An Introduction to Geometrical Physics

691 Pages·1995·4.706 MB·English

by  R. Aldrovandi, J. G. Pereira

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
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PREAMBLE: SPACE AND GEOMETRY
What stuff’tis made of, whereof it is born,
I am to learn.
MerchantofVenice
The simplest geometrical setting used — consciously or not — by physi-
cists in their everyday work is the 3-dimensional euclidean space E3. It con-
sists of the set R3 of ordered triples of real numbers such as p = (p1,p2,p3), q
= (q1,q2,q3), etc, and is endowed with a very special characteristic, a metric
defined by the distance function
" #1/2
3
X
d(p,q) = (pi −qi)2 .
i=1
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 E3. 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 R3 of triples above, but with a
different distance function, such as
3
X
d(p,q) = |pi −qi|.
i=1
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
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any other space given by any distance function with R3 as set point. It so
happens, however, thatNaturehaschosentheformerandnotthelatterspace
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
E3, 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: E3 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/.
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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 E3 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.
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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
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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