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An Introduction to GEOMETRICAL PHYSICS R. Aldrovandi & J.G. Pereira Instituto de F´ısica Teórica State University of São Paulo – UNESP São Paulo — Brazil To our parents Nice, Dina, José and Tito i ii 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 consists 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 everyday 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 iii iv 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 experience.” 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/. 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 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 knowledge, 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 geometrical 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 experiment. 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