Quantum Field Theory III: Gauge Theory



Eberhard Zeidler

Quantum Field
Theory III:
Gauge Theory
A Bridge between Mathematicians
and Physicists


Eberhard Zeidler
Max Planck Institute
for Mathematics in the Sciences
Inselstr. 22-26
04103 Leipzig
Germany

ISBN 978-3-642-22420-1
e-ISBN 978-3-642-22421-8
DOI 10.1007/978-3-642-22421-8
Springer Heidelberg Dordrecht London New York
Library of Congress Control Number: 2006929535
Mathematics Subject Classification (2010): 35-XX, 47-XX, 49-XX, 51-XX, 55-XX, 81-XX, 82-XX
© Springer-Verlag Berlin Heidelberg 2011
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TO KRZYSZTOF MAURIN
IN GRATITUDE



Preface

Sein Geist drang in die tiefsten Geheimnisse der Zahl, des Raumes und
der Natur; er maß den Lauf der Gestirne, die Gestalt und die Kră
afte der
Erde; die Entwicklung der mathematischen Wissenschaft eines kommenden
Jahrhunderts trug er in sich.1
Lines under the portrait of Carl Friedrich Gauss (1777–1855)
in the German Museum in Munich
Force equals curvature.
The basic principle of modern physics
A theory is the more impressive, the simpler are its premises, the more
distinct are the things it connects, and the broader is the range of applicability.
Albert Einstein (1879–1955)

Textbooks should be attractive by showing the beauty of the subject.
Johann Wolfgang von Goethe (1749–1832)
The present book is the third volume of a comprehensive introduction to the mathematical and physical aspects of modern quantum field theory which comprises the
following six volumes:
Volume
Volume
Volume
Volume
Volume
Volume

I: Basics in Mathematics and Physics
II: Quantum Electrodynamics
III: Gauge Theory
IV: Quantum Mathematics
V: The Physics of the Standard Model
VI: Quantum Gravitation and String Theory.

It is our goal to build a bridge between mathematicians and physicists based on
challenging questions concerning the fundamental forces in
• the macrocosmos (the universe) and
• the microcosmos (the world of elementary particles).
1

His mind pierced the deepest secrets of numbers, space, and nature; he measured
the orbits of the planets, the form and the forces of the earth; in his mind he
carried the mathematical science of a coming century.
VII



VIII

Preface

The six volumes address a broad audience of readers, including both undergraduate
and graduate students, as well as experienced scientists who want to become familiar
with quantum field theory, which is a fascinating topic in modern mathematics and
physics, full of many crucial open questions.
For students of mathematics, detailed knowledge of the physical background
helps to enliven mathematical subjects and to discover interesting interrelationships between quite different mathematical topics. For students of physics, fairly
advanced mathematical subjects are presented that go beyond the usual curriculum
in physics. The strategies and the structure of the six volumes are thoroughly discussed in the Prologue to Volume I. In particular, we will try to help the reader to
understand the basic ideas behind the technicalities. In this connection, the famous
ancient story of Ariadne’s thread is discussed in the Preface to Volume I:
In terms of this story, we want to put the beginning of Ariadne’s thread in
quantum field theory into the hands of the reader.
There are four fundamental forces in the universe, namely,
•
•
•
•

gravitation,
electromagnetic interaction (e.g., light),
strong interaction (e.g., the binding force of the proton),
weak interaction (e.g., radioactive decay).

In modern physics, these four fundamental forces are described by
• Einstein’s theory of general relativity (gravitation), and
• the Standard Model in elementary particle physics (electromagnetic, strong, and

weak interaction).
The basic mathematical framework is provided by gauge theory:
The main idea is to describe the four fundamental forces by the curvature
of appropriate fiber bundles.
In this way, the universal principle force equals curvature is implemented. There are
many open questions:
• A mathematically rigorous quantum field theory for the quantized version of the
Standard Model in elementary particles has yet to be found.
• We do not know how to combine gravitation with the Standard Model in elementary particle physics (the challenge of quantum gravitation).
• Astrophysical observations show that 96 percent of the universe consists of both
dark matter and dark energy. However, both the physical structure and the
mathematical description of dark matter and dark energy are unknown.
One of the greatest challenges of the human intellect is the discovery of
a unified theory for the four fundamental forces in nature based on first
principles in physics and rigorous mathematics.
In the present volume, we concentrate on the classical aspects of gauge theory
related to curvature. These have to be supplemented by the crucial, but elusive
quantization procedure. The quantization of the Maxwell–Dirac system leads to
quantum electrodynamics (see Vol. II). The quantization of both the full Standard
Model in elementary particle physics and the quantization of gravitation will be
studied in the volumes to come.
One cannot grasp modern physics without understanding gauge theory,
which tells us that the fundamental interactions in nature are based on
parallel transport, and in which forces are described by curvature, which
measures the path-dependence of the parallel transport.


Preface

IX


Gauge theory is the result of a fascinating long-term development in both mathematics and physics. Gauge transformations correspond to a change of potentials,
and physical quantities measured in experiments are invariants under gauge transformations. Let us briefly discuss this.
Gauss discovered that the curvature of a two-dimensional surface is an intrinsic
property of the surface. This means that the Gaussian curvature of the surface can
be determined by using measurements on the surface (e.g., on the earth) without
using the surrounding three-dimensional space. The precise formulation is provided
by Gauss’ theorema egregium (the egregious theorem). Bernhard Riemann (1826–
´
1866) and Elie
Cartan (1859–1951) formulated far-reaching generalizations of the
theorema egregium which lie at the heart of
• modern differential geometry (the curvature of general fiber bundles), and
• modern physics (gauge theories).
Interestingly enough, in this way,
• Einstein’s theory of general relativity (the curvature of the four-dimensional
space-time manifold), and
• the Standard Model in elementary particle physics (the curvature of a specific
fiber bundle with the symmetry group U (1) × SU (2) × SU (3))
can be traced back to Gauss’ theorema egregium.
In classical mechanics, a large class of forces can be described by the differentiation of potentials. This simplifies the solution of Newton’s equation of motion
and leads to the concept of potential energy together with energy conservation (for
the sum of kinetic and potential energy). In the 1860s, Maxwell determined that
the computation of electromagnetic fields can be substantially simplified by introducing potentials for both the electric and the magnetic field (the electromagnetic
four-potential).
Gauge theory generalizes this by describing forces (interactions) by the
differentiation of generalized potentials (also called connections).
The point is that gauge transformations change the generalized potentials, but not
the essential physical effects.
Physical quantities, which can be measured in experiments, have to be invariant under gauge transformations.

Parallel to this physical situation, in mathematics the Riemann curvature tensor can
be described by the differentiation of the Christoffel symbols (also called connection
coefficients or geometric potentials). The notion of the Riemann curvature tensor
was introduced by Riemann in order to generalize Gauss’ theorema egregium to
higher dimensions. In 1915, Einstein discovered that the Riemann curvature tensor
of a four-dimensional space-time manifold can be used to describe gravitation in
the framework of the theory of general relativity.
The basic idea of gauge theory is the transport of physical information
along curves (also called parallel transport).
This generalizes the parallel transport of vectors in the three-dimensional Euclidean
space of our intuition.
In 1917, it was discovered by Levi-Civita that the study of curved manifolds
in differential geometry can be based on the notion of parallel transport of
tangent vectors (velocity vectors).


X

Preface

In particular, curvature can be measured intrinsically by transporting a tangent
´
vector along a closed path. This idea was further developed by Elie
Cartan in
the 1920s (the method of moving frames) and by Ehresmann in the 1950s (the
connection of both principal fiber bundles and their associated vector bundles).
The very close relation between
• gauge theory in modern physics (the transport of local SU (2)-phase factors investigated by Yang and Mills in 1954), and
• the formulation of differential geometry in terms of fiber bundles in modern
mathematics

was only noticed by physicists in 1975 (see T. Wu and C. Yang, Concept of nonintegrable phase factors and global formulation of gauge fields, Phys. Rev. D12
(1975), 3845–3857).
The present Volume III on gauge theory and the following Volume IV on quantum mathematics form a unified whole. The two volumes cover the following topics:

Volume III: Gauge Theory
Part I: The Euclidean Manifold as a Paradigm
Chapter 1: The Euclidean Space E3 (Hilbert Space and Lie Algebra Structure)
Chapter 2: Algebras and Duality (Tensor Algebra, Grassmann Algebra, Clifford
Algebra, Lie Algebra)
Chapter 3: Representations of Symmetries in Mathematics and Physics
Chapter 4: The Euclidean Manifold E3
Chapter 5: The Lie Group U (1) as a Paradigm in Harmonic Analysis and Geometry
Chapter 6: Infinitesimal Rotations and Constraints in Physics
Chapter 7: Rotations, Quaternions, the Universal Covering Group, and the Electron Spin
Chapter 8: Changing Observers – A Glance at Invariant Theory Based on the
Principle of the Correct Index Picture
Chapter 9: Applications of Invariant Theory to the Rotation Group
Chapter 10: Temperature Fields on the Euclidean Manifold E3
Chapter 11: Velocity Vector Fields on the Euclidean Manifold E3
Chapter 12: Covector Fields on the Euclidean Manifold E3 and Cartan’s Exterior
Differential – the Beauty of Differential Forms
Part II: Ariadne’s Thread in Gauge Theory
Chapter 13: The Commutative Weyl U (1)-Gauge Theory and the Electromagnetic
Field
Chapter 14: Symmetry Breaking
Chapter 15: The Noncommutative Yang–Mills SU (N )-Gauge Theory
Chapter 16: Cocycles and Observers
Chapter 17: The Axiomatic Geometric Approach to Vector Bundles and Principal
Bundles
Part III: Einstein’s Theory of Special Relativity

Chapter 18: Inertial Systems and Einstein’s Principle of Special Relativity
Chapter 19: The Relativistic Invariance of the Maxwell Equations
Chapter 20: The Relativistic Invariance of the Dirac Equations and the Electron
Spin


Preface

XI

Part IV: Ariadne’s Thread in Cohomology
Chapter 21: Exact Sequences
Chapter 22: Electrical Circuits as a Paradigm in Homology and Cohomology
Chapter 23: The Electromagnetic Field and the de Rham Cohomology.

Volume IV: Quantum Mathematics
Part I: The Hydrogen Atom as a Paradigm
Chapter 1: The Non-Relativistic Hydrogen Atom via Lie Algebra, Gauss’s Hypergeometric Functions, von Neuman’s Functional Analytic Approach, the Weyl–
Kodaira Theory, Gelfand’s Generalized Eigenfunctions, and Supersymmetry
Chapter 2: The Dirac Equation and the Relativistic Hydrogen Atom via the Clifford Algebra of the Minkowski Space
Part II: The Four Fundamental Forces in the Universe
Chapter 3: Relativistic Invariance and the Energy–Momentum Tensor in Classical
Field Theories
Chapter 4: The Standard Model for Electroweak and Strong Interaction in Particle
Physics
Chapter 5: Gravitation, Einstein’s Theory of General Relativity, and the Standard
Model in Cosmology
Part III: Lowest-Order Radiative Corrections in Quantum Electrodynamics (QED)
Chapter 6: Dimensional Regularization for the Feynman Propagators in QED
(Quantum Electrodynamics)

Chapter 7: The Electron in an External Electromagnetic Field (Renormalization
of Electron Mass and Electron Charge)
Chapter 8: The Lamb Shift
Part IV: Conformal Symmetry
Chapter 9: Conformal Transformations According to Gauss, Riemann, and Lichtenstein
Chapter 10: Compact Riemann Surfaces
Chapter 11: Minimal Surfaces
Chapter 12: Strings and the Graviton
Chapter 13: Complex Function Theory and Conformal Quantum Field Theory
Part V: Models in Quantum Field Theory
Part VI: Distributions and the Epstein–Glaser Approach to Perturbative Quantum
Field Theory
Part VII: Nets of Operator Algebras and the Haag–Kastler Approach to Quantum
Field Theory
Part VIII: Symmetry and Quantization – the BRST Approach to Quantum Field
Theory
Part IX: Topology, Quantization, and the Global Structure of Physical Fields
Part X: Quantum Information.


XII

Preface

Readers who want to understand modern differential geometry and modern physics
as quickly as possible should glance at the Prologue of the present volume and at
Chaps. 13 through 17 on Ariadne’s thread in gauge theory.
Cohomology plays a fundamental role in modern mathematics and physics.
It turns out that cohomology and homology have their roots in the rules for
electrical circuits formulated by Kirchhoff in 1847.

This helps to explain why the Maxwell equations in electrodynamics are closely
related to cohomology, namely, de Rham cohomology based on Cartan’s calculus
for differential forms and the corresponding Hodge duality on the Minkowski space.
Since the Standard Model in particle physics is obtained from the Maxwell equations
by replacing the commutative gauge group U (1) with the noncommutative gauge
group U (1) × SU (2) × SU (3), it should come as no great surprise that de Rham
cohomology also plays a key role in the Standard Model in particle physics via
the theory of characteristic classes (e.g., Chern classes which were invented by
Shing-Shen Chern in 1945 in order to generalize the Gauss–Bonnet theorem for
two-dimensional manifolds to higher dimensions).
It is our goal to show that the gauge-theoretical formulation of modern physics
is closely related to important long-term developments in mathematics pioneered by
Gauss, Riemann, Poincar´e and Hilbert, as well as Grassmann, Lie, Klein, Cayley,
´
Elie
Cartan and Weyl. The prototype of a gauge theory in physics is Maxwell’s
theory of electromagnetism. The Standard Model in particle physics is based on the
principle of local symmetry. In contrast to Maxwell’s theory of electromagnetism,
the gauge group of the Standard Model in particle physics is a noncommutative
Lie group. This generates additional interaction forces which are mathematically
described by Lie brackets.
We also emphasize the methods of invariant theory. In terms of physics, different observers measure different values in their experiments. However, physics does
not depend on the choice of observers. Therefore, one needs both an invariant approach and the passage to coordinate systems which correspond to the observers, as
emphasized by Einstein in the theory of general relativity and by Dirac in quantum
mechanics. The appropriate mathematical tool is provided by invariant theory.
Acknowledgments. In 2003, Jă
urgen Tolksdorf initiated a series of four International Workshops on the state of the art in quantum field theory and the search
for a unified theory concerning the four fundamental interactions in nature. I am
very grateful to Felix Finster, Olaf Mă
uller, Marc Nardmann, and Jă

urgen Tolksdorf
for organizing the workshop Quantum Field Theory and Gravity, Regensburg, 2010.
The following three volumes contain survey articles written by leading experts:
F. Finster, O. Mă
uller, M. Nardmann, J. Tolksdorf, and E. Zeidler (Eds.),
Quantum Field Theory and Gravity: Conceptual and Mathematical Advances in the Search for a Unied Framework, Birkhă
auser, Basel (to appear).
B. Fauser, J. Tolksdorf, and E. Zeidler (Eds.), Quantum Field Theory
Competitive Methods, Birkhă
auser, Basel, 2008.
B. Fauser, J. Tolksdorf, and E. Zeidler (Eds.), Quantum Gravitation:
Mathematical Models and Experimental Bounds, Birkhă
auser, Basel, 2006.
These three volumes are recommended as supplements to the material contained
in the present monograph. For stimulating discussions and guidance, I would like
to thank Sergio Albeverio, Christian Bă
ar, Helga Baum, Christian Brouder, Romeo
Brunetti, Detlef Buchholz, Christopher Deninger, Michael Dă
utsch, Claudia Eberlein, Kurusch Ebrahimi-Fard, William Farris, Bertfried Fauser, Joel Feldman, Chris
Fewster, Felix Finster, Christian Fleischhack, Hans Fă
ollmer, Alessandra Frabetti,


Preface

XIII

Klaus Fredenhagen, Harald Fritzsch, Jă
org Fră
ohlich, Peter Gilkey, Jose GraciaBonda, Ivor Grattan-Guiness, Harald Grosse, Stefan Hollands, Arthur Jae, Jă

urgen
Jost, Hans Kastrup, Jerzy Kijowski, Klaus Kirsten, Dirk Kreimer, Elliott Lieb,
Jan Louis, Dieter Lă
ust, Kishore Marathe, Matilde Marcolli, Michael Oberguggenberger, Robert Oeckl, Norbert Ortner, Frederic Patras, Frank Pfă
ae, Klaus Rehren,
Vincent Rivasseau, Hartmann Ră
omer, Gerd Rudolph, Manfred Salmhofer, Erhard
Scholz, Klaus Sibold, Harold Steinacker, Jă
urgen Tolksdorf, Armin Uhlmann, Rainer
Verch, Peter Wagner, Raimar Wulkenhaar, Hans Wußing, and Jacob Yngvason.
I am very grateful to the staff of the Max-Planck Institute for Mathematics in
the Sciences, Leipzig, for kindly supporting me in various aspects. In particular, I
would like to cordially thank my secretary Regine Lă
ubke (for her invaluable support), Kerstin Fă
olting (for her meticulous production of the graphics), Katarzyna
Baier, Ingo Bră
uggemann, and the library team (for their perfect cooperation), Oliver
Heller, Rainer Kleinrensing, and the computer team (for their continued support),
as well as, Michaela Krieger–Hauwede (for patiently answering my LATEX questions). Thanks also to the staff of the Springer Publishing House in Heidelberg,
Ruth Allewelt, Joachim Heinze, and Martin Peters, for their harmonious collaboration. The LATEX file was written by myself; therefore, any errors should be laid at
my door.
This volume is gratefully dedicated to Professor Krzysztof Maurin in Warsaw.
As a young man, I learned from him that mathematics, physics, and philosophy
form a unity; they represent marvellous tools for the human intellect in order to
approximate step by step the better understanding of the real world, and they have
to serve the well-being of human society.
My hometown, Leipzig, is full of the music composed by Johann Sebastian Bach,
who worked in Leipzig’s Saint Thomas church from 1723 until his death in 1750.
In the Preface of his book Electroweak and Strong Interaction: An Introduction to
Theoretical Particle Physics, Springer, Berlin, 1996, my colleague Florian Scheck

from Mainz University adapted Bach’s dedication to his “Well-Tempered Clavier”
from 1722:
Written and composed for the benefit and use of young physicists and for
the particular diversion of those already advanced in this study.
I would like to use the same quotation, replacing ‘physicists’ with ‘mathematicians
and physicists.’
I hope that readers will get a feel for the unity of mathematics and the unity
of science. In 1915, John Dewey wrote in his book The School and Society, The
University of Chicago Press, Chicago, Illinois: “We do not have a series of stratified
earths, one of which is mathematical, another physical, another historical, and so
on. We should not be able to live very long in any one taken by itself. We live in
a world where all sides are bound together; all studies grow out of relations in the
one great common world.”
Leipzig, Spring 2011

Eberhard Zeidler



Contents

Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Part I. The Euclidean Manifold as a Paradigm
1.

The Euclidean Space E3 (Hilbert Space and Lie Algebra
Structure) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1 A Glance at History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Algebraic Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Symmetrization and Antisymmetrization . . . . . . . . . .
1.2.2 Cramer’s Rule for Systems of Linear Equations . . . .
1.2.3 Determinants and the Inverse Matrix . . . . . . . . . . . . .
1.2.4 The Hilbert Space Structure . . . . . . . . . . . . . . . . . . . . .
1.2.5 Orthogonality and the Dirac Calculus . . . . . . . . . . . .
1.2.6 The Lie Algebra Structure . . . . . . . . . . . . . . . . . . . . . .
1.2.7 The Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.8 The Volume Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.9 Grassmann’s Alternating Product . . . . . . . . . . . . . . . .
1.2.10 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 The Skew-Field H of Quaternions . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 The Field C of Complex Numbers . . . . . . . . . . . . . . . .
1.3.2 The Galois Group Gal(C|R) and Galois Theory . . . .
1.3.3 A Glance at the History of Hamilton’s Quaternions .
1.3.4 Pauli’s Spin Matrices and the Lie Algebras su(2)
and sl(2, C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.5 Cayley’s Matrix Approach to Quaternions . . . . . . . . .
1.3.6 The Unit Sphere U (1, H) and the Electroweak Gauge
Group SU (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.7 The Four-Dimensional Extension of the Euclidean
Space E3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.8 Hamilton’s Nabla Operator . . . . . . . . . . . . . . . . . . . . . .
1.3.9 The Indefinite Hilbert Space H and the Minkowski
Space M4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Riesz Duality between Vectors and Covectors . . . . . . . . . . . . .

69
69

71
72
73
75
78
81
82
85
85
86
87
89
90
91
94
99
101
102
103
104
104
104
XV


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Contents

1.5

1.6
2.

The Heisenberg Group, the Heisenberg Algebra, and
Quantum Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
The Heisenberg Group Bundle and Gauge Transformations . 112

Algebras and Duality (Tensor Algebra, Grassmann
Algebra, Clifford Algebra, Lie Algebra) . . . . . . . . . . . . . . . . . .
2.1 Multilinear Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 The Graded Algebra of Polynomials . . . . . . . . . . . . . .
2.1.2 Products of Multilinear Functionals . . . . . . . . . . . . . .
2.1.3 Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Grassmann Algebra (Alternating Algebra) . . . . . . . .
2.1.5 Symmetric Tensor Algebra . . . . . . . . . . . . . . . . . . . . . .
2.1.6 The Universal Property of the Tensor Product . . . . .
2.1.7 Diagram Chasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The Clifford Algebra (E1 ) of the One-Dimensional
Euclidean Space E1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Algebras of the Two-Dimensional Euclidean Space E2 . . . . .
2.3.1 The Clifford Algebra (E2 ) and Quaternions . . . . . .
2.3.2 The Cauchy–Riemann Differential Equations
in Complex Function Theory . . . . . . . . . . . . . . . . . . . .
2.3.3 The Grassmann Algebra (E2 ) . . . . . . . . . . . . . . . . . .
2.3.4 The Grassmann Algebra (E2d ) . . . . . . . . . . . . . . . . . .
2.3.5 The Symplectic Structure of E2 . . . . . . . . . . . . . . . . . .
2.3.6 The Tensor Algebra (E2 ) . . . . . . . . . . . . . . . . . . . . .
2.3.7 The Tensor Algebra (E2d ) . . . . . . . . . . . . . . . . . . . . .
2.4 Algebras of the Three-Dimensional Euclidean Space E3 . . . .
2.4.1 Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4.2 Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Grassmann Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 Clifford Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Algebras of the Dual Euclidean Space E3d . . . . . . . . . . . . . . . .
2.5.1 Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Grassmann Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 The Mixed Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 The Hilbert Space Structure of the Grassmann Algebra
(Hodge Duality) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.1 The Hilbert Space (E3 ) . . . . . . . . . . . . . . . . . . . . . . .
2.7.2 The Hilbert Space (E3d ) . . . . . . . . . . . . . . . . . . . . . . .
2.7.3 Multivectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 The Cliord Structure of the Grassmann Algebra
(ExteriorInterior Kă
ahler Algebra) . . . . . . . . . . . . . . . . . . . . . .
2.8.1 The Kă
ahler Algebra (E3 ) . . . . . . . . . . . . . . . . . . . . .
2.8.2 The Kă
ahler Algebra (E3d )∨ . . . . . . . . . . . . . . . . . . . .
∗
2.9 The C -Algebra End(E3 ) of the Euclidean Space . . . . . . . . .
2.10 Linear Operator Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115
115
115
118
120
121
121

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124
126
127
128
129
131
132
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133
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133
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134
134
135
135
135
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138
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144
144
145
145
146



Contents

2.10.1 The Prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10.2 The Grassmann Theorem . . . . . . . . . . . . . . . . . . . . . . .
2.10.3 The Superposition Principle . . . . . . . . . . . . . . . . . . . . .
2.10.4 Duality and the Fredholm Alternative . . . . . . . . . . . .
2.10.5 The Language of Matrices . . . . . . . . . . . . . . . . . . . . . . .
2.10.6 The Gaussian Elimination Method . . . . . . . . . . . . . . .
2.11 Changing the Basis and the Cobasis . . . . . . . . . . . . . . . . . . . . .
2.11.1 Similarity of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11.2 Volume Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11.3 The Determinant of a Linear Operator . . . . . . . . . . . .
2.11.4 The Reciprocal Basis in Crystallography . . . . . . . . . .
2.11.5 Dual Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11.6 The Trace of a Linear Operator . . . . . . . . . . . . . . . . . .
2.11.7 The Dirac Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.12 The Strategy of Quotient Algebras and Universal Properties
2.13 A Glance at Division Algebras . . . . . . . . . . . . . . . . . . . . . . . . . .
2.13.1 From Real Numbers to Cayley’s Octonions . . . . . . . .
2.13.2 Uniqueness Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.13.3 The Fundamental Dimension Theorem . . . . . . . . . . . .
3.

Representations of Symmetries in Mathematics and
Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 The Symmetric Group as a Prototype . . . . . . . . . . . . . . . . . . .
3.2 Incredible Cancellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 The Symmetry Strategy in Mathematics and Physics . . . . . .
3.4 Lie Groups and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5 Basic Notions of Representation Theory . . . . . . . . . . . . . . . . .
3.5.1 Linear Representations of Groups . . . . . . . . . . . . . . . .
3.5.2 Linear Representations of Lie Algebras . . . . . . . . . . . .
3.6 The Reflection Group Z2 as a Prototype . . . . . . . . . . . . . . . . .
3.6.1 Representations of Z2 . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.2 Parity of Elementary Particles . . . . . . . . . . . . . . . . . . .
3.6.3 Reflections and Chirality in Nature . . . . . . . . . . . . . . .
3.6.4 Parity Violation in Weak Interaction . . . . . . . . . . . . .
3.6.5 Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Permutation of Elementary Particles . . . . . . . . . . . . . . . . . . . .
3.7.1 The Principle of Indistinguishability of Quantum
Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7.2 The Pauli Exclusion Principle . . . . . . . . . . . . . . . . . . .
3.7.3 Entangled Quantum States . . . . . . . . . . . . . . . . . . . . . .
3.8 The Diagonalization of Linear Operators . . . . . . . . . . . . . . . . .
3.8.1 The Theorem of Principal Axes in Geometry and
in Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.2 The Schur Lemma in Linear Representation Theory
3.8.3 The Jordan Normal Form of Linear Operators . . . . .

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165

166
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XVIII Contents

3.8.4

3.9
3.10
3.11
3.12
3.13

3.14

3.15

3.16

3.17

3.18

The Standard Maximal Torus of the Lie Group SU (n)
and the Standard Cartan Subalgebra
of the Lie Algebra su(n) . . . . . . . . . . . . . . . . . . . . . . . .
3.8.5 Eigenvalues and the Operator Strategy for Lie

Algebras (Adjoint Representation) . . . . . . . . . . . . . . .
The Action of a Group on a Physical State Space, Orbits,
and Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Intrinsic Symmetry of a Group . . . . . . . . . . . . . . . . . . . . .
Linear Representations of Finite Groups and the Hilbert
Space of Functions on the Group . . . . . . . . . . . . . . . . . . . . . . . .
The Tensor Product of Representations and Characters . . . .
Applications to the Symmetric Group Sym(n) . . . . . . . . . . . .
3.13.1 The Characters of the Symmetric Group Sym(2) . . .
3.13.2 The Characters of the Symmetric Group Sym(3) . . .
3.13.3 Partitions and Young Frames . . . . . . . . . . . . . . . . . . . .
3.13.4 Young Tableaux and the Construction of a Complete
System of Irreducible Representations . . . . . . . . . . . .
Application to the Standard Model in Elementary Particle
Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.14.1 Quarks and Baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.14.2 Antiquarks and Mesons . . . . . . . . . . . . . . . . . . . . . . . . .
3.14.3 The Method of Highest Weight for Composed
Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.14.4 The Pauli Exclusion Principle and the Color
of Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Complexification of Lie Algebras . . . . . . . . . . . . . . . . . . . .
3.15.1 Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.15.2 The Complex Lie Algebra slC (3, C) and Root
Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.15.3 Representations of the Complex Lie Algebra slC (3, C)
and Weight Functionals . . . . . . . . . . . . . . . . . . . . . . . . .
Classification of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.16.1 Simplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.16.2 Direct Product and Semisimplicity . . . . . . . . . . . . . . .

3.16.3 Solvablity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.16.4 Semidirect Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Classification of Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.17.1 The Classification of Complex Simple Lie Algebras .
3.17.2 Semisimple Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . .
3.17.3 Solvability and the Heisenberg Algebra in Quantum
Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.17.4 Semidirect Product and the Levi Decomposition . . .
3.17.5 The Casimir Operators . . . . . . . . . . . . . . . . . . . . . . . . .
Symmetric and Antisymmetric Functions . . . . . . . . . . . . . . . .

204
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225
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Contents

3.18.1
3.18.2
3.18.3
3.18.4
3.18.5
3.18.6

3.19

3.20
3.21
3.22


3.23

3.24
3.25

3.26
3.27
4.

The
4.1
4.2
4.3
4.4
4.5

Symmetrization and Antisymmetrization . . . . . . . . . .
Elementary Symmetric Polynomials . . . . . . . . . . . . . .
Power Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Completely Symmetric Polynomials . . . . . . . . . . . . . .
Symmetric Schur Polynomials . . . . . . . . . . . . . . . . . . .
Raising Operators and the Creation and
Annihilation of Particles . . . . . . . . . . . . . . . . . . . . . . . .
Formal Power Series Expansions and Generating Functions .
3.19.1 The Fundamental Frobenius Character Formula . . . .
3.19.2 The Pfaffian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Frobenius Algebras and Frobenius Manifolds . . . . . . . . . . . . .
Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.22.1 Graduation in Nature . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.22.2 General Strategy in Mathematics . . . . . . . . . . . . . . . .
3.22.3 The Super Lie Algebra of the Euclidean Space . . . . .
Artin’s Braid Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.23.1 The Braid Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.23.2 The Yang–Baxter Equation . . . . . . . . . . . . . . . . . . . . .
3.23.3 The Geometric Meaning of the Braid Group . . . . . . .
3.23.4 The Topology of the State Space of n Indistinguishable Particles in the Plane . . . . . . . . . . . . . . . . . . . . . .
The HOMFLY Polynomials in Knot Theory . . . . . . . . . . . . . .
Quantum Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.25.1 Quantum Mechanics as a Deformation . . . . . . . . . . . .
3.25.2 Manin’s Quantum Planes R2q and C2q . . . . . . . . . . . . .
3.25.3 The Coordinate Algebra of the Lie Group SL(2, C) .
3.25.4 The Quantum Group SLq (2, C) . . . . . . . . . . . . . . . . . .
3.25.5 The Quantum Algebra slq (2, C) . . . . . . . . . . . . . . . . . .
3.25.6 The Coaction of the Quantum Group SLq (2, C)
on the Quantum Plane C2q . . . . . . . . . . . . . . . . . . . . . . .
3.25.7 Noncommutative Euclidean Geometry and Quantum
Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additive Groups, Betti Numbers, Torsion Coefficients, and
Homological Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lattices and Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Euclidean Manifold E3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Velocity Vectors and the Tangent Space . . . . . . . . . . . . . . . . .
Duality and Cotangent Spaces . . . . . . . . . . . . . . . . . . . . . . . . . .
Parallel Transport and Acceleration . . . . . . . . . . . . . . . . . . . . .
Newton’s Law of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bundles Over the Euclidean Manifold . . . . . . . . . . . . . . . . . . .
4.5.1 The Tangent Bundle and Velocity Vector Fields . . . .
4.5.2 The Cotangent Bundle and Covector Fields . . . . . . .


XIX

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XX

Contents

4.6

5.

6.

4.5.3 Tensor Bundles and Tensor Fields . . . . . . . . . . . . . . . .
4.5.4 The Frame Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Newton and Leibniz . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.2 The Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.3 The Dirac Delta Function and Laurent Schwartz’s
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6.4 The Algebraization of the Calculus . . . . . . . . . . . . . . .
4.6.5 Formal Power Series Expansions and the Ritt
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.6 Differential Rings and Derivations . . . . . . . . . . . . . . . .
4.6.7 The p-adic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.8 The Local–Global Principle in Mathematics . . . . . . .
4.6.9 The Global Adelic Ring . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.10 Solenoids, Foliations, and Chaotic Dynamical
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.11 Period Three Implies Chaos . . . . . . . . . . . . . . . . . . . . .
4.6.12 Differential Calculi, Noncommutative Geometry, and
the Standard Model in Particle Physics . . . . . . . . . . .
4.6.13 BRST-Symmetry, Cohomology, and the Quantization of Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.14 Itˆ
o’s Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . .

326
327
327
327
329
330
330
331
331
332
336
337
339
345

346
347
348

The Lie Group U (1) as a Paradigm in Harmonic Analysis
and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Linearization and the Lie Algebra u(1) . . . . . . . . . . . . . . . . . .
5.2 The Universal Covering Group of U (1) . . . . . . . . . . . . . . . . . .
5.3 Left-Invariant Velocity Vector Fields on U (1) . . . . . . . . . . . . .
5.3.1 The Maurer–Cartan Form of U (1) . . . . . . . . . . . . . . . .
5.3.2 The Maurer–Cartan Structural Equation . . . . . . . . . .
5.4 The Riemannian Manifold U (1) and the Haar Measure . . . .
5.5 The Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 The Hilbert Space L2 (U (1)) . . . . . . . . . . . . . . . . . . . . .
5.5.2 Pseudo–Differential Operators . . . . . . . . . . . . . . . . . . .
5.5.3 The Sobolev Space W2m (U (1)) . . . . . . . . . . . . . . . . . . .
5.6 The Group of Motions on the Gaussian Plane . . . . . . . . . . . .
5.7 Rotations of the Euclidean Plane . . . . . . . . . . . . . . . . . . . . . . .
5.8 Pontryagin Duality for U (1) and Quantum Groups . . . . . . . .

355
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356
356
357
358
358
359
359
360

361
361
362
369

Infinitesimal Rotations and Constraints in Physics . . . . . . .
6.1 The Group U (E3 ) of Unitary Transformations . . . . . . . . . . . .
6.2 Euler’s Rotation Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 The Lie Algebra of Infinitesimal Rotations . . . . . . . . . . . . . . .
6.4 Constraints in Classical Physics . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1 Archimedes’ Lever Principle . . . . . . . . . . . . . . . . . . . . .

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373
374
375
375


Contents

6.4.2
6.4.3
6.4.4

6.5

6.6


6.7

6.8
7.

d’Alembert’s Principle of Virtual Power . . . . . . . . . . .
d’Alembert’s Principle of Virtual Work . . . . . . . . . . .
The Gaussian Principle of Least Constraint and
Constraining Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.5 Manifolds and Lagrange’s Variational Principle . . . .
6.4.6 The Method of Perturbation Theory . . . . . . . . . . . . . .
6.4.7 Further Reading on Perturbation Theory and
its Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Application to the Motion of a Rigid Body . . . . . . . . . . . . . . .
6.5.1 The Center of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.2 Moving Orthonormal Frames and Infinitesimal
Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.3 Kinetic Energy and the Inertia Tensor . . . . . . . . . . . .
6.5.4 The Equations of Motion – the Existence and
Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.5 Euler’s Equation of the Spinning Top . . . . . . . . . . . . .
6.5.6 Equilibrium States and Torque . . . . . . . . . . . . . . . . . .
6.5.7 The Principal Bundle R3 × SO(3) – the Position
Space of a Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . .
A Glance at Constraints in Quantum Field Theory . . . . . . . .
6.6.1 Gauge Transformations and Virtual Degrees
of Freedom in Gauge Theory . . . . . . . . . . . . . . . . . . . .
6.6.2 Elimination of Unphysical States (Ghosts) . . . . . . . . .
6.6.3 Degenerate Minimum Problems . . . . . . . . . . . . . . . . . .
6.6.4 Variation of the Action Functional . . . . . . . . . . . . . . .

6.6.5 Degenerate Lagrangian and Constraints . . . . . . . . . . .
6.6.6 Degenerate Legendre Transformation . . . . . . . . . . . . .
6.6.7 Global and Local Symmetries . . . . . . . . . . . . . . . . . . . .
6.6.8 Quantum Symmetries and Anomalies . . . . . . . . . . . . .
Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7.1 Topological Constraints in Maxwell’s Theory
of Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7.2 Constraints in Einstein’s Theory of General
Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7.3 Hilbert’s Algebraic Theory of Relations (Syzygies) .
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Rotations, Quaternions, the Universal Covering Group,
and the Electron Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 Quaternions and the Cayley–Hamilton Rotation Formula . .
7.2 The Universal Covering Group SU (2) . . . . . . . . . . . . . . . . . . .
7.3 Irreducible Unitary Representations of the Group SU (2) and
the Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.1 The Spin Quantum Numbers . . . . . . . . . . . . . . . . . . . .
7.3.2 The Addition Theorem for the Spin . . . . . . . . . . . . . .

XXI

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384
385
388

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391
393
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399
400
401
404
408
408
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414
417
417
417
417
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426
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XXII

Contents

7.4
8.

7.3.3 The Model of Homogeneous Polynomials . . . . . . . . . . 435
7.3.4 The Clebsch–Gordan Coefficients . . . . . . . . . . . . . . . . . 436
Heisenberg’s Isospin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

Changing Observers – A Glance at Invariant Theory
Based on the Principle of the Correct Index Picture . . . . .
8.1 A Glance at the History of Invariant Theory . . . . . . . . . . . . .
8.2 The Basic Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 The Mnemonic Principle of the Correct Index Picture . . . . .
8.4 Real-Valued Physical Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.1 The Chain Rule and the Key Duality Relation . . . . .
8.4.2 Linear Differential Operators . . . . . . . . . . . . . . . . . . . .
8.4.3 Duality and Differentials . . . . . . . . . . . . . . . . . . . . . . . .
8.4.4 Admissible Systems of Observers . . . . . . . . . . . . . . . . .
8.4.5 Tensorial Families and the Construction of Invariants
via the Basic Trick of Index Killing . . . . . . . . . . . . . . .
8.4.6 Orientation, Pseudo-Tensorial Families, and
the Levi-Civita Duality . . . . . . . . . . . . . . . . . . . . . . . . .
8.5 Differential Forms (Exterior Product) . . . . . . . . . . . . . . . . . . .
8.5.1 Cartan Families and the Cartan Differential . . . . . . .
8.5.2 Hodge Duality, the Hodge Codifferential, and
the Laplacian (Hodge’s Star Operator) . . . . . . . . . . . .
8.6 The Kă

ahlerCliord Calculus and the Dirac Operator
(Interior Product) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6.1 The Exterior Differential Algebra . . . . . . . . . . . . . . . .
8.6.2 The Interior Differential Algebra . . . . . . . . . . . . . . . . .
8.6.3 Kă
ahler Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6.4 Applications to Fundamental Differential Equations
in Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6.5 The Potential Equation and the Importance
of the de Rham Cohomology . . . . . . . . . . . . . . . . . . . .
8.6.6 Tensorial Differential Forms . . . . . . . . . . . . . . . . . . . . .
8.7 Integrals over Differential Forms . . . . . . . . . . . . . . . . . . . . . . . .
8.8 Derivatives of Tensorial Families . . . . . . . . . . . . . . . . . . . . . . . .
8.8.1 The Lie Algebra of Linear Differential Operators and
the Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.8.2 The Inverse Index Principle . . . . . . . . . . . . . . . . . . . . .
8.8.3 The Covariant Derivative (Weyl’s Affine Connection) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.9 The Riemann–Weyl Curvature Tensor . . . . . . . . . . . . . . . . . . .
8.9.1 Second-Order Covariant Partial Derivatives . . . . . . . .
8.9.2 Local Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.9.3 The Method of Differential Forms (Cartan’s Structural Equations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.9.4 The Operator Method . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents XXIII

8.10 The Riemann–Christoffel Curvature Tensor . . . . . . . . . . . . . .
8.10.1 The Levi-Civita Metric Connection . . . . . . . . . . . . . . .

8.10.2 Levi-Civita’s Parallel Transport . . . . . . . . . . . . . . . . . .
8.10.3 Symmetry Properties of the Riemann–Christoffel
Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.10.4 The Ricci Curvature Tensor and the Einstein Tensor
8.10.5 The Conformal Weyl Curvature Tensor . . . . . . . . . . .
8.10.6 The Hodge Codifferential and the Covariant Partial
Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.10.7 The Weitzenbă
ock Formula for the Hodge Laplacian .
8.10.8 The One-Dimensional σ-Model and Affine Geodesics
8.11 The Beauty of Connection-Free Derivatives . . . . . . . . . . . . . . .
8.11.1 The Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.11.2 The Cartan Derivative . . . . . . . . . . . . . . . . . . . . . . . . . .
8.11.3 The Weyl Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.12 Global Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.13 Summary of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.14 Two Strategies in Invariant Theory . . . . . . . . . . . . . . . . . . . . .
8.15 Intrinsic Tangent Vectors and Derivations . . . . . . . . . . . . . . . .
8.16 Further Reading on Symmetry and Invariants . . . . . . . . . . . .
9.

Applications of Invariant Theory to the Rotation Group .
9.1 The Method of Orthonormal Frames on the Euclidean
Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.1 Hamilton’s Quaternionic Analysis . . . . . . . . . . . . . . . .
9.1.2 Transformation of Orthonormal Frames . . . . . . . . . . .
9.1.3 The Coordinate-Dependent Approach (SO(3)-Tensor
Calculus) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.4 The Coordinate-Free Approach . . . . . . . . . . . . . . . . . .
9.1.5 Hamilton’s Nabla Calculus . . . . . . . . . . . . . . . . . . . . . .

9.1.6 Rotations and Cauchy’s Invariant Functions . . . . . . .
9.2 Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.1 Local Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.2 The Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.3 The Volume Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.4 Special Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3 The Index Principle of Mathematical Physics . . . . . . . . . . . . .
9.3.1 The Basic Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.2 Applications to Vector Analysis . . . . . . . . . . . . . . . . . .
9.4 The Euclidean Connection and Gauge Theory . . . . . . . . . . . .
9.4.1 Covariant Partial Derivative . . . . . . . . . . . . . . . . . . . . .
9.4.2 Curves of Least Kinectic Energy (Affine Geodesics) .
9.4.3 Curves of Minimal Length . . . . . . . . . . . . . . . . . . . . . . .
9.4.4 The Gauss Equations of Moving Frames . . . . . . . . . .

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XXIV Contents

9.4.5


9.5

9.6

Parallel Transport of a Velocity Vector and Cartan’s
Propagator Equation . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.6 The Dual Cartan Equations of Moving Frames . . . . .
9.4.7 Global Parallel Transport on Lie Groups and
the Maurer–Cartan Form . . . . . . . . . . . . . . . . . . . . . . .
9.4.8 Cartan’s Global Connection Form
on the Frame Bundle of the Euclidean Manifold . . . .
9.4.9 The Relation to Gauge Theory . . . . . . . . . . . . . . . . . .
9.4.10 The Reduction of the Frame Bundle
to the Orthonormal Frame Bundle . . . . . . . . . . . . . . .
The Sphere as a Paradigm in Riemannian Geometry and
Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.5.1 The Newtonian Equation of Motion and Levi-Civita’s
Parallel Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.5.2 Geodesic Triangles and the Gaussian Curvature . . . .
9.5.3 Geodesic Circles and the Gaussian Curvature . . . . . .
9.5.4 The Spherical Pendulum . . . . . . . . . . . . . . . . . . . . . . . .
9.5.5 Geodesics and Gauge Transformations . . . . . . . . . . . .
9.5.6 The Local Hilbert Space Structure . . . . . . . . . . . . . . .
9.5.7 The Almost Complex Structure . . . . . . . . . . . . . . . . . .
9.5.8 The Levi-Civita Connection on the Tangent Bundle
and the Riemann Curvature Tensor . . . . . . . . . . . . . .
9.5.9 The Components of the Riemann Curvature Tensor
and Gauge Fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.5.10 Computing the Riemann Curvature Operator via
Parallel Transport Along Loops . . . . . . . . . . . . . . . . . .

9.5.11 The Connection on the Frame Bundle and Parallel
Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.5.12 Poincar´e’s Topological No-Go Theorem for Velocity
Vector Fields on a Sphere . . . . . . . . . . . . . . . . . . . . . . .
Gauss’ Theorema Egregium . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.6.1 The Natural Basis and Cobasis . . . . . . . . . . . . . . . . . .
9.6.2 Intrinsic Metric Properties . . . . . . . . . . . . . . . . . . . . . .
9.6.3 The Extrinsic Definition of the Gaussian Curvature
9.6.4 The Gauss–Weingarten Equations for Moving
Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.6.5 The Integrability Conditions and the Riemann
Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.6.6 The Intrinsic Characterization of the Gaussian
Curvature (Theorema Egregium) . . . . . . . . . . . . . . . . .
9.6.7 Differential Invariants and the Existence and
Uniqueness Theorem of Classical Surface Theory . . .
9.6.8 Gauss’ Theorema Elegantissimum and the
Gauss–Bonnet Theorem . . . . . . . . . . . . . . . . . . . . . . . . .

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