the index theorem in terms
of supersymmetric quantum mechanics. There are also some rearrangements of
material in many places. I have learned from publications and internet homepages
that the first edition of the book has been read by students and researchers from a
wide variety of fields, not only in physics and mathematics but also in philosophy,

chemistry, geodesy and oceanology among others. This is one of the reasons
why I did not specialize this book to the forefront of recent developments. I
hope to publish a separate book on the recent fascinating application of quantum
field theory to low dimensional topology and number theory, possibly with a
mathematician or two, in the near future.
The first edition of the book has been used in many classes all over the world.
Some of the lecturers gave me valuable comments and suggestions. I would like
to thank, in particular, Jouko Mikkelsson for constructive suggestions. Kazuhiro
Sakuma, my fellow mathematician, joined me to translate the first edition of the
book into Japanese. He gave me valuable comments and suggestions from a
mathematician’s viewpoint. I also want to thank him for frequent discussions
and for clarifying many of my questions. I had a chance to lecture on the material
of the book while I was a visiting professor at Helsinki University of Technology
during fall 2001 through spring 2002. I would like to thank Martti Salomaa for
warm hospitality at his materials physics laboratory. Sami Virtanen was the course
assisitant whom I would like to thank for his excellent work. I would also like to
thank Juha Vartiainen, Antti Laiho, Teemu Ojanen, Teemu Keski-Kuha, Markku
Stenberg, Juha Heiskala, Tuomas Hytăonen, Antti Niskanen and Ville Bergholm
for helping me to find typos and errors in the manuscript and also for giving me
valuable comments and questions.
Jim Revill and Tom Spicer of IOP Publishing have always been generous
in forgiving me for slow revision. I would like to thank them for their generosity
and patience. I also want to thank Simon Laurenson for arranging the copyediting,
typesetting and proofreading and Sarah Plenty for arranging the printing, binding

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and scheduling. The first edition of the book was prepared using an old NEC
computer whose operating system no longer exists. I hesitated to revise the

book mainly because I was not so courageous as to type a more-than-500-page
book again. Thanks to the progress of information technology, IOP Publishing
scanned all the pages of the book and supplied me with the files, from which I
could extract the text files with the help of optical character recognition (OCR)
software. I would like to thank the technical staff of IOP Publishing for this
painstaking work. The OCR is not good enough to produce the LATEX codes for
equations. Mariko Kamada edited the equations from the first version of the book.
I would like to thank Yukitoshi Fujimura of Peason Education Japan for frequent
TEX-nical assistance. He edited the Japanese translation of the first edition of the
present book and produced an excellent LATEX file, from which I borrowed many
LATEX definitions, styles, diagrams and so on. Without the Japanese edition, the
publication of this second edition would have been much more difficult.
Last but not least, I would thank my family to whom this book is dedicated.
I had to spend an awful lot of weekends on this revision. I wish to thank my
wife, Fumiko, and daughters, Lisa and Yuri, for their patience. I hope my
little daughters will someday pick up this book in a library or a bookshop and
understand what their dad was doing at weekends and late after midnight.
Mikio Nakahara
Nara, December 2002

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HOW TO READ THIS BOOK

As the author of this book, I strongly wish that this book is read in order. However,
I admit that the book is thick and the materials contained in it are diverse. Here
I want to suggest some possibilities when this book is used for a course in
mathematics or mathematical physics.
(1) A one year course on mathematical physics: chapters 1 through 10.

Chapters 11 and 12 are optional.
(2) A one-year course on geometry and topology for mathematics students:
chapters 2 through 12. Chapter 2 may be omitted if students are familiar with
elementary topology. Topics from physics may be omitted without causing
serious problems.
(3) A single-semester course on geometry and topology: chapters 2 through
7. Chapter 2 may be omitted if the students are familiar with elementary
topology. Chapter 8 is optional.
(4) A single-semester course on differential geometry for general relativity:
chapters 2, 5 and 7.
(5) A single-semester course on advanced mathematical physics: sections 1.1–
1.7 and sections 12.9 and 12.10, assuming that students are familiar with
Riemannian geometry and fibre bundles. This makes a self-contained course
on the path integral and its application to index theorem.
Some repetition of the material or a summary of the subjects introduced in
the previous part are made to make these choices possible.

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NOTATION AND CONVENTIONS
The symbols Ỉ , , É , Ê and
denote the sets of natural numbers, integers,
rational numbers, real numbers and complex numbers, respectively. The set of
quaternions is defined by

À = {a + bi + c j + d k| a, b, c, d ∈ Ê}
where (1, i, j, k) is a basis such that i · j = − j · i = k, j · k = −k · j = i,
k · i = −i · k = j , i 2 = j 2 = k2 = −1. Note that i, j and k have the 2×2 matrix
representations i = iσ3 , j = iσ2 , k = iσ1 where σi are the Pauli spin matrices

σ1 =

0 1
1 0

σ2 =

0 −i
i 0

σ3 =

1 0
0 −1

.

The imaginary part of a complex number z is denoted by Im z while the real part
is Re z.
We put c (speed of light) = h¯ (Planck’s constant/2π) = kB (Boltzmann’s
constant) = 1, unless otherwise stated explicitly. We employ the Einstein
summation convention: if the same index appears twice, once as a superscript
and once as a subscript, then the index is summed over all possible values. For
example, if µ runs from 1 to m, one has
A µ Bµ =

m

A µ Bµ .


µ=1

The Euclid metric is gµν = δµν = diag(+1, . . . , +1) while the Minkowski metric
is gµν = ηµν = diag(−1, +1, . . . , +1).
The symbol £ denotes ‘the end of a proof’.

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1
QUANTUM PHYSICS
A brief introduction to path integral quantization is presented in this chapter.
Physics students who are familiar with this subject and mathematics students who
are not interested in physics may skip this chapter and proceed directly to the next
chapter. Our presentation is sketchy and a more detailed account of this subject
is found in Bailin and Love (1996), Cheng and Li (1984), Huang (1982), Das
(1993), Kleinert (1990), Ramond (1989), Ryder (1986) and Swanson (1992). We
closely follow Alvarez (1995), Bertlmann (1996), Das (1993), Nakahara (1998),
Rabin (1995), Sakita (1985) and Swanson (1992).
1.1 Analytical mechanics
We introduce some elementary principles of Lagrangian and Hamiltonian
formalisms that are necessary to understand quantum mechanics.
1.1.1 Newtonian mechanics
Let us consider the motion of a particle m in three-dimensional space and let x(t)
denote the position of m at time t.1 Suppose this particle is moving under an
external force F(x). Then x(t) satisfies the second-order differential equation
m

d2 x(t)
= F(x(t))

dt 2

(1.1)

called Newton’s equation or the equation of motion.
If force F(x) is expressed in terms of a scalar function V (x) as F(x) =
−∇V (x), the force is called a conserved force and the function V (x) is called
the potential energy or simply the potential. When F is a conserved force, the
combination
m dx 2
+ V (x)
(1.2)
E=
2 dt
is conserved. In fact,
dE
=
dt

m
k=x,y,z

dx k d2 x k
∂ V dx k
+
dt dt 2
∂ x k dt

=


m
k

1 We call a particle with mass m simply ‘a particle m’.

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d2 x k
∂V
+
dt 2
∂ xk

dx k
=0
dt


where use has been made of the equation of motion. The function E, which is
often the sum of the kinetic energy and the potential energy, is called the energy.
Example 1.1. (One-dimensional harmonic oscillator) Let x be the coordinate
and suppose the force acting on m is F(x) = −kx, k being a constant. This force
is conservative. In fact, V (x) = 12 kx 2 yields F(x) = −dV (x)/dx = −kx.
In general, any one-dimensional force F(x) which is a function of x only is
conserved and the potential is given by
V (x) = −

x

F(ξ ) dξ.


An example of a force that is not conserved is friction F = −η dx/dt. We
will be concerned only with conserved forces in the following.
1.1.2 Lagrangian formalism
Newtonian mechanics has the following difficulties;
1.
2.
3.
4.

This formalism is based on a vector equation (1.1) which is not very easy to
handle unless an orthogonal coordinate system is employed.
The equation of motion is a second-order equation and the global properties
of the system cannot be figured out easily.
The analysis of symmetries is not easy.
Constraints are difficult to take into account.

Furthermore, quantum mechanics cannot be derived directly from
Newtonian mechanics. The Lagrangian formalism is now introduced to overcome
these difficulties.
Let us consider a system whose state (the position of masses for example)
is described by N parameters {qi } (1 ≤ i ≤ N). The parameter is an element
of some space M.2 The space M is called the configuration space and the {qi }
are called the generalized coordinates. If one considers a particle on a circle, for
example, the generalized coordinate q is an angle θ and the configuration space
M is a circle. The generalized velocity is defined by q˙i = dqi /dt.
The Lagrangian L(q, q)
˙ is a function to be defined in Hamilton’s
principle later. We will restrict ourselves mostly to one-dimensional space but
generalization to higher-dimensional space should be obvious. Let us consider

a trajectory q(t) (t ∈ [ti , t f ]) of a particle with conditions q(ti ) = qi and
q(t f ) = q f . Consider a functional3
tf

S[q(t), q(t)]
˙
=

L(q, q)
˙ dt

(1.3)

ti
2 A manifold, to be more precise, see chapter 5.
3 A functional is a function of functions. A function f (•) produces a number f (x) for a given number

x. Similarly, a functional F[•] assigns a number F[ f ] to a given function f (x).

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called the action. Given a trajectory q(t) and q(t),
˙
the action S[q, q]
˙ produces
a real number. Hamilton’s principle, also known as the principle of the least
action, claims that the physically realized trajectory corresponds to an extremum
of the action. Now the Lagrangian must be chosen so that Hamilton’s principle is
fulfilled.

It turns out to be convenient to write Hamilton’s principle in a local form
as a differential equation. Suppose q(t) is a path realizing an extremum of S.
Consider a variation δq(t) of the trajectory such that δq(ti ) = δq(t f ) = 0. The
action changes under this variation by
tf

δS =
ti

ti

L(q, q)
˙ dt

ti
tf

=

tf

L(q + δq, q˙ + δ q)
˙ dt −
d ∂L
∂L
−
∂q
dt ∂ q˙

δq dt


(1.4)

which must vanish because q yields an extremum of S. Since this is true for any
δq, the integrand of the last line of (1.4) must vanish. Thus, the Euler–Lagrange
equation
d ∂L
∂L
−
=0
(1.5)
∂q
dt ∂ q˙
has been obtained. If there are N degrees of freedom, one obtains
d ∂L
∂L
−
=0
∂qk
dt ∂ q˙k

(1 ≤ k ≤ N).

(1.6)

If we introduce the generalized momentum conjugate to the coordinate qk
by
pk =

∂L

∂ q˙k

(1.7)

the Euler–Lagrange equation takes the form
∂L
d pk
=
.
dt
∂qk

(1.8)

By requiring this equation to reduce to Newton’s equation, one quickly finds the
possible form of the Lagrangian in the ordinary mechanics of a particle. Let us
put L = 12 m q˙ 2 − V (q). By substituting this Lagrangian into the Euler–Lagrange
equation, it is easily shown that it reduces to Newton’s equation of motion,
m qăk +

V
= 0.
qk

(1.9)

Let us consider the one-dimensional harmonic oscillator for example. The
Lagrangian is
L(x, x)
˙ = 12 m x˙ 2 − 12 kx 2

(1.10)

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from which one finds m xă + kx = 0.
It is convenient for later purposes to introduce the notion of a functional
derivative. Let us consider the case with a single degree of freedom for simplicity.
Define the functional derivative of S with respect to q by
{S[q(t) + εδ(t − s), q(t)
˙ + ε dtd δ(t − s)] − S[q(t), q(t)]}
˙
δS[q, q]
˙
≡ lim
.
ε→0
δq(s)
ε
(1.11)
Since
S q(t) + εδ(t − s), q(t)
˙ +ε

d
δ(t − s)
dt

d
δ(t − s)

dt
∂L d
∂L
δ(t − s) +
δ(t − s) + Ç(ε2 )
= dt L(q, q)
˙ + ε dt
∂q
∂ q˙ dt
d ∂L
∂L
(s) −
(s) + Ç(ε2 ),
= S[q, q]
˙ +ε
∂q
dt ∂ q˙
=

dt L q(t) + εδ(t − s), q(t)
˙ +ε

the Euler–Lagrange equation may be written as
∂L
d
δS
=
(s) −
δq(s)
∂q

dt

∂L
∂ q˙

(s) = 0.

(1.12)

Let us next consider symmetries in the context of the Lagrangian formalism.
Suppose the Lagrangian L is independent of a certain coordinate qk .4 Such
a coordinate is called cyclic. The momentum which is conjugate to a cyclic
coordinate is conserved. In fact, the condition ∂ L/∂qk = 0 leads to
d ∂L
∂L
d pk
=
=
= 0.
dt
dt ∂ q˙k
∂qk

(1.13)

This argument can be mathematically elaborated as follows. Suppose the
Lagrangian L has a symmetry, which is continuously parametrized. This means,
more precisely, that the action S = dt L is invariant under the symmetry
operation on qk (t). Let us consider an infinitesimal symmetry operation qk (t) →
qk (t) + δqk (t) on the path qk (t).5 This implies that if qk (t) is a path producing

an extremum of the action, then qk (t) → qk (t) + δqk (t) also corresponds to an
extremum. Since S is invariant under this change, it follows that
tf

δS =
ti

δqk
k

d ∂L
∂L
−
∂qk
dt ∂ q˙k

+

δqk
k

∂L
∂ q˙k

tf

= 0.

ti


4 Of course, L may depend on q˙ . Otherwise, the coordinate q is not our concern at all.
k
k
5 Since the symmetry is continuous, it is always possible to define such an infinitesimal operation.

Needless to say, δq(ti ) and δq(t f ) do not, in general, vanish in the present case.

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The first term in the middle expression vanishes since q is a solution to the Euler–
Lagrange equation. Accordingly, we obtain
δqk (ti ) pk (ti ) =
k

δqk (t f ) pk (t f )

(1.14)

k

where use has been made of the definition pk = ∂ L/∂ q˙k . Since ti and t f
are arbitrary, this equation shows that the quantity k δqk (t) pk (t) is, in fact,
independent of t and hence conserved.
Example 1.2. Let us consider a particle m moving under a force produced by a
spherically symmetric potential V (r ), where r, θ, φ are three-dimensional polar
coordinates. The Lagrangian is given by
L = 12 m[˙r 2 + r 2 (θ˙ 2 + sin2 θ φ˙ 2 )] − V (r ).
Note that qk = φ is cyclic, which leads to the conservation law
δφ


∂L
∝ mr 2 sin2 θ φ˙ = constant.
∂ φ˙

This is nothing but the angular momentum around the z axis. Similar arguments
can be employed to show that the angular momenta around the x and y axes are
also conserved.
A few remarks are in order:
•

Let Q(q) be an arbitrary function of q. Then the Lagrangians L and
L + dQ/dt yield the same Euler–Lagrange equation. In fact,
∂
∂qk

d
dQ
dQ
∂
−
L+
dt
dt ∂ q˙k
dt
d ∂L
∂ dQ
d ∂
∂L
−

+
−
=
∂qk
∂qk dt
dt ∂ q˙k
dt ∂ q˙k
L+

j

∂Q
q˙ j
∂q j

∂ dQ
d ∂Q
=
−
= 0.
∂qk dt
dt ∂qk
•

An interesting observation is that Newtonian mechanics is realized as an
extremum of the action but the action itself is defined for any trajectory. This
fact plays an important role in path integral formation of quantum theory.

1.1.3 Hamiltonian formalism
The Lagrangian formalism yields a second-order ordinary differencial equation

(ODE). In contrast, the Hamiltonian formalism gives equations of motion which
are first order in the time derivative and, hence, we may introduce flows in the

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phase space defined later. What is more important, however, is that we can make
the symplectic structure manifest in the Hamiltonian formalism, which will be
shown in example 5.12 later.
Suppose a Lagrangian L is given. Then the corresponding Hamiltonian is
introduced via Legendre transformation of variables as
H (q, p) ≡

pk q˙k − L(q, q),
˙

(1.15)

k

where q˙ is eliminated in the left-hand side (LHS) in favour of p by making use of
˙ q˙k . For this transformation to
the definition of the momentum pk = ∂ L(q, q)/∂
be defined, the Jacobian must satisfy
det

∂ pi
∂ q˙ j

= det


∂2 L
∂ q˙i q˙ j

= 0.

The space with coordinates (qk , pk ) is called the phase space.
Let us consider an infinitesimal change in the Hamiltonian induced by δqk
and δpk ,
δH =

δpk q˙k + pk δ q˙k −
k

δpk q˙k −

=
k

∂L
∂L
δqk −
δ q˙k
∂qk
∂ q˙k

∂L
δqk .
∂qk


It follows from this relation that
∂H
= q˙k ,
∂ pk

∂H
∂L
=−
∂qk
∂qk

(1.16)

which are nothing more than the replacements of independent variables.
Hamilton’s equations of motion are obtained from these equations if the Euler–
Lagrange equation is employed to replace the LHS of the second equation,
q˙k =

∂H
∂ pk

p˙ k = −

∂H
.
∂qk

(1.17)

Example 1.3. Let us consider a one-dimensional harmonic oscillator with the

Lagrangian L = 12 m q˙ 2 − 12 mω2 q 2 , where ω2 = k/m. The momentum conjugate
to q is p = ∂ L/∂ q˙ = m q,
˙ which can be solved for q˙ to yield q˙ = p/m. The
Hamiltonian is
H (q, p) = p q˙ − L(q, q)
˙ =

p2
1
+ mω2 q 2 .
2m
2

(1.18)

Hamilton’s equations of motion are:
dp
= −mω2 q
dt

p
dq
= .
dt
m

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(1.19)