Itiswellknownthat,inrelativistictheory,thereisarelationbetweenmomentum p andtheenergy ε = ε(p)ofafreeparticle,
where c isthespeedoflight,and m isthemassoftheparticle.Ifthefieldcandescribe particles,itmusttakeintoaccounttherelation(1.1)betweenenergyandmomentum. Letustrytoclarifyhowtherelation(1.1)canbeimplementedforthefield.Let ˆ φ(t, x) bethefieldoperator,associatedwiththefreeparticle.Wecanwritetheexpansionas aFourierintegral,
where isthePlanckconstant.Accordingtothestandardinterpretation,thevector p istreatedasthemomentumofaparticle,andthequantity ε astheenergyofthe particle.Then,since,foreachFouriermode, ε and p arerelatedbyEq.(1.1),the quantity ε undertheintegral(1.2)isnotanindependentvariablebutisafunctionof p.Inordertosatisfythiscondition,onecanwrite
where ˆ φ∗(p)dependsonlyon p.Asaresult,wearriveattherepresentation
Whenweconsidertherelativistichigh-energyquantumphenomenainthefundamentalquantumphysicsofelementaryparticles,itisnaturaltoemploytheunits relatedtothefundamentalconstantsofnature.Thismeansthatwehavetochoose thesystemofunitswherethespeedoflightis c =1,andthePlanckconstant(which hasthedimensionoftheaction)is =1.Asaresult,weobtainthenaturalsystemofunitsbasedonlyonthefundamentalconstantsofnature.Intheseunits,the actionisdimensionless,thespeedisdimensionlessandthedimensionsofenergyand momentumcoincide.Asinquantumtheory,thereisaPlanckformula,relatingenergy
andfrequencyas ε ∼ ω,and ω ∼ 1 t , where t istime,andthedimensionssatisfythe relation
ε]=[p]=[m]=[l] 1 =[t] 1 (1.7)
Thus,wehaveonlyoneremainingdimensionalquantity,theunitofenergy.Usually, theenergyinhigh-energyphysicsismeasuredinelectron-volts,suchthattheunitof energyis1 eV ,or1 GeV =109 eV .Thedimensionsoflengthandtimeareidentical.In whatfollows,weshallusethisapproachandassumethenaturalunitsofmeasurements describedabove,with = c =1.
3)Thereexistsamaximalspeedofpropagationofaphysicalsignal.Thismaximal speedcoincideswiththespeedoflight.Inallinertialreferenceframesthespeedof lighthasthesamevalue, c
Let P1 and P2 betwoinfinitesimallyseparatedeventsthatarepointsinspacetime. Insomeinertialreferenceframe,thefour-dimensionalcoordinatesoftheseeventsare xµ and xµ + dxµ.Theintervalbetweenthesetwoeventsisdefinedas
Inanotherinertialreferenceframe,thesametwoeventshavethecoordinates x′µ and x′µ + dx′µ.Thecorrespondingintervalis
Eq.(2.3)enablesonetofindtherelationbetweenthecoordinates x′α and xµ . Let x′α = f α(x),withsomeunknownfunction f α(x).Substitutingthisrelationinto Eq.(2.3),onegetsanequationforthefunction f α(x)thatcanbesolvedinageneral form.Asaresult,
wherethematrix R =(Ri j )transformsonlyspacecoordinates, x′i = Ri j xj .Substitutingeq.(2.8)intothebasicrelation(2.7),weobtaintheorthogonalitycondition
where 13 isathree-dimensionalunitmatrixwithelements δij .Relation(2.9)defines thethree-dimensionalrotations
Ifmatrix R satisfiesEq.(2.9),thenthetransformation(2.10)istheLorentztransformation.Thus,thethree-dimensionalrotationsrepresentaparticularcaseofLorentz transformation.
Asubset H ⊂ G issaidtobeasubgroupofgroup G if H itselfisthegroupunder thesamemultiplicationruleasgroup G.Inparticular,thismeansif h1,h2 ∈ H,then h1h2 ∈ H.Also, e ∈ H,andif h ∈ H,then h 1 ∈ H
1. Let G beasetof n × n realmatrices M suchthatdet M =0.Itisevidentthat if M1,M2 ∈ G,thendet M1M2 =det M1 det M2 =0andhence M1,M2 ∈ G.Thus, thissetformsagroupundertheusualmatrixmultiplication.Theunitelementisthe unitmatrix E,andtheelementinversetothematrix M istheinversematrix M 1.We knowthatthemultiplicationofmatricesisassociative.Thus,allgroupconditionsare fulfilled.Thisgroupiscalledagenerallinear n-dimensionalrealgroupandisdenoted as GL(n|R).Considerasubset H ⊂ GL(n|R)consistingofmatrices N thatsatisfythe conditiondet N =1.Itisevidentthatdet(N1N2)=det N1 det N2 =1.Hence N1,N2 ∈ H =⇒ N1N2 ∈ H.
Considerotherpropertiesofthisgroup.Itisevidentthat E ∈ H.Onthetopofthis, N ∈ G =⇒ det N 1 =(det N ) 1 =1.
Thelastmeans N 1 ∈ H.Hence H isasubgroupofthegroup GL(n|R).Group H is calledaspeciallinear n-dimensionalrealgroupandisdenotedas SL(n|R).Inasimilar way,onecanintroducegeneralandspecialcomplexgroups GL(n|C)and SL(n|C), respectively,where C isasetofcomplexnumbers.
2. Let G beasetofcomplex n × n matrices U suchthat U +U = UU + = E,where E istheunit n × n matrix.Here,asusual,(U +)ab =(U ∗)ba or U † =(U ∗)T ,where ∗ meanstheoperationofcomplexconjugation,and T meanstransposition.Evidently, E ∈ G and,forany U1,U2 ∈ G,thefollowingrelationstakeplace: (U1U2)+(U1U2)= U + 2 (U + 1 U1)U2 = U + 2 U2 = E, (U1U2)(U1U2)+ = U1(U2U + 2 )U + 1 = U1U + 1 = E. (2.18)
Inaddition,if U ∈ G,then(U 1)+U 1 =(UU +) 1 = U 1(U 1)+ =(U +U ) 1 = E.Therefore,if U ∈ G,then U 1 ∈ G too.Asaresult,thesetofmatricesunder considerationformagroup.Thisgroupiscalledthe n-dimensionalunitarygroup U (n).
Thecondition U +U = E leadsto | det U |2 =1.Hencedet U = eiα,where α ∈ R
Onecanalsoconsiderasubsetofmatrices U ∈ U (n),thatsatisfytherelationdet U = 1.Thissubsetformsaspecialunitarygroupandisdenoted SU (n).
Sincethemultiplicationofmatricesis,ingeneral,anon-commutativeoperation, thematrixgroups GL(n, R), SL(n, R), U (n)and SU (n)are,ingeneral,non-Abelian.
Agroup G iscalledtheLiegroupifeachofitselementisadifferentiablefunction ofthefinitenumberofparameters,andtheproductofanytwogroupelementsisa differentiablefunctionofparametersofeachofthefactors.Thatis,consider, ∀g ∈ G, andfor g1 = g ξ(1) 1 ,...,ξ(1) N and g2 = g ξ(2) 1 ,...,ξ(2) N , g = g1g2 =
(ξ1,...,ξN ). Then
where I =1, 2,...,N arethedifferentiablefunctionsoftheparameters ξ(1) 1 ,...,ξ(1) N ,ξ(2) 1 ,...,ξ(2) N .TheLiegroupiscalledcompactiftheparameters ξ1,...,ξN varywithinacompactdomain.Onecanprovethattheparameters ξ1,...,ξN canbe choseninsuchawaythat g(0,..., 0)= e,where e istheunitelementofthegroup.
Thetwogroups G and G′ arecalledhomomorphicifthereexistsamap f ofthe group G intothegroup G′ suchthat,foranytwoelements g1,g2 ∈ G,thefollowing conditionstakeplace: f (g1g2)= f (g1)f (g2),andif f (g)= g′,then f (g 1)= g′−1 , where g′−1 isaninverseelementinthegroup G′.Suchamapiscalledhomomorphism. Onecanprovethat f (e)= e′,where e′ istheunitelementofthegroup G′.One-toonehomomorphismiscalledisomorphism,andthecorrespondinggroupsarecalled isomorphic.Wewillwrite,inthiscase, G = G′
Let G besomegroup,and V bearealorcomplexlinearspace.Consideramap R suchthat, ∀g ∈ G,thereexistsaninvertibleoperator DR(g)actinginthespace V Furthermore,lettheoperators DR(g)satisfythefollowingconditions:
1) DR(e)= I,where I isaunitoperatorinthespace V ;and2) ∀g1, g2 ∈ G,we have DR(g1g2)= DR(g1)DR(g2).
Themap R iscalledarepresentationofthegroup G inthelinearspace V .Operators DR(g)arecalledtheoperatorsofrepresentation,andthespace V iscalledthespace oftherepresentation.Onecanprovethat, ∀g ∈ G,thereis DR(g 1)= D 1 R (g),where D 1 R (g)istheinverseoperatorfor DR(g).Thus,thesetofoperators DR(g)formsa groupwhereamultiplicationruleistheusualoperatorproduct.
Let R bearepresentationofthegroup G inthelinearspace V ,and V bea subspacein V ,i.e., V ⊂ V .Weassumethat,foranyvector˜ v ∈ V andforany operator DR(g),thecondition DR(g)˜ v ∈ V takesplace.Then,thesubspace V is calledtheinvariantsubspaceoftherepresentation R.Anyrepresentationalwayshas twoinvariantsubspaces,whicharecalledtrivial.Thesearethesubspace ˜ V = V ,and thesubspace ˜ V = {0},whichconsistsofasinglezeroelement.Allotherinvariant
Relativisticsymmetry
subspaces,iftheyexist,arecallednon-trivial.Arepresentation R iscalledreducible ifithasnon-trivialinvariantsubspaces,andirreducibleifitdoesnot.Inotherwords, therepresentation R iscalledirreducibleifithasonlytrivialinvariantsubspaces. Arepresentationiscalledcompletelyirreducibleifallrepresentationmatrices DR(g) havetheblock-diagonalform.Thismeansthat,inacertainbasis,
Theoperators TRI arecalledthegeneratorsofthegroup G intherepresentation R Onecanshowthatanyoperator DR(ξ)whichisobtainedbythecontinuousdeformationfromtheunitelementcanbewrittenas
R(ξ)= eiξI TRI . (2.21)
Iftheoperator DR(ξ)isunitary,i.e., DR(ξ)D+ R (ξ)= D+ R (ξ)DR(ξ)= 1,thenthe generators TRI areHermitian,i.e., TRI = TR + I .Thegeneratorsofourinterestsatisfy thefollowingrelationintermsofcommutators:
where fIJ K arethestructureconstantsoftheLiegroup G.Itisevidentthat fIJ K = fJI K .Ingeneral,theformofthematrices TRI dependsontherepresentation.However,onecanprovethatthestructureconstantsdonotdependonthe representation.Thus,theseconstantscharacterizethegroup G itself.
ThegroupgeneratorsarecloselyrelatedtothenotionofLiealgebra.Let A bea realorcomplexlinearspacewiththeelements a1, a2,... .Alinearspace A iscalled Liealgebra,ifforeachtwoelements a1,a2 ∈ A,thereexistsacompositionlaw(also calledmultiplicationortheLieproduct)[a1,a2],suchthat
TheLorentzandPoincar´egroups 15
1)[a1,a2] ∈ A,
2)[a1,a2]= [a2,a1], 3)[c1a1 +
Here, c1 and c2 arearbitraryrealorcomplexnumbers,and a3 ∈ A.Thecomposition law[a1,a2]iscalledtheLiebracket,orthecommutator.Property4iscalledtheJacobi identity.
Itiseasytocheckthatthecommutatorofthegenerators(2.22)oftherepresentationoftheLiegroup G satisfiesallpropertiesofthecompositionlawfortheLie algebra.Therefore,thegenerators TRI formtheLiealgebrawhichiscalledtheLie algebraassociatedwithagivenLiegroup G.Tobemoreprecise,theyformarepresentationoftheLiealgebra.Itmeansthatonecandefinethemap T (a)oftheLie algebraintoalinearspaceofoperatorssuchthat
ALiealgebraiscalledcommutative,orAbelian,if,foranytwoelements a1,a2 ∈ A, [a1,a2]=0.Intheoppositecase,theLiealgebraiscallednon-commutative,ornonAbelian.OnecanprovethattheLiealgebraassociatedwithanAbelianLiegroupis Abelian.
wherethematrixΛsatisfiesthebasicrelation(2.7).LetusshowthatLorentztransformationsformagroup.ConsiderthesetofallLorentztransformationsor,equivalently, thesetofallmatricesΛsatisfyingΛT ηΛ= η
