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1、AnIntroductiontoGEOMETRICALPHYSICSR.Aldrovandi&J.G.PereiraInstitutodeF´ısicaTe´oricaStateUniversityofS˜aoPaulo–UNESPS˜aoPaulo—BrazilToourparentsNice,Dina,Jos´eandTitoiiiPREAMBLE:SPACEANDGEOMETRYWhatstuff’tismadeof,whereofitisborn,Iamtolearn.MerchantofVeniceThesimplest
2、geometricalsettingused—consciouslyornot—byphysi-cistsintheireverydayworkisthe3-dimensionaleuclideanspaceE3.Itcon-sistsofthesetR3oforderedtriplesofrealnumberssuchasp=(p1,p2,p3),q=(q1,q2,q3),etc,andisendowedwithaveryspecialcharacteristic,ametricdefinedbythedistancefunct
3、ion"#1/2X3ii2d(p,q)=(p−q).i=1Itisthespaceofordinaryhumanexperienceandthestartingpointofourgeometricintuition.Studiedfortwo-and-a-halfmillenia,ithasbeentheobjectofcelebratedcontroversies,themostfamousconcerningtheminimumnumberofpropertiesnecessarytodefineitcompletely.F
4、romAristotletoNewton,throughGalileoandDescartes,theverywordspacehasbeenreservedtoE3.Onlyinthe19-thcenturyhasitbecomeclearthatother,differentspacescouldbethoughtof,andmathematicianshavesincegreatlyamusedthemselvesbyinventingallkindsofthem.Forphysi-cists,theage-longdeba
5、teshiftedtoanotherquestion:howcanwerecognize,amongstsuchinnumerablepossiblespaces,thatrealspacechosenbyNatureasthestage-setofitsprocesses?Forexample,supposethespaceofourev-erydayexperienceconsistsofthesamesetR3oftriplesabove,butwithadifferentdistancefunction,suchasX3i
6、id(p,q)=
7、p−q
8、.i=1Thiswoulddefineadifferentmetricspace,inprincipleasgoodasthatgivenabove.Wereitonlyamatterofprinciple,itwouldbeasgoodasiiiivanyotherspacegivenbyanydistancefunctionwithR3assetpoint.Itsohappens,however,thatNaturehaschosentheformerandnotthelatterspaceforust
9、olivein.Toknowwhichoneistherealspaceisnotasimplequestionofprinciple—somethingelseisneeded.Whatelse?Theanswermayseemrathertrivialinthecaseofourhomespace,thoughlesssoinotherspacessingledoutbyNatureinthemanydifferentsituationswhichareobjectsofphysicalstudy.ItwasgivenbyRi
10、emanninhisfamousInauguralAddress1:“...thosepropertieswhichdistinguishSpacefromothercon-ceivabletriplyextendedquantitiescanonlybeded
