An Introduction to GEOMETRICAL PHYSICS docx

An Introduction to

GEOMETRICAL PHYSICS

R. Aldrovandi & J.G. Pereira

Instituto de F´ısica Te´orica

State University of S˜ao Paulo – UNESP

S˜ao Paulo — Brazil

To our parents

Nice, Dina, Jos´e and Tito

i

ii

PREAMBLE: SPACE AND GEOMETRY

What stuff ’tis made of, whereof it is born,

I am to learn.

Merchant of Venice

The simplest geometrical setting used — consciously or not — by physi￾cists in their everyday work is the 3-dimensional euclidean space E

3

. It con￾sists of the set R

3 of ordered triples of real numbers such as p = (p

1

, p2

, p3

), q

= (q

1

, q2

, q3

), etc, and is endowed with a very special characteristic, a metric

defined by the distance function

d(p, q) = "X

3

i=1

(p

i − q

i

)

2

#1/2

.

It is the space of ordinary human experience and the starting point of our

geometric intuition. Studied for two-and-a-half millenia, it has been the

object of celebrated controversies, the most famous concerning the minimum

number of properties necessary to define it completely.

From Aristotle to Newton, through Galileo and Descartes, the very word

space has been reserved to E

3

. Only in the 19-th century has it become clear

that other, different spaces could be thought of, and mathematicians have

since greatly amused themselves by inventing all kinds of them. For physi￾cists, the age-long debate shifted to another question: how can we recognize,

amongst such innumerable possible spaces, that real space chosen by Nature

as the stage-set of its processes? For example, suppose the space of our ev￾eryday experience consists of the same set R

3 of triples above, but with a

different distance function, such as

d(p, q) = X

3

i=1

|p

i − q

i

|.

This would define a different metric space, in principle as good as that

given above. Were it only a matter of principle, it would be as good as

iii

iv

any other space given by any distance function with R

3 as set point. It so

happens, however, that Nature has chosen the former and not the latter space

for us to live in. To know which one is the real space is not a simple question

of principle — something else is needed. What else? The answer may seem

rather trivial in the case of our home space, though less so in other spaces

singled out by Nature in the many different situations which are objects of

physical study. It was given by Riemann in his famous Inaugural Address1

:

“ ... those properties which distinguish Space from other con￾ceivable triply extended quantities can only be deduced from expe￾rience.”

Thus, from experience! It is experiment which tells us in which space we

actually live in. When we measure distances we find them to be independent

of the direction of the straight lines joining the points. And this isotropy

property rules out the second proposed distance function, while admitting

the metric of the euclidean space.

In reality, Riemann’s statement implies an epistemological limitation: it

will never be possible to ascertain exactly which space is the real one. Other

isotropic distance functions are, in principle, admissible and more experi￾ments are necessary to decide between them. In Riemann’s time already

other geometries were known (those found by Lobachevsky and Boliyai) that

could be as similar to the euclidean geometry as we might wish in the re￾stricted regions experience is confined to. In honesty, all we can say is that

E

3

, as a model for our ambient space, is strongly favored by present day

experimental evidence in scales ranging from (say) human dimensions down

to about 10−15 cm. Our knowledge on smaller scales is limited by our ca￾pacity to probe them. For larger scales, according to General Relativity, the

validity of this model depends on the presence and strength of gravitational

fields: E

3

is good only as long as gravitational fields are very weak.

“ These data are — like all data — not logically necessary,

but only of empirical certainty . . . one can therefore investigate

their likelihood, which is certainly very great within the bounds of

observation, and afterwards decide upon the legitimacy of extend￾ing them beyond the bounds of observation, both in the direction of

the immeasurably large and in the direction of the immeasurably

small.”

1 A translation of Riemann’s Address can be found in Spivak 1970, vol. II. Clifford’s

translation (Nature, 8 (1873), 14-17, 36-37), as well as the original transcribed by David

R. Wilkins, can be found in the site http://www.emis.de/classics/Riemann/.

v

The only remark we could add to these words, pronounced in 1854, is

that the “bounds of observation” have greatly receded with respect to the

values of Riemann times.

“ . . . geometry presupposes the concept of space, as well as

assuming the basic principles for constructions in space .”

In our ambient space, we use in reality a lot more of structure than

the simple metric model: we take for granted a vector space structure, or

an affine structure; we transport vectors in such a way that they remain

parallel to themselves, thereby assuming a connection. Which one is the

minimum structure, the irreducible set of assumptions really necessary to

the introduction of each concept? Physics should endeavour to establish on

empirical data not only the basic space to be chosen but also the structures

to be added to it. At present, we know for example that an electron moving

in E

3 under the influence of a magnetic field “feels” an extra connection (the

electromagnetic potential), to which neutral particles may be insensitive.

Experimental science keeps a very special relationship with Mathemat￾ics. Experience counts and measures. But Science requires that the results

be inserted in some logically ordered picture. Mathematics is expected to

provide the notion of number, so as to make countings and measurements

meaningful. But Mathematics is also expected to provide notions of a more

qualitative character, to allow for the modeling of Nature. Thus, concerning

numbers, there seems to be no result comforting the widespread prejudice

by which we measure real numbers. We work with integers, or with rational

numbers, which is fundamentally the same. No direct measurement will sort

out a Dedekind cut. We must suppose, however, that real numbers exist:

even from the strict experimental point of view, it does not matter whether

objects like “π” or “e” are simple names or are endowed with some kind of an

sich reality: we cannot afford to do science without them. This is to say that

even pure experience needs more than its direct results, presupposes a wider

background for the insertion of such results. Real numbers are a minimum

background. Experience, and “logical necessity”, will say whether they are

sufficient.

From the most ancient extant treatise going under the name of Physics2

:

“When the objects of investigation, in any subject, have first

principles, foundational conditions, or basic constituents, it is

through acquaintance with these that knowledge, scientific knowl￾edge, is attained. For we cannot say that we know an object before

2 Aristotle, Physics I.1.

vi

we are acquainted with its conditions or principles, and have car￾ried our analysis as far as its most elementary constituents.”

“The natural way of attaining such a knowledge is to start

from the things which are more knowable and obvious to us and

proceed towards those which are clearer and more knowable by

themselves . . .”

Euclidean spaces have been the starting spaces from which the basic geo￾metrical and analytical concepts have been isolated by successive, tentative,

progressive abstractions. It has been a long and hard process to remove the

unessential from each notion. Most of all, as will be repeatedly emphasized,

it was a hard thing to put the idea of metric in its due position.

Structure is thus to be added step by step, under the control of experi￾ment. Only once experiment has established the basic ground will internal

coherence, or logical necessity, impose its own conditions.

Contents

I MANIFOLDS 1

1 GENERAL TOPOLOGY 3

1.0 INTRODUCTORY COMMENTS . . . . . . . . . . . . . . . . . 3

1.1 TOPOLOGICAL SPACES . . . . . . . . . . . . . . . . . . . . 5

1.2 KINDS OF TEXTURE . . . . . . . . . . . . . . . . . . . . . . 15

1.3 FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.4 QUOTIENTS AND GROUPS . . . . . . . . . . . . . . . . . . . 36

1.4.1 Quotient spaces . . . . . . . . . . . . . . . . . . . . . . 36

1.4.2 Topological groups . . . . . . . . . . . . . . . . . . . . 41

2 HOMOLOGY 49

2.1 GRAPHS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.1.1 Graphs, first way . . . . . . . . . . . . . . . . . . . . . 50

2.1.2 Graphs, second way . . . . . . . . . . . . . . . . . . . . 52

2.2 THE FIRST TOPOLOGICAL INVARIANTS . . . . . . . . . . . 57

2.2.1 Simplexes, complexes & all that . . . . . . . . . . . . . 57

2.2.2 Topological numbers . . . . . . . . . . . . . . . . . . . 64

3 HOMOTOPY 73

3.0 GENERAL HOMOTOPY . . . . . . . . . . . . . . . . . . . . . 73

3.1 PATH HOMOTOPY . . . . . . . . . . . . . . . . . . . . . . . . 78

3.1.1 Homotopy of curves . . . . . . . . . . . . . . . . . . . . 78

3.1.2 The Fundamental group . . . . . . . . . . . . . . . . . 85

3.1.3 Some Calculations . . . . . . . . . . . . . . . . . . . . 92

3.2 COVERING SPACES . . . . . . . . . . . . . . . . . . . . . . 98

3.2.1 Multiply-connected Spaces . . . . . . . . . . . . . . . . 98

3.2.2 Covering Spaces . . . . . . . . . . . . . . . . . . . . . . 105

3.3 HIGHER HOMOTOPY . . . . . . . . . . . . . . . . . . . . . 115

vii

viii CONTENTS

4 MANIFOLDS & CHARTS 121

4.1 MANIFOLDS . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.1.1 Topological manifolds . . . . . . . . . . . . . . . . . . . 121

4.1.2 Dimensions, integer and other . . . . . . . . . . . . . . 123

4.2 CHARTS AND COORDINATES . . . . . . . . . . . . . . . . 125

5 DIFFERENTIABLE MANIFOLDS 133

5.1 DEFINITION AND OVERLOOK . . . . . . . . . . . . . . . . . 133

5.2 SMOOTH FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . 135

5.3 DIFFERENTIABLE SUBMANIFOLDS . . . . . . . . . . . . . . 137

II DIFFERENTIABLE STRUCTURE 141

6 TANGENT STRUCTURE 143

6.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.2 TANGENT SPACES . . . . . . . . . . . . . . . . . . . . . . . . 145

6.3 TENSORS ON MANIFOLDS . . . . . . . . . . . . . . . . . . . 154

6.4 FIELDS & TRANSFORMATIONS . . . . . . . . . . . . . . . . 161

6.4.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.4.2 Transformations . . . . . . . . . . . . . . . . . . . . . . 167

6.5 FRAMES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.6 METRIC & RIEMANNIAN MANIFOLDS . . . . . . . . . . . . 180

7 DIFFERENTIAL FORMS 189

7.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 189

7.2 EXTERIOR DERIVATIVE . . . . . . . . . . . . . . . . . . . 197

7.3 VECTOR-VALUED FORMS . . . . . . . . . . . . . . . . . . 210

7.4 DUALITY AND CODERIVATION . . . . . . . . . . . . . . . 217

7.5 INTEGRATION AND HOMOLOGY . . . . . . . . . . . . . . 225

7.5.1 Integration . . . . . . . . . . . . . . . . . . . . . . . . 225

7.5.2 Cohomology of differential forms . . . . . . . . . . . . 232

7.6 ALGEBRAS, ENDOMORPHISMS AND DERIVATIVES . . . . . 239

8 SYMMETRIES 247

8.1 LIE GROUPS . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

8.2 TRANSFORMATIONS ON MANIFOLDS . . . . . . . . . . . . . 252

8.3 LIE ALGEBRA OF A LIE GROUP . . . . . . . . . . . . . . . 259

8.4 THE ADJOINT REPRESENTATION . . . . . . . . . . . . . 265

CONTENTS ix

9 FIBER BUNDLES 273

9.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 273

9.2 VECTOR BUNDLES . . . . . . . . . . . . . . . . . . . . . . . 275

9.3 THE BUNDLE OF LINEAR FRAMES . . . . . . . . . . . . . . 277

9.4 LINEAR CONNECTIONS . . . . . . . . . . . . . . . . . . . . 284

9.5 PRINCIPAL BUNDLES . . . . . . . . . . . . . . . . . . . . . 297

9.6 GENERAL CONNECTIONS . . . . . . . . . . . . . . . . . . 303

9.7 BUNDLE CLASSIFICATION . . . . . . . . . . . . . . . . . . 316

III FINAL TOUCH 321

10 NONCOMMUTATIVE GEOMETRY 323

10.1 QUANTUM GROUPS — A PEDESTRIAN OUTLINE . . . . . . 323

10.2 QUANTUM GEOMETRY . . . . . . . . . . . . . . . . . . . . 326

IV MATHEMATICAL TOPICS 331

1 THE BASIC ALGEBRAIC STRUCTURES 333

1.1 Groups and lesser structures . . . . . . . . . . . . . . . . . . . . 334

1.2 Rings and fields . . . . . . . . . . . . . . . . . . . . . . . . . . 338

1.3 Modules and vector spaces . . . . . . . . . . . . . . . . . . . . . 341

1.4 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

1.5 Coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

2 DISCRETE GROUPS. BRAIDS AND KNOTS 351

2.1 A Discrete groups . . . . . . . . . . . . . . . . . . . . . . . . . 351

2.2 B Braids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

2.3 C Knots and links . . . . . . . . . . . . . . . . . . . . . . . . . 363

3 SETS AND MEASURES 371

3.1 MEASURE SPACES . . . . . . . . . . . . . . . . . . . . . . . . 371

3.2 ERGODISM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

4 TOPOLOGICAL LINEAR SPACES 379

4.1 Inner product space . . . . . . . . . . . . . . . . . . . . . . . 379

4.2 Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

4.3 Normed vector spaces . . . . . . . . . . . . . . . . . . . . . . . 380

4.4 Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

4.5 Banach space . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

4.6 Topological vector spaces . . . . . . . . . . . . . . . . . . . . . 382

x CONTENTS

4.7 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 383

5 BANACH ALGEBRAS 385

5.1 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

5.2 Banach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 387

5.3 *-algebras and C*-algebras . . . . . . . . . . . . . . . . . . . . 389

5.4 From Geometry to Algebra . . . . . . . . . . . . . . . . . . . . 390

5.5 Von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . 393

5.6 The Jones polynomials . . . . . . . . . . . . . . . . . . . . . . 397

6 REPRESENTATIONS 403

6.1 A Linear representations . . . . . . . . . . . . . . . . . . . . . 404

6.2 B Regular representation . . . . . . . . . . . . . . . . . . . . . 408

6.3 C Fourier expansions . . . . . . . . . . . . . . . . . . . . . . . 409

7 VARIATIONS & FUNCTIONALS 415

7.1 A Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

7.1.1 Variation of a curve . . . . . . . . . . . . . . . . . . . . 415

7.1.2 Variation fields . . . . . . . . . . . . . . . . . . . . . . 416

7.1.3 Path functionals . . . . . . . . . . . . . . . . . . . . . . 417

7.1.4 Functional differentials . . . . . . . . . . . . . . . . . . 418

7.1.5 Second-variation . . . . . . . . . . . . . . . . . . . . . 420

7.2 B General functionals . . . . . . . . . . . . . . . . . . . . . . . 421

7.2.1 Functionals . . . . . . . . . . . . . . . . . . . . . . . . 421

7.2.2 Linear functionals . . . . . . . . . . . . . . . . . . . . 422

7.2.3 Operators . . . . . . . . . . . . . . . . . . . . . . . . . 423

7.2.4 Derivatives – Fr´echet and Gateaux . . . . . . . . . . . 423

8 FUNCTIONAL FORMS 425

8.1 A Exterior variational calculus . . . . . . . . . . . . . . . . . . . 426

8.1.1 Lagrangian density . . . . . . . . . . . . . . . . . . . . 426

8.1.2 Variations and differentials . . . . . . . . . . . . . . . . 427

8.1.3 The action functional . . . . . . . . . . . . . . . . . . 428

8.1.4 Variational derivative . . . . . . . . . . . . . . . . . . . 428

8.1.5 Euler Forms . . . . . . . . . . . . . . . . . . . . . . . . 429

8.1.6 Higher order Forms . . . . . . . . . . . . . . . . . . . 429

8.1.7 Relation to operators . . . . . . . . . . . . . . . . . . 429

8.2 B Existence of a lagrangian . . . . . . . . . . . . . . . . . . . . 430

8.2.1 Inverse problem of variational calculus . . . . . . . . . 430

8.2.2 Helmholtz-Vainberg theorem . . . . . . . . . . . . . . . 430

8.2.3 Equations with no lagrangian . . . . . . . . . . . . . . 431

CONTENTS xi

8.3 C Building lagrangians . . . . . . . . . . . . . . . . . . . . . . 432

8.3.1 The homotopy formula . . . . . . . . . . . . . . . . . . 432

8.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 434

8.3.3 Symmetries of equations . . . . . . . . . . . . . . . . . 436

9 SINGULAR POINTS 439

9.1 Index of a curve . . . . . . . . . . . . . . . . . . . . . . . . . . 439

9.2 Index of a singular point . . . . . . . . . . . . . . . . . . . . . 442

9.3 Relation to topology . . . . . . . . . . . . . . . . . . . . . . . 443

9.4 Basic two-dimensional singularities . . . . . . . . . . . . . . . 443

9.5 Critical points . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

9.6 Morse lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

9.7 Morse indices and topology . . . . . . . . . . . . . . . . . . . 446

9.8 Catastrophes . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

10 EUCLIDEAN SPACES AND SUBSPACES 449

10.1 A Structure equations . . . . . . . . . . . . . . . . . . . . . . 450

10.1.1 Moving frames . . . . . . . . . . . . . . . . . . . . . . 450

10.1.2 The Cartan lemma . . . . . . . . . . . . . . . . . . . . 450

10.1.3 Adapted frames . . . . . . . . . . . . . . . . . . . . . . 450

10.1.4 Second quadratic form . . . . . . . . . . . . . . . . . . 451

10.1.5 First quadratic form . . . . . . . . . . . . . . . . . . . 451

10.2 B Riemannian structure . . . . . . . . . . . . . . . . . . . . . 452

10.2.1 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . 452

10.2.2 Connection . . . . . . . . . . . . . . . . . . . . . . . . 452

10.2.3 Gauss, Ricci and Codazzi equations . . . . . . . . . . . 453

10.2.4 Riemann tensor . . . . . . . . . . . . . . . . . . . . . . 453

10.3 C Geometry of surfaces . . . . . . . . . . . . . . . . . . . . . . 455

10.3.1 Gauss Theorem . . . . . . . . . . . . . . . . . . . . . . 455

10.4 D Relation to topology . . . . . . . . . . . . . . . . . . . . . . 457

10.4.1 The Gauss-Bonnet theorem . . . . . . . . . . . . . . . 457

10.4.2 The Chern theorem . . . . . . . . . . . . . . . . . . . . 458

11 NON-EUCLIDEAN GEOMETRIES 459

11.1 The old controversy . . . . . . . . . . . . . . . . . . . . . . . . 459

11.2 The curvature of a metric space . . . . . . . . . . . . . . . . . 460

11.3 The spherical case . . . . . . . . . . . . . . . . . . . . . . . . . 461

11.4 The Boliyai-Lobachevsky case . . . . . . . . . . . . . . . . . . 464

11.5 On the geodesic curves . . . . . . . . . . . . . . . . . . . . . . 466

11.6 The Poincar´e space . . . . . . . . . . . . . . . . . . . . . . . . 467

xii CONTENTS

12 GEODESICS 471

12.1 Self–parallel curves . . . . . . . . . . . . . . . . . . . . . . . . 472

12.1.1 In General Relativity . . . . . . . . . . . . . . . . . . . 472

12.1.2 The absolute derivative . . . . . . . . . . . . . . . . . . 473

12.1.3 Self–parallelism . . . . . . . . . . . . . . . . . . . . . . 474

12.1.4 Complete spaces . . . . . . . . . . . . . . . . . . . . . 475

12.1.5 Fermi transport . . . . . . . . . . . . . . . . . . . . . . 475

12.1.6 In Optics . . . . . . . . . . . . . . . . . . . . . . . . . 476

12.2 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

12.2.1 Jacobi equation . . . . . . . . . . . . . . . . . . . . . . 476

12.2.2 Vorticity, shear and expansion . . . . . . . . . . . . . . 480

12.2.3 Landau–Raychaudhury equation . . . . . . . . . . . . . 483

V PHYSICAL TOPICS 485

1 HAMILTONIAN MECHANICS 487

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

1.2 Symplectic structure . . . . . . . . . . . . . . . . . . . . . . . 488

1.3 Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 490

1.4 Canonical transformations . . . . . . . . . . . . . . . . . . . . 491

1.5 Phase spaces as bundles . . . . . . . . . . . . . . . . . . . . . 494

1.6 The algebraic structure . . . . . . . . . . . . . . . . . . . . . . 496

1.7 Relations between Lie algebras . . . . . . . . . . . . . . . . . . 498

1.8 Liouville integrability . . . . . . . . . . . . . . . . . . . . . . . 501

2 MORE MECHANICS 503

2.1 Hamilton–Jacobi . . . . . . . . . . . . . . . . . . . . . . . . . 503

2.1.1 Hamiltonian structure . . . . . . . . . . . . . . . . . . 503

2.1.2 Hamilton-Jacobi equation . . . . . . . . . . . . . . . . 505

2.2 The Lagrange derivative . . . . . . . . . . . . . . . . . . . . . 507

2.2.1 The Lagrange derivative as a covariant derivative . . . 507

2.3 The rigid body . . . . . . . . . . . . . . . . . . . . . . . . . . 510

2.3.1 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

2.3.2 The configuration space . . . . . . . . . . . . . . . . . 511

2.3.3 The phase space . . . . . . . . . . . . . . . . . . . . . . 511

2.3.4 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 512

2.3.5 The “space” and the “body” derivatives . . . . . . . . 513

2.3.6 The reduced phase space . . . . . . . . . . . . . . . . . 513

2.3.7 Moving frames . . . . . . . . . . . . . . . . . . . . . . 514

2.3.8 The rotation group . . . . . . . . . . . . . . . . . . . . 515

CONTENTS xiii

2.3.9 Left– and right–invariant fields . . . . . . . . . . . . . 515

2.3.10 The Poinsot construction . . . . . . . . . . . . . . . . . 518

3 STATISTICS AND ELASTICITY 521

3.1 A Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . 521

3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 521

3.1.2 General overview . . . . . . . . . . . . . . . . . . . . . 522

3.2 B Lattice models . . . . . . . . . . . . . . . . . . . . . . . . . 526

3.2.1 The Ising model . . . . . . . . . . . . . . . . . . . . . . 526

3.2.2 Spontaneous breakdown of symmetry . . . . . . . . . 529

3.2.3 The Potts model . . . . . . . . . . . . . . . . . . . . . 531

3.2.4 Cayley trees and Bethe lattices . . . . . . . . . . . . . 535

3.2.5 The four-color problem . . . . . . . . . . . . . . . . . . 536

3.3 C Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

3.3.1 Regularity and defects . . . . . . . . . . . . . . . . . . 537

3.3.2 Classical elasticity . . . . . . . . . . . . . . . . . . . . 542

3.3.3 Nematic systems . . . . . . . . . . . . . . . . . . . . . 547

3.3.4 The Franck index . . . . . . . . . . . . . . . . . . . . . 550

4 PROPAGATION OF DISCONTINUITIES 553

4.1 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 553

4.2 Partial differential equations . . . . . . . . . . . . . . . . . . . 554

4.3 Maxwell’s equations in a medium . . . . . . . . . . . . . . . . 558

4.4 The eikonal equation . . . . . . . . . . . . . . . . . . . . . . . 561

5 GEOMETRICAL OPTICS 565

5.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565

5.1 The light-ray equation . . . . . . . . . . . . . . . . . . . . . . 566

5.2 Hamilton’s point of view . . . . . . . . . . . . . . . . . . . . . 567

5.3 Relation to geodesics . . . . . . . . . . . . . . . . . . . . . . . 568

5.4 The Fermat principle . . . . . . . . . . . . . . . . . . . . . . . 570

5.5 Maxwell’s fish-eye . . . . . . . . . . . . . . . . . . . . . . . . . 571

5.6 Fresnel’s ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . 572

6 CLASSICAL RELATIVISTIC FIELDS 575

6.1 A The fundamental fields . . . . . . . . . . . . . . . . . . . . . 575

6.2 B Spacetime transformations . . . . . . . . . . . . . . . . . . . 576

6.3 C Internal transformations . . . . . . . . . . . . . . . . . . . . 579

6.4 D Lagrangian formalism . . . . . . . . . . . . . . . . . . . . . 579

xiv CONTENTS

7 GAUGE FIELDS 589

7.1 A The gauge tenets . . . . . . . . . . . . . . . . . . . . . . . . 590

7.1.1 Electromagnetism . . . . . . . . . . . . . . . . . . . . . 590

7.1.2 Nonabelian theories . . . . . . . . . . . . . . . . . . . . 591

7.1.3 The gauge prescription . . . . . . . . . . . . . . . . . . 593

7.1.4 Hamiltonian approach . . . . . . . . . . . . . . . . . . 594

7.1.5 Exterior differential formulation . . . . . . . . . . . . . 595

7.2 B Functional differential approach . . . . . . . . . . . . . . . . 596

7.2.1 Functional Forms . . . . . . . . . . . . . . . . . . . . . 596

7.2.2 The space of gauge potentials . . . . . . . . . . . . . . 598

7.2.3 Gauge conditions . . . . . . . . . . . . . . . . . . . . . 601

7.2.4 Gauge anomalies . . . . . . . . . . . . . . . . . . . . . 602

7.2.5 BRST symmetry . . . . . . . . . . . . . . . . . . . . . 603

7.3 C Chiral fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

8 GENERAL RELATIVITY 605

8.1 Einstein’s equation . . . . . . . . . . . . . . . . . . . . . . . . 605

8.2 The equivalence principle . . . . . . . . . . . . . . . . . . . . . 608

8.3 Spinors and torsion . . . . . . . . . . . . . . . . . . . . . . . . 612

9 DE SITTER SPACES 615

9.1 General characteristics . . . . . . . . . . . . . . . . . . . . . . 615

9.2 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619

9.3 Geodesics and Jacobi equations . . . . . . . . . . . . . . . . . 620

9.4 Some qualitative aspects . . . . . . . . . . . . . . . . . . . . . 621

9.5 Wigner-In¨on¨u contraction . . . . . . . . . . . . . . . . . . . . 621

10 SYMMETRIES ON PHASE SPACE 625

10.1 Symmetries and anomalies . . . . . . . . . . . . . . . . . . . . 625

10.2 The Souriau momentum . . . . . . . . . . . . . . . . . . . . . 628

10.3 The Kirillov form . . . . . . . . . . . . . . . . . . . . . . . . . 629

10.4 Integrability revisited . . . . . . . . . . . . . . . . . . . . . . . 630

10.5 Classical Yang-Baxter equation . . . . . . . . . . . . . . . . . 631

VI Glossary and Bibliography 635