An Introduction to GEOMETRICAL PHYSICS R. Aldrovandi & J.G. Pereira Instituto de F´ısica Te´orica State Univers ity 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 , p 2 , p 3 ), q = (q 1 , q 2 , q 3 ), etc, and is endowed with a very special characteristic, a metric defined by the distance function d(p, q) =  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) = 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 Address 1 : “ 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 Physics 2 : “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 GENE RAL 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 DIFFE RENTIABLE 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 DIFFE RENTIAL FORMS 189 7.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 189 7.2 EXTERIOR DERIVATIVE . . . . . . . . . . . . . . . . . . . 197 7.3 VECTOR-VALUE D 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 SYMMETR IES 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 NONCOMMUTATIV E 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 Module s 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 [...]... spacelike and timelike vectors The fact is that we do not know the real topology of spacetime We would like to retain euclidean properties both in the space sector and on the time axis Zeeman4 has proposed an appealing topology: it is defined as the finest topology defined on R4 which induces an E3 topology on the space sector and an E1 topology on the time axis It is not first-countable and, consequently, cannot... nonmetric topology may have finer topologies which are metric, and a metric topology can have finer non-metric topologies And a non-metric topology may have weaker topologies which are metric, and a metric topology can have weaker non-metric topologies § 1.1.19 Topological product Given two topological spaces A and B, their topological product (or cartesian product) A×B is the set of pairs (p, q) with p ∈ A and... small; and so on Something makes a sphere deeply different from a torus and both different from a plane, and that independently of any measure, scale or proportion A hyperboloid sheet is quite distinct from the sphere and the torus, and also from the plane E2 , but less so for the latter: we feel that it can be somehow unfolded without violence into a plane A sphere can be stretched so as to become an ellipsoid... insight, to think about open disks, open triangles and open rectangles on the euclidean plane E2 No two distinct topologies may have a common basis, but a fixed topology may have many different basis On E2 , for instance, we could take the open disks, or the open squares or yet rectangles, or still the open ellipses We would say intuitively that all these different basis lead to the same topology and we... a first trial to classify topological spaces Topology frequently resorts to this kind of practice, trying to place the space in some hierarchy In the study of the anatomy of a topological space, some variations are sometimes helpful An example is a small change in the concept of a basis, leading to the idea of a ’network’ A network is a collection N of subsets such that any member of T can be obtained... (see as an example the definition of a continuous function in section 1.3.4) 1.1 TOPOLOGICAL SPACES 13 § 1.1.13 Consider two topologies T1 and T2 defined on the same point set S We say that T1 is weaker than T2 if every member of T1 belongs also to T2 The topology T1 is also said to be coarser than T2 , and T2 is finer than T1 (or T2 is a refinement of T1 , or still T2 is stronger than T1 ) The topology... correct As a topology is most frequently introduced via a basis, it is useful to have a criterium to check whether or not two basis correspond to the same topology This is provided by another theorem: 10 CHAPTER 1 GENERAL TOPOLOGY B and B are basis defining the same topology iff, for every Uα ∈ B and every p ∈ Uα , there exists some Uβ ∈ B such that p ∈ Bβ ⊂ Uα and vice-versa Again, it is instructive to give... properties, i.e., many different possible topologies Each such family will make of S a different topological space Rigour would require that a name or symbol be attributed to the family (say, T ) and the topological space be given name and surname, being denoted by the pair (S, T ) Some well known topological spaces have historical names When we say “euclidean space”, the set Rn with the usual topology of open... plane A sphere can be stretched so as to become an ellipsoid but cannot be made into a plane without losing something of its “spherical character” Topology is that primitive structure which will be the same for spheres and ellipsoids; which will be another one for planes and hyperboloid sheets; and still another, quite different, for toruses It will be that set of qualities of a space which is preserved... It follows that ∅ and S are closed (and open!) sets in all topological spaces § 1.1.16 Closedness is a relative concept: a subset C of a topological subspace Y of S can be closed in the induced topology even if open in S; for instance, Y itself will be closed (and open) in the induced topology, even if Y is an open set of S Retain that “closed”, just as “open”, depends on the chosen topology A set which . An Introduction to GEOMETRICAL PHYSICS R. Aldrovandi & J.G. Pereira Instituto de F´ısica Te´orica State Univers. sphere and the torus, and also from the plane E 2 , but less so for the latter: we feel that it can be somehow unfolded without violence into a plane. A