HAL Id: hal-00735107https://hal.archives-ouvertes.fr/hal-00735107v1
Submitted on 25 Sep 2012 (v1), last revised 1 Feb 2014 (v2)
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
Larchive ouverte pluridisciplinaire HAL, estdestine au dpt et la diffusion de documentsscientifiques de niveau recherche, publis ou non,manant des tablissements denseignement et derecherche franais ou trangers, des laboratoirespublics ou privs.
Mathematics for theoretical physicsJean Claude Dutailly
To cite this version:
Jean Claude Dutailly. Mathematics for theoretical physics. 2012.
https://hal.archives-ouvertes.fr/hal-00735107v1https://hal.archives-ouvertes.fr
Mathematics for theoretical physics
Jean Claude.Dutailly
Paris
September 25, 2012
Abstract
This book intends to give the main definitions and theorems in math-ematics which could be useful for workers in theoretical physics. It givesan extensive and precise coverage of the subjects which are addressed, ina consistent and intelligible manner.The first part addresses the Foun-dations (mathematical logic, set theory, categories), the second Alge-bra (algebraic strucutes, groups, vector spaces tensors, matrices, Clif-ford algebra). The third Analysis (general topology, measure theory, Ba-nach Spaces, Spectral theory). The fourth Differential Geometry (deriva-tives, manifolds, tensorial bundle, pseudo-riemannian manifolds, symplec-tic manifolds). The fifth Lie Algebras, Lie Groups.and representation the-ory. The sixth Fiber bundles and jets. The last one Functional Analy-sis (differential operators, distributions, ODE, PDE, variational calculus).Several signficant new results are presented (distributions over vector bun-dles, functional derivative, spin bundle and manifolds with boundary).
The purpose of this book is to give a comprehensive collection of precisedefinitions and results in advanced mathematics, which can be useful to workersin mathematic or physics.
The specificities of this book are :- it is self contained : any definition or notation used can be found within- it is precise : any theorem lists the precise conditions which must be met
for its use- it is easy to use : the book proceeds from the simple to the most advanced
topics, but in any part the necessary definitions are reminded so that the readercan enter quickly into the subject
- it is comprehensive : it addresses the basic concepts but reaches most ofthe advanced topics which are required nowodays
- it is pedagogical : the key points and usual misunderstandings are under-lined so that the reader can get a strong grasp of the tools which are presented.
The first option is unusual for a book of this kind. Usually a book starts withthe assumption that the reader has already some background knowledge. Theproblem is that nobody has the same background. So a great deal is dedicatedto remind some basic stuff, in an abbreviated way, which does not left muchscope to their understanding, and is limited to specific cases. In fact, starting
1
from the very beginning, it has been easy, step by step, to expose each conceptin the most general settings. And, by proceeding this way, to extend the scopeof many results so that they can be made available to the - unavoidable - specialcase that the reader may face. Overall it gives a fresh, unified view of the math-ematics, but still affordable because it avoids as far as possible the sophisticatedlanguage which is fashionable. The goal is that the reader understands clearlyand effortlessly, not to prove the extent of the authors knowledge.
The definitions choosen here meet the generally accepted definitions inmathematics. However, as they come in many flavors according to the authorsand their field of interest, we have striven to take definitions which are both themost general and the most easy to use.
Of course this cannot be achieved with some drawbacks. So many demon-strations are omitted. More precisely the chosen option is the following :
- whenever a demonstration is short, it is given entirely, at least as an ex-ample of how it works
- when a demonstation is too long and involves either technical or specificconditions, a precise reference to where the demonstation can be found is given.Anyway the theorem is written in accordance with the notations and defini-tions of this book, and a special attention has been given that they match thereference.
- exceptionnaly, when this is a well known theorem, whose demonstrationcan be found easily in any book on the subject, there is no reference.
The bibliography is short. Indeed due to the scope which is covered it couldbe enormous. So it is strictly limited to the works which are referenced in thetext, with a priority to the most easily available sources.
This is not mainly a research paper, even if the unification of the concepts is,in many ways, new, but some significant results appear here for the first time,to my knowledge.
- distributions over vector bundles- a rigorous definitition of functional derivatives- a manifold with boundary can be defined by a unique functionand several other results about Clifford algebras, spin bundles and differen-
tial geometry.
1
[email protected]
2
CONTENTS
PART 1 : FOUNDATIONS
LOGICPropositional logic 10Predicates 14Formal theories 16SET THEORYAxiomatic 20Maps 23Binary relations 27CATEGORIESCategory 32Functors 37
3
PART 2 : ALGEBRA
USUAL ALGEBRAIC STRUCTURESFrom monoids to fields 43From vector spaces to algebras 47GROUPSDefinitions 52Finite groups 57VECTOR SPACESDefinitions 62Linear maps 66Scalar product on vector spaces 78Symplectic vector space 82Complex vector space 85Affine space 92TENSORSTensorial product 100Symmetric and antisymmetric tensors 107Tensor product of maps 118MATRICESOperations with matrices 126Eigen values 131Matrix calculus 134CLIFFORD ALGEBRAMain operations in Clifford algebras 144Pin and Spin groups 149Classification of Clifford algebras 155
4
PART 3 : ANALYSIS
GENERAL TOPOLOGYTopological spaces 166Maps on topological spaces 178Metric and semi-metric spaces 184Algebraic topology 193MEASUREMeasurable spaces 201Measured spaces 204Integral 212Probability 216BANACH SPACESTopological vector spaces 223Normed vector spaces 228Banach spaces 236Normed algebras 246Hilbert Spaces 257SPECTRAL THEORYRepresentation of algebras 273Spectral theory 277
5
PART 4 : DIFFERENTIAL GEOMETRY
DERIVATIVESDifferentiables maps 286Higher order derivatives 294Extremum of a function 298Implicit maps 301Holomorphic maps 302MANIFOLDSManifolds 310Differentiable maps 286Tangent bundle 320Submanifolds 333TENSORIAL BUNDLETensor fields 346Lie derivative 352Exterior algebra 357Covariant derivative 360INTEGRALOrientation of a manifold 371Integral 373Cohomology 380COMPLEX MANIFOLDS 384PSEUDO RIEMANNIAN MANIFOLDSGeneral properties 388Levi Civita connection 394Submanifolds 400SYMPLECTIC MANIFOLDS 405
6
PART 5 : LIE ALGEBRAS AND LIE GROUPS
LIE ALGEBRASLie algebras : definitions 413Sum and product of Lie algebras 417Classification of Lie algebras 420LIE GROUPSGeneral definitions and results 434Structure of Lie groups 450Integration 463CLASSICAL LINEAR GROUPS AND ALGEBRASGeneral results 468List of classical linear groups and algebras 471REPRESENTATION THEORYDefinitions and general results 481Representation of Lie groups 491Representation of Lie algebras 502Representation of classical groups 506
7
PART 6 : FIBER BUNDLES
FIBER BUNDLESGeneral fiber bundles 520Vector bundles 532Principal bundles 545Associated bundles 553JETS 567CONNECTIONSGeneral connections 580Connections on vector bundles 587Connections on associated bundles 601BUNDLE FUNCTORS 614
8
PART 7 : FUNCTIONAL ANALYSIS
SPACES of FUNCTIONSPreliminaries 626Spaces of bounded or continuous maps 633Spaces of integrable maps 636Spaces of differentiables maps 644DISTRIBUTIONSSpaces of functionals 650Distributions on functions 653Extension of distributions 669FOURIER TRANSFORMFourier series 676Fourier integrals 678Fourier transform of distributions 681DIFFERENTIAL OPERATORSLinear differential operators 688Laplacian 705Heat kernel 716Pseudo-differential operators 719
+
DIFFERENTIAL EQUATIONSOrdinary differential equations 726Partial differential equations 732VARIATIONAL CALCULUS 749
BIBLIOGRAPHY
766
9
Part I
PART1 : FOUNDATIONS
In this first part we start with what makes the real foundations of today mathe-matics : logic, set theory and categories. The two last subsections are natural inthis book, and they will be mainly dedicated to a long list of definitions, manda-tory to fix the language that is used in the rest of the book. A section aboutlogic seems appropriate, even if it gives just an overview of the topic, becausethis is a subject that is rarely addressed, except in specialized publications, andshould give some matter for reflection, notably to physicists.
1 LOGIC
For a mathematician logic can be addressed from two points of view :- the conventions and rules that any mathematical text should follow in order
to be deemed right- the consistency and limitations of any formal theory using these logical
rules.It is the scope of a branch of mathematics of its own : mathematical logicIndeed logic is not limited to a bylaw for mathematicians : there are also
theorems in logic. To produce these theorems one distinguishes the object of thei
