物理学中的群论
出版时间:
2011-01
版次:
1
ISBN:
9787510029554
定价:
49.00
装帧:
平装
开本:
24开
纸张:
胶版纸
页数:
344页
正文语种:
英语
103人买过
-
grouptheoryprovidesthenaturalmathematicallanguagetoformulatesymmetryprinciplesandtoderivetheirconsequencesinmathematicsandinphysics.the"specialfunctions"ofmathematicalphysics，whichpervademathematicalanalysis,classicalphysics,andquantummechanics,invariablyoriginatefromunderlyingsymmetriesoftheproblemalthoughthetraditionalpresentationofsuchtopicsmaynotexpresslyemphasizethisuniversalfeature.moderndevelopmentsinallbranchesofphysicsareputtingmoreandmoreemphasisontheroleofsymmetriesoftheunderlyingphysicalsystems.thustheuseofgrouptheoryhasbecomeincreasinglyimportantinrecentyears.however,theincorporationofgrouptheoryintotheundergraduateorgraduatephysicscurriculumofmostuniversitieshasnotkeptupwiththisdevelopment.atbest,thissubjectisofferedasaspecialtopiccourse,cateringtoarestrictedclassofstudents.symptomaticofthisunfortunategapisthelackofsuitabletextbooksongeneralgroup-theoreticalmethodsinphysicsforallseriousstudentsofexperimentalandtheoreticalphysicsatthebeginninggraduateandadvancedundergraduatelevel.thisbookiswrittentomeetpreciselythisneed.
therealreadyexist,ofcourse,manybooksongrouptheoryanditsapplicationsinphysics.foremostamongthesearetheoldclassicsbyweyl,wigner,andvanderwaerden.forapplicationstoatomicandmolecularphysics,andtocrystallatticesinsolidstateandchemicalphysics,therearemanyelementarytextbooksemphasizingpointgroups,spacegroups,andtherotationgroup.reflectingtheimportantroleplayedbygrouptheoryinmodernelementaryparticletheory,manycurrentbooksexpoundonthetheoryofliegroupsandliealgebraswithemphasissuitableforhighenergytheoreticalphysics.finally,thereareseveralusefulgeneraltextsongrouptheoryfeaturingcomprehensivenessandmathematicalrigorwrittenforthemoremathematicallyorientedaudience.experienceindicates,however,thatformoststudents,itisdifficulttofindasuitablemodernintroductorytextwhichisbothgeneralandreadilyunderstandable. preface
chapter1introduction
1.1particleonaone-dimensionallattice
1.2representationsofthediscretetranslationoperators
1.3physicalconsequencesoftranslationalsymmetry
1.4therepresentationfunctionsandfourieranalysis
1.5symmetrygroupsofphysics
chapter2basicgrouptheory
2.1basicdefinitionsandsimpleexamples
2.2furtherexamples,subgroups
2.3therearrangementlemmaandthesymmetric（permutation）group
2.4classesandinvariantsubgroups
2.5cosetsandfactor（quotient）groups
2.6homomorphisms
2.7directproductsproblems
chapter3grouprepresentations
3.1representations
3.2irreducible,inequivalentrepresentations
3.3unitaryrepresentations
3.4schur'slemmas
3.5orthonormalityandcompletenessrelationsofirreduciblerepresentationmatrices
3.6orthonormalityandcompletenessrelationsofirreduciblecharacters
3.7theregularrepresentation
3.8directproductrepresentations,clebsch-gordancoefficientsproblems
chapter4generalpropertiesofirreduciblevectorsandoperators
4.1irreduciblebasisvectors
4.2thereductionofvectors——projectionoperatorsforirreduciblecomponents
4.3irreducibleoperatorsandthewigner-eckarttheoremproblems
chapter5representationsofthesymmetricgroups
5.1one-dimensionalrepresentations
5.2partitionsandyoungdiagrams
5.3symmetrizersandanti-symmetrizersofyoungtableaux
5.4irreduciblerepresentationsofsn
5.5symmetryclassesoftensorsproblems
chapter6one-dimensionalcontinuousgroups
6.1therotationgroupso（2）
6.2thegeneratorofso（2）
6.3irreduciblerepresentationsofso（2）
6.4invariantintegrationmeasure,orthonormalityandcompletenessrelations
6.5multi-valuedrepresentations
6.6continuoustranslationalgroupinonedimension
6.7conjugatebasisvectorsproblems
chapter7rotationsinthree-dimensionalspace——thegroupso（3）
7.1descriptionofthegroupso（3）
7.1.1theangle-and-axisparameterization
7.1.2theeulerangles
7.2oneparametersubgroups,generators,andtheliealgebra
7.3irreduciblerepresentationsoftheso（3）liealgebra
7.4propertiesoftherotationalmatricesdj（a,fl,7）
7.5applicationtoparticleinacentralpotential
7.5.1characterizationofstates
7.5.2asymptoticplanewavestates
7.5.3partialwavedecomposition
7.5.4summary
7.6transformationpropertiesofwavefunctionsandoperators
7.7directproductrepresentationsandtheirreduction
7.8irreducibletensorsandthewigner-eckarttheoremproblems
chapter8thegroupsu（2）andmoreaboutso（3）
8.1therelationshipbetweenso（3）andsu（2）
8.2invariantintegration
8.30rthonormalityandcompletenessrelationsofdj
8.4projectionoperatorsandtheirphysicalapplications
8.4.1singleparticlestatewithspill
8.4.2twoparticlestateswithspin
8.4.3partialwaveexpansionfortwoparticlescatteringwithspin
8.5differentialequationssatisfiedbythedj-functions
8.6grouptheoreticalinterpretationofsphericalharmonics
8.6.1transformationunderrotation
8.6.2additiontheorem
8.6.3decompositionofproductsofyimwiththesamearguments
8.6.4recursionformulas
8.6.5symmetryinm
8.6.60rthonormalityandcompleteness
8.6.7summaryremarks
8.7multipoleradiationoftheelectromagneticfieldproblems
chapter9euclideangroupsintwo-andthree-dimensionalspace
9.1theeuclideangroupintwo-dimensionalspacee2
9.2unitaryirreduciblerepresentationsofe2——theangular-momentumbasis
9.3theinducedrepresentationmethodandtheplane-wavebasis
9.4differentialequations,recursionformulas,andadditiontheoremofthebesselfunction
9.5groupcontraction——so（3）ande2
9.6theeuclideangroupinthreedimensions:e3
9.7unitaryirreduciblerepresentationsofe3bytheinducedrepresentationmethod
9.8angularmomentumbasisandthesphericalbesselfunctionproblems
chapter10thelorentzandpoincariegroups,andspace-timesymmetries
10.1thelorentzandpoincaregroups
10.1.1homogeneouslorentztransformations
10.1.2theproperlorentzgroup
10.1.3decompositionoflorentztransformations
10.1.4relationoftheproperlorentzgrouptosl（2）
10.1.5four-dimensionaltranslationsandthepoincaregroup
10.2generatorsandtheliealgeebra
10.3irreduciblerepresentationsoftheproperlorentzgroup
10.3.1equivalenceoftheliealgebratosu（2）xsu（2）
10.3.2finitedimensionalrepresentations
10.3.3unitaryrepresentations
10.4unitaryirreduciblerepresentationsofthepoincaregroup
10.4.1nullvectorcase（pu=0）
10.4.2time-likevectorcase（c1＞30）
10.4.3thesecondcasimiroperator
10.4.4light-likecase（c1=0）
10.4.5space-likecase（c1＜0）
10.4.6covariantnormalizationofbasisstatesandintegrationmeasure
10.5relationbetweenrepresentationsofthelorentzandpoincaregroups-relativisticwavefunctions,fields,andwaveequations
10.5.1wavefunctionsandfieldoperators
10.5.2relativisticwaveequationsandtheplanewaveexpansion
10.5.3thelorentz-poincareconnection
10.5.4"deriving"relativisticwaveequationsproblems
chapter11spaceinversioninvariance
11.1spaceinversionintwo-dimensionaleuclideanspace
11.1.ithegroup0（2）
11.1.2irreduciblerepresentationsof0（2）
11.1.3theextendedeuclideangroupe2anditsirreduciblerepresentations
11.2spaceinversioninthree-dimensionaleuclideanspace
11.2.1thegroup0（3）anditsirreduciblerepresentations
11.2.2theextendedeuclideangroupe3anditsirreduciblerepresentations
11.3spaceinversioninfour-dimensionalminkowskispace
11.3.1thecompletelorentzgroupanditsirreduciblerepresentations
11.3.2theextendedpoincaregroupanditsirreduciblerepresentations
11.4generalphysicalconsequencesofspaceinversion
11.4.1eigenstatesofangularmomentumandparity
11.4.2scatteringamplitudesandelectromagneticmultipoletransitionsproblems
chapter12timereversalinvariance
12.1preliminarydiscussion
12.2timereversalinvarianceinclassicalphysics
12.3problemswithlinearrealizationoftimereversaltransformation
12.4theanti-unitarytimereversaloperator
12.5irreduciblerepresentationsofthefullpoincaregroupinthetime-likecase
12.6irreduciblerepresentationsinthelight-likecase（c1=c2=0）
12.7physicalconsequencesoftimereversalinvariance
12.7.1timereversalandangularmomentumeigenstates
12.7.2time-reversalsymmetryoftransitionamplitudes
12.7.3timereversalinvarianceandperturbationamplitudesproblems
chapter13finite-dimensionalrepresentationsoftheclassicalgroups
13.1gl（m）:fundamentalrepresentationsandtheassociatedvectorspaces
13.2tensorsinvxv,contraction,andgl（m）transformations
13.3irreduciblerepresentationsofgl（m）onthespaceofgeneraltensors
13.4irreduciblerepresentationsofotherclassicallineargroups
13.4.1unitarygroupsu（m）andu（m+,m_）
13.4.2speciallineargroupssl（m）andspecialunitarygroupssu（m+,m_）
13.4.3therealorthogonalgroupo（m+,m_;r）andthespecialrealorthogonalgroupso（m+,m_;r）
13.5concludingremarksproblemsappendixinotationsandsymbols
i.1summationconvention
i.2vectorsandvectorindices
i.3matrixindices
appendixiisummaryoflinearvectorspaces
ii.1linearvectorspace
ii.2lineartransformations（operators）onvectorspaces
ii.3matrixrepresentationoflinearoperators
ii.4dualspace,adjointoperators
ii.5inner（scalar）productandinnerproductspace
ii.6lineartransformations（operators）oninnerproductspaces
appendixillgroupalgebraandthereductionofregularrepresentation
iii.1groupalgebra
1ii.2leftideals,projectionoperators
iii.3idempotents
iii.4completereductionoftheregularrepresentation
appendixivsupplementstothetheoryofsymmetricgroupssn
appependixvclebsch-gordancoefficientsandsphericalharmonics
appendixvirotationalandlorentzspinors
appendixviiunitaryrepresentationsoftheproperlorentzgroup
appendixviiianti-linearoperators
referencesandbibliography
index
-
内容简介:
grouptheoryprovidesthenaturalmathematicallanguagetoformulatesymmetryprinciplesandtoderivetheirconsequencesinmathematicsandinphysics.the"specialfunctions"ofmathematicalphysics，whichpervademathematicalanalysis,classicalphysics,andquantummechanics,invariablyoriginatefromunderlyingsymmetriesoftheproblemalthoughthetraditionalpresentationofsuchtopicsmaynotexpresslyemphasizethisuniversalfeature.moderndevelopmentsinallbranchesofphysicsareputtingmoreandmoreemphasisontheroleofsymmetriesoftheunderlyingphysicalsystems.thustheuseofgrouptheoryhasbecomeincreasinglyimportantinrecentyears.however,theincorporationofgrouptheoryintotheundergraduateorgraduatephysicscurriculumofmostuniversitieshasnotkeptupwiththisdevelopment.atbest,thissubjectisofferedasaspecialtopiccourse,cateringtoarestrictedclassofstudents.symptomaticofthisunfortunategapisthelackofsuitabletextbooksongeneralgroup-theoreticalmethodsinphysicsforallseriousstudentsofexperimentalandtheoreticalphysicsatthebeginninggraduateandadvancedundergraduatelevel.thisbookiswrittentomeetpreciselythisneed.
therealreadyexist,ofcourse,manybooksongrouptheoryanditsapplicationsinphysics.foremostamongthesearetheoldclassicsbyweyl,wigner,andvanderwaerden.forapplicationstoatomicandmolecularphysics,andtocrystallatticesinsolidstateandchemicalphysics,therearemanyelementarytextbooksemphasizingpointgroups,spacegroups,andtherotationgroup.reflectingtheimportantroleplayedbygrouptheoryinmodernelementaryparticletheory,manycurrentbooksexpoundonthetheoryofliegroupsandliealgebraswithemphasissuitableforhighenergytheoreticalphysics.finally,thereareseveralusefulgeneraltextsongrouptheoryfeaturingcomprehensivenessandmathematicalrigorwrittenforthemoremathematicallyorientedaudience.experienceindicates,however,thatformoststudents,itisdifficulttofindasuitablemodernintroductorytextwhichisbothgeneralandreadilyunderstandable.
-
目录:
preface
chapter1introduction
1.1particleonaone-dimensionallattice
1.2representationsofthediscretetranslationoperators
1.3physicalconsequencesoftranslationalsymmetry
1.4therepresentationfunctionsandfourieranalysis
1.5symmetrygroupsofphysics
chapter2basicgrouptheory
2.1basicdefinitionsandsimpleexamples
2.2furtherexamples,subgroups
2.3therearrangementlemmaandthesymmetric（permutation）group
2.4classesandinvariantsubgroups
2.5cosetsandfactor（quotient）groups
2.6homomorphisms
2.7directproductsproblems
chapter3grouprepresentations
3.1representations
3.2irreducible,inequivalentrepresentations
3.3unitaryrepresentations
3.4schur'slemmas
3.5orthonormalityandcompletenessrelationsofirreduciblerepresentationmatrices
3.6orthonormalityandcompletenessrelationsofirreduciblecharacters
3.7theregularrepresentation
3.8directproductrepresentations,clebsch-gordancoefficientsproblems
chapter4generalpropertiesofirreduciblevectorsandoperators
4.1irreduciblebasisvectors
4.2thereductionofvectors——projectionoperatorsforirreduciblecomponents
4.3irreducibleoperatorsandthewigner-eckarttheoremproblems
chapter5representationsofthesymmetricgroups
5.1one-dimensionalrepresentations
5.2partitionsandyoungdiagrams
5.3symmetrizersandanti-symmetrizersofyoungtableaux
5.4irreduciblerepresentationsofsn
5.5symmetryclassesoftensorsproblems
chapter6one-dimensionalcontinuousgroups
6.1therotationgroupso（2）
6.2thegeneratorofso（2）
6.3irreduciblerepresentationsofso（2）
6.4invariantintegrationmeasure,orthonormalityandcompletenessrelations
6.5multi-valuedrepresentations
6.6continuoustranslationalgroupinonedimension
6.7conjugatebasisvectorsproblems
chapter7rotationsinthree-dimensionalspace——thegroupso（3）
7.1descriptionofthegroupso（3）
7.1.1theangle-and-axisparameterization
7.1.2theeulerangles
7.2oneparametersubgroups,generators,andtheliealgebra
7.3irreduciblerepresentationsoftheso（3）liealgebra
7.4propertiesoftherotationalmatricesdj（a,fl,7）
7.5applicationtoparticleinacentralpotential
7.5.1characterizationofstates
7.5.2asymptoticplanewavestates
7.5.3partialwavedecomposition
7.5.4summary
7.6transformationpropertiesofwavefunctionsandoperators
7.7directproductrepresentationsandtheirreduction
7.8irreducibletensorsandthewigner-eckarttheoremproblems
chapter8thegroupsu（2）andmoreaboutso（3）
8.1therelationshipbetweenso（3）andsu（2）
8.2invariantintegration
8.30rthonormalityandcompletenessrelationsofdj
8.4projectionoperatorsandtheirphysicalapplications
8.4.1singleparticlestatewithspill
8.4.2twoparticlestateswithspin
8.4.3partialwaveexpansionfortwoparticlescatteringwithspin
8.5differentialequationssatisfiedbythedj-functions
8.6grouptheoreticalinterpretationofsphericalharmonics
8.6.1transformationunderrotation
8.6.2additiontheorem
8.6.3decompositionofproductsofyimwiththesamearguments
8.6.4recursionformulas
8.6.5symmetryinm
8.6.60rthonormalityandcompleteness
8.6.7summaryremarks
8.7multipoleradiationoftheelectromagneticfieldproblems
chapter9euclideangroupsintwo-andthree-dimensionalspace
9.1theeuclideangroupintwo-dimensionalspacee2
9.2unitaryirreduciblerepresentationsofe2——theangular-momentumbasis
9.3theinducedrepresentationmethodandtheplane-wavebasis
9.4differentialequations,recursionformulas,andadditiontheoremofthebesselfunction
9.5groupcontraction——so（3）ande2
9.6theeuclideangroupinthreedimensions:e3
9.7unitaryirreduciblerepresentationsofe3bytheinducedrepresentationmethod
9.8angularmomentumbasisandthesphericalbesselfunctionproblems
chapter10thelorentzandpoincariegroups,andspace-timesymmetries
10.1thelorentzandpoincaregroups
10.1.1homogeneouslorentztransformations
10.1.2theproperlorentzgroup
10.1.3decompositionoflorentztransformations
10.1.4relationoftheproperlorentzgrouptosl（2）
10.1.5four-dimensionaltranslationsandthepoincaregroup
10.2generatorsandtheliealgeebra
10.3irreduciblerepresentationsoftheproperlorentzgroup
10.3.1equivalenceoftheliealgebratosu（2）xsu（2）
10.3.2finitedimensionalrepresentations
10.3.3unitaryrepresentations
10.4unitaryirreduciblerepresentationsofthepoincaregroup
10.4.1nullvectorcase（pu=0）
10.4.2time-likevectorcase（c1＞30）
10.4.3thesecondcasimiroperator
10.4.4light-likecase（c1=0）
10.4.5space-likecase（c1＜0）
10.4.6covariantnormalizationofbasisstatesandintegrationmeasure
10.5relationbetweenrepresentationsofthelorentzandpoincaregroups-relativisticwavefunctions,fields,andwaveequations
10.5.1wavefunctionsandfieldoperators
10.5.2relativisticwaveequationsandtheplanewaveexpansion
10.5.3thelorentz-poincareconnection
10.5.4"deriving"relativisticwaveequationsproblems
chapter11spaceinversioninvariance
11.1spaceinversionintwo-dimensionaleuclideanspace
11.1.ithegroup0（2）
11.1.2irreduciblerepresentationsof0（2）
11.1.3theextendedeuclideangroupe2anditsirreduciblerepresentations
11.2spaceinversioninthree-dimensionaleuclideanspace
11.2.1thegroup0（3）anditsirreduciblerepresentations
11.2.2theextendedeuclideangroupe3anditsirreduciblerepresentations
11.3spaceinversioninfour-dimensionalminkowskispace
11.3.1thecompletelorentzgroupanditsirreduciblerepresentations
11.3.2theextendedpoincaregroupanditsirreduciblerepresentations
11.4generalphysicalconsequencesofspaceinversion
11.4.1eigenstatesofangularmomentumandparity
11.4.2scatteringamplitudesandelectromagneticmultipoletransitionsproblems
chapter12timereversalinvariance
12.1preliminarydiscussion
12.2timereversalinvarianceinclassicalphysics
12.3problemswithlinearrealizationoftimereversaltransformation
12.4theanti-unitarytimereversaloperator
12.5irreduciblerepresentationsofthefullpoincaregroupinthetime-likecase
12.6irreduciblerepresentationsinthelight-likecase（c1=c2=0）
12.7physicalconsequencesoftimereversalinvariance
12.7.1timereversalandangularmomentumeigenstates
12.7.2time-reversalsymmetryoftransitionamplitudes
12.7.3timereversalinvarianceandperturbationamplitudesproblems
chapter13finite-dimensionalrepresentationsoftheclassicalgroups
13.1gl（m）:fundamentalrepresentationsandtheassociatedvectorspaces
13.2tensorsinvxv,contraction,andgl（m）transformations
13.3irreduciblerepresentationsofgl（m）onthespaceofgeneraltensors
13.4irreduciblerepresentationsofotherclassicallineargroups
13.4.1unitarygroupsu（m）andu（m+,m_）
13.4.2speciallineargroupssl（m）andspecialunitarygroupssu（m+,m_）
13.4.3therealorthogonalgroupo（m+,m_;r）andthespecialrealorthogonalgroupso（m+,m_;r）
13.5concludingremarksproblemsappendixinotationsandsymbols
i.1summationconvention
i.2vectorsandvectorindices
i.3matrixindices
appendixiisummaryoflinearvectorspaces
ii.1linearvectorspace
ii.2lineartransformations（operators）onvectorspaces
ii.3matrixrepresentationoflinearoperators
ii.4dualspace,adjointoperators
ii.5inner（scalar）productandinnerproductspace
ii.6lineartransformations（operators）oninnerproductspaces
appendixillgroupalgebraandthereductionofregularrepresentation
iii.1groupalgebra
1ii.2leftideals,projectionoperators
iii.3idempotents
iii.4completereductionoftheregularrepresentation
appendixivsupplementstothetheoryofsymmetricgroupssn
appependixvclebsch-gordancoefficientsandsphericalharmonics
appendixvirotationalandlorentzspinors
appendixviiunitaryrepresentationsoftheproperlorentzgroup
appendixviiianti-linearoperators
referencesandbibliography
index
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