A Physicist’s Introduction to Algebraic Structures Algebraicstructuresplayanimportantroleinunderstanding crucialconcepts ofmodern physics. Starting with mathematical logic, sets and functions, this text presents short descriptions of different algebraic structures such as metric spaces, rings, fields, Boolean algebra, groups, vector spaces and measurespaces. A review of the basic ideas of vector space, linear independence of vectors, basis, dual space, operators on finite dimensional vectorspaces,andsoon,isfollowed upbyadetaileddiscussion ofthetypesofmatrices, that are especially important for physics, for example, Hermitian and unitary matrices. Infinite dimensionalvectorspaces,alsoknown asfunctionspaces,andoperatorsonsuch spacesarediscussed. Group Theory, starting with general properties of groups and their representations, is elaborated. Finite groups, as well as representation of finite groups, are discussed in detail:permutationgroupstakethecentralroleinthediscussion. Somefinitegroupsthat have easy geometry, such as symmetry groups of polygons and polyhedrons as well as crystal symmetry groups, are interpreted. Likewise, continuous groups or Lie groups in particular are discussed. The representation of unitary groups and orthogonal groups is elaborated,followedbyachapterontheLorentzgroup.SomeotherLiegroups,suchasthe symplecticgroupsandexceptionalgroups,arediscussedbriefly.Weightsandrootvectors, withenumerationofthenumberofcompactLiealgebrasusingtheDynkindiagrams,are presented. The last part of the text deals with topology. Continuity of functions, and the idea of topological spaces, is taken up. The idea of homotopy of maps is introduced and the fundamentalgroupoftopologicalspacesisdiscussedindetail.Higher homotopygroups andhomologygroupshavealsobeenvisited. Thisbookwillbeusefulforgraduatestudentsandresearchersinterestedinpickingup mathematicalconcepts,whichareusefulinmanybranchesofphysics. PalashB.PalretiredfromtheTheoryDivision,SahaInstituteofNuclearPhysics,Kolkata, India. Currently, he isanEmeritusProfessor atthe University of Calcutta. He hastaught courses on mathematical methods, particle physics, quantum field theory, theoretical physics and classical field theory at the graduate level. He has published more than 100 papers in journals of international repute. His current research includes elementary particle physics, with specialization in neutrinos, grand unified theories and particles in electromagneticfields. A Physicist’s Introduction to Algebraic Structures Vector Spaces, Groups, Topological Spaces and More Palash B. Pal UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,vic3207,Australia 314to321,3rdFloor,PlotNo.3,SplendorForum,JasolaDistrictCentre,NewDelhi110025,India 79AnsonRoad,#06-04/06,Singapore079906 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learningandresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781108492201 (cid:13)c PalashB.Pal2019 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2019 PrintedinIndia AcataloguerecordforthispublicationisavailablefromtheBritishLibrary ISBN978-1-108-49220-1Hardback ISBN978-1-108-72911-6Paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication, anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. To the memory of my friend Darwin Chang who would have loved to see this book Contents ListofFigures xix Preface xxi A GENERAL INTRODUCTION 1 1 RulesofLogic 3 1.1 Sentences 3 1.2 BinaryRelationsonSentences 3 1.3 LogicalEquivalenceandImplication 7 1.4 PredicateLogic 11 1.4.1 Thenecessityforusingpredicates 11 1.4.2 Quantifiers 12 1.5 RulesofInference 13 1.5.1 Rulesofinferenceforpropositionallogic 14 1.5.2 Rulesofinferenceforquantifiers 15 1.5.3 Combiningrulesoftheprevioustwotypes 16 1.5.4 Proofsusingsequences 16 2 SetsandFunctions 18 2.1 SetTheory 18 2.1.1 Fundamentals 18 2.1.2 Basicoperationsonsets 20 2.1.3 Subsets 23 2.1.4 Productsets 23 2.2 Functions 24 2.2.1 Definition 24 2.2.2 Imageandpre-imageofsets 25 2.2.3 Binaryoperations 28 2.3 CountableandUncountableSets 30 vii viii Contents 2.4 SetswithRelations 32 2.4.1 Setswithequivalencerelations 33 2.4.2 Setswithorderrelations 34 3 AlgebraicStructures 37 3.1 WhatareAlgebraicStructures? 37 3.2 MetricSpace 38 3.3 Group 40 3.4 Ring 40 3.4.1 Generalconsiderations 40 3.4.2 Modularnumbers 45 3.4.3 Algebraicintegers 47 3.5 Field 49 3.6 VectorSpace 52 3.7 Algebra 54 3.8 BooleanAlgebra 55 3.9 Arithmetic 56 3.10 GrassmannArithmetic 57 3.10.1 Grassmanniannumbers 57 3.10.2 Grassmannianmatrices 59 3.11 MeasureSpace 60 3.11.1 σ-algebra 60 3.11.2 Measurespace 61 3.11.3 Lebesgueintegral 62 B VECTOR SPACES 65 4 Basics 67 4.1 DefinitionandExamples 67 4.2 LinearIndependenceofVectors 68 4.3 DualSpace 69 4.4 Subspaces 71 4.5 VectorSpaceswithExtraStructure 72 4.5.1 Normedvectorspaces 72 4.5.2 Innerproductspaces 74 4.6 FurtherPropertiesofInnerProductSpaces 75 4.6.1 Propertiesofthenullvector 76 4.6.2 Cauchy–Schwarzinequality 76 4.6.3 Equivalencebetweenvectorsandlinearmaps 78 4.7 OrthonormalBasis 80 4.7.1 Findinganorthonormalbasis 80