Wigner_On Unitary Representations of the Inhomogeneous Lorentz Group

On Unitary Representations of the Inhomogeneous Lorentz Group E. Wigner The Annals of Mathematics , 2nd Ser., Vol. 40, No. 1. (Jan., 1939), pp. 149-204. Stable URL: http://links.jstor.org/sici?sici=0003-486X%28193901%292%3A40%3A1%3C149%3AOUROTI%3E2.0.CO%3B2-X The Annals of Mathematics is currently published by Annals of Mathematics. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html . JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/annals.html . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact [email protected]. http://www.jstor.org Fri Mar 23 09:56:35 2007 dooua oFMarmeuaric. Vol 0,No, Jeary, 1989 ON UNITARY REPRESENTATIONS OF THE INHOMOGENEOUS LORENTZ GROUP* Br E. Wioxer (Received December 22, 1997) 1,ORIGIN AND CHARACTERIZATION OFTHE PROBLEM Itisperhaps themost fundamental principle ofQuantum Mechanics that the system ofstates forms alinear manifold,’ inwhich aunitary scalar product is defined.’ Thestates aregenerally represented bywave functions’ insuch away that »and constant multiples ofyrepresent the same physical state. Itis possible, therefore, tonormalize the wave function, ie., tomultiply itbya constant factor such that itssealar product with itself becomes 1.Then, only a constant factor ofmodulus 1,the so-called phase, will beleft undetermined inthe wave function. The linear character ofthe wave function iscalled the superposition principle. The square ofthe modulus ofthe unitary sealarproduct(¥,¢)oftwonormalized wavefunctions yand¢iscalledthetransitionprobability from thestate yinto y,orconversely. This issupposed togive the probability that anexperiment performed onasystem inthestate ,tosee whether ornotthestate is¥,gives theresult that itisy. Ifthere aretwo or more different experiments todecide this (e.g., essentially thesame experiment, *Parts ofthe present paper were presented atthe Pittsburgh Symposium onGroup‘TheoryandQuantum Mechanics. Cf.Bull.Amer.Math.Soc.,41,p.306,1935. "The possibility ofafuture non linear character ofthe quantum mechanics must be admitted, ofcourse. Anindication inthis direction isgiven bythetheory ofthepositron,asdeveloped byP.A.M.Dirac(Proc.Camb.Phil.Soc.$0,150,1934,ef.alsoW.Heisenberg, Zeits.£.Phys. 90,209, 1984; 9f,623, 1984;W.Heisenberg andH.Euler,ibid.98,714,1998 andR.Serber,Phys.Rev.48,49,1935;49,545,1936)whichdoesnotusewavefunctions and is anon linear theorCf.P,A.M.Dirac,ThPrinciples ofQuantum Mechanics, Oxford 198,ChaptersTand 11;J.v,Neumann, Mathematische Grundlagen derQuantenmechanik, Berlin 1982, pages 19°24 +The wave functions represent throughout this paper states inthesense ofthe“Heisen- berg picture,”i.e.asinglewavefunctionrepresentsthestateforallpastandfuture.On the other hand, the operator which refers toameasurement atacertain time ¢contains this ¢asaparameter. (Cf. e.g. Dirac,le.ref.2,pages115-123).Oneobtainsthewave function g4(0) ofthe Schrodinger picture from the wave function eyofthe Heisenberg picture byy(t) =exp (—iHi/A)ey The operator oftheHeisenberg picture isQW) = exp(iHt/h) Qexp(—iHt/h), whereQitheoperatorintheSchrodinger picturewhichdoesnot depend ontime. Cf. alsoE.Schrédinger, Sitz.d.Kén.Preuss.Akad.p.418,1990 ‘The wave functions arecomplex quantities and theundetermined factors inthem are complex also. Recently attempts have been made toward atheory with real wave func-tions.Cf.E.Majorans, NuovoCim.14,171,1987andP.A.M.Dirac,inprint. 49 150 BE.WIGNER: performed atdifferent times) they areallsupposed togive thesame result, ice,,thetransition probability hesaninvariant physical sense. ‘The wave functions form adescription ofthephysical state, notaninvariant however, since thesame state willbedescribed indifferent codrdinate systems bydifferent wave functions. Inorder toputthis into evidence, weshall affix anindextoourwavefunctions, denoting theLorentzframeofreference forwhichthewave function isgiven. Thus yandgyrepresent thesame state, butthey are different functions. The first isthe wave function ofthe state inthe co- ordinate system 1,the second inthe codrdinate system U’. Ifg=ywthe state ¢behaves inthecoordinate system lexactly asybehaves inthecodrdinate system’. Ifgris given, allgyaredetermined uptoaconstant factor.Because oftheinvariance ofthetransition probability wehave a) Her w=Lees we) anditcanbeshown‘ that theaforementioned constants inthegcanbechosen insuch away that thegvareobtained from theg,byalinear unitary operation, depending, ofcourse, onland U” (2) vv=DV’, Dev. The unitary operators Daredetermined bythephysical content ofthetheory upto®constant factor again, which ean depend onland I’. Apart from this, constant however, theoperations D(V’, 1)andD(i,, h)must beidentical ifU’ arises from Jbythesame Lorentz transformation, bywhich I;arises from ,. Ifthis were not true, there would beareal difference between theframes of reference Iand . Thus theunitary operator D(V’, 1)=D(L) isinevery Lorentz invariant quantum mechanical theory (apart from theconstant factor which has nophysical significance) completely determined bythe Lorentz transformation Lwhich carries 1into U’=Ll. One can write, instead of(2) (2a) vu=Dijon. Bygoing over from afirst system ofreference !toasecond!’ =Inland then toa third 1”=LaL,1 ordirectly tothethird 1”=(Z_l,)l, onemust obtain—apart from the above mentioned constant—the came set ofwave functions. Hence from ev=DO" U)DU, Der eu =DO", Det itfollows @) DU UD, D=wD0", D +E, Wigner, Gruppentheorie und ihre Anwendungen avfdieQuantenmechanik derAtoms pekiren, Braunschweig 1931, pages 251-254, UNITARY REPRESENTATIONS OFLORENTZ GROUP 151 or (Ba) D(L,)D(L1) =#D(Lala), where wisanumber ofmodulus 1and ean depend onLzand L,. Thus the D(L) form, uptoafactor, arepresentation oftheinhomogeneous Lorentz group bylinear, unitary operators, ‘Wescethus’ that there corresponds toevery invariant quantum mechanical system ofequations such arepresentation oftheinhomogeneous Lorentz group. ‘This representation, ontheother hand, though notsufficient toreplace the quantum mechanical equations entirely, canreplace them toalarge extent Ifweknew, e.g.,theoperator Kcorresponding tothemeasurement ofaphysical quantity atthetime t=0,wecould follow upthechange ofthis quantity throughout time. Inorder toobtain itsvalue forthetime =4,wecould transform theoriginal wave function gbyD(V', 1)toacoérdinate system U’ thetime scale ofwhich begins atime f,later. ‘Themeasurement ofthequantity inquestion inthiscodrdinate system forthetime 0isgiven—as intheoriginal one—by theoperator K.This measurement isindentical, however, with the measurement ofthequantity attime f;intheoriginal system. Onecansaythat therepresentation canreplace theequation ofmotion, itcannot replace, how- ever, connections holding between operators atoneinstant oftime. Itmay bementioned, finally, that these developments apply notonly in quantum mechanics, butalsotoalllinear theories, e.g., theMaxwell equations inempty space. Theonly difference isthat there isnoarbitrary factor inthe description andthewcanbeomittedin(3a)andoneisledtorealrepresentations instead ofrepresentations uptoafactor. Ontheother hand, theunitary char- acter oftherepresentation isnotaconsequence ofthebasic assumptions. ‘The increase ingenerality, obtained bythepresent calculus, ascompared with theusual tensor theory, consists inthat noassumptions regarding the field nature oftheunderlying equations arenecessary. Thus more general equations, asfarasthey exist (e.g., inwhich thecodrdinate isquantized, ete.) arealso included inthepresent treatment. Itmust berealized, however, that some assumptions concerning thecontinuity ofspace have been made by assuming Lorentz frames ofreference intheclassical sense. Weshould liketo mention, ontheother hand, that theprevious remarks concerning thetime- parameter intheobservables, have only anexplanatory character, andwedonot, make assumptions ofthekind that measurements canbeperformed instan- taneously. Weshall endeavor, intheensuing sections, todetermine allthecontinuous" unitary representations uptoafactor oftheinhomogeneous Lorentz group, i.e.,allcontinuous systems oflinear, unitary operators satisfying (3a) +BE,Wigner,Le.ChapterXX. «Theexactdefinitionofthecontinuouscharacterofarepresentation uptoafactorwill begiven inSection 5A. ‘Thedefinition oftheinhomogeneous Lorentz group iscontained inSection 4A. 152 F.WIGNER 2.Companisox Wirt Previous Treatments axp Somm Iuweprate Simpiirications A. Previous treatments Therepresentations oftheLorentz group have been investigated repeatedly. ‘Thefirstinvestigation ixduetoMajorana,’ whoinfactfound allrepresentations oftheclass tobedealt with inthepresent work excepting twosetsofrepresenta- tions. “Dirac® andProca* gave more elegant derivations ofMajorana’s results and brought them into aform which eanbehandled more easily. Klein's work" does notendeavor toderive irreducible representations andseems tobe inalesselose connection with thepresent work The difference between thepresent paper and that ofMajorana and Dirac lies—apart from thefinding ofnew representations—mainly initsgreater mathematical rigor. Majorana and Dirae freely usethenotion ofinfinitesimal operators and asetoffunctions toallmembers ofwhich every infinitesimal operator canbeapplied. This procedure cannot bemathematically justified atpresent, andnosuch assumption willbeused inthepresent paper. Also the conditions ofreducibility andirreducibility could be,ingeneral, somewhat more complicated than assumed byMajorana and Dirac. Finally, theprevious treatments assume from theoutset that thespace and time coordinates willbe continuous variables ofthewave function intheusual way. This willnotbe done, ofcourse, inthepresent work B.Some immediate simplifications Two representations are physically equivalent ifthere isaone toone cor- respondence between the states ofboth which is1.invariant under Lorentz transformations and 2.ofsuch acharacter that thetransition probabilities between corresponding states arethesame, Itfollows from thesecond condition’ thatthere either exists aunitary operator Sbywhich thewave functions ©ofthesecond representation canbeobtained from thecorresponding wave functions #”ofthefirstrepresentation @) 3? =sa” orthat this istrue fortheconjugate imaginary of# Although, inthe latter ease, thetworepresentations arestillequivalent physically, weshall, in keeping with themathematical convention, notcallthem equivalent. ‘Thefirstcondition now means that ifthestates 6”, &®=Sé" correspond toeach other inonecodrdinate system, thestates D”(L)®” andD®(L)#” correspond toeach other also. We have then (4a) D°(L)®® =SD"(L)@” =SD(L) S48", +E, Majorana, Nuovo Cim. 9,335, 1992, *P.A.M, Dirac, Proc. Roy. Soc.A.156,447,1996;Al.Proca,J.dePhys.Rad.7,347, 1938. °Klein, Arkivf.Matem.Astr.ochFysik,#4,No.15,1998.IamindebtedtoMr. Darling foraninteresting conversation onthis paper. UNITARY REPRESENTATIONS OF LORENTZ GROUP 153 Asthisshall hold forevery ®®, theexistence ofaunitary Swhich transforms D"into D®isthecondition fortheequivalence ofthese two representations. Equivalent representations are not considered tobereally different and itwill besufficient tofind one sample from every infinite class ofequivalent repre- sentations. Ifthere isaclosed linear manifold ofstates which isinvariant under all Lorentz, transformations, i.e.which contains D(L)y ifitcontains y.the linear manifold perpendicular tothis one will beinvariant also. Infact, ifybelongs tothe second manifold, D(L)g will be,onaccount ofthe unitary character of D(L), perpendicular toD(L)W/ ify"belongs tothe first manifold. However, D(L-)y belongs tothefirstmanifold if¥does andthus D(L)¢ willbeorthogonaltoD(L)D(L)y =wWi.e.toallmembers ofthefirstmanifold andhelongitselftothesecond manifold also. The original representation then “decomposes” into two representations, corresponding tothe two linear manifolds, Itis clear that, conversely, one can form arepresentation, bysimply “adding” several other representations together, i.e. byconsidering asstates linear combinations ofthestates ofseveral representations and assume that thestates which originate from different representations areperpendicular toeach other. Representations which areequivalent tosums ofalready known representa- tions arenot really new and, inorder tomaster allrepresentations, itwill be sufficient todetermine those, outofwhich allothers canbeobtained by“adding” afinite orinfinite number ofthem together. Two simple theorems shall bementioned here which will beproved later (Sections 7Aand 8Crespectively). The first one refers tounitary representa- tions ofany closed group, the second toirreducible unitary representations of any (closed oropen) group. ‘The representations ofaclosed group byunitary operators canbetransformed. into the sum ofunitary representations with matrices offinite dimensions. Given two non equivalent irreducible unitary representations ofanarbitrary group. Ifthescalar product between thewave functions isinvariant under the operations ofthegroup, thewave functions belonging’ tothe first represerita- tion areorthogonal toallwave functions belonging tothesecond representation C.Classification ofunitary representations according tovon Neumann andMurray” Given the operators D(L) ofaunitary representations, or@representation uptoafactor, one can consider thealgebra ofthese operators, i.e.alllinear combinations a,D(La) +aD(La) +asD(La) ++ ofthe D(L) and alllimits ofsuch linear combinations which are bounded operators. According tothe properties ofthis representation algebra, three classes ofunitary representations can bedistinguished. 18. J.Murray andJ.v.Neumann, Ann.ofMath.$7,116,1996;J.v.Neumann,tobe publishedsoon 154 E.WIGNER, ‘The first class ofirreducible representations has arepresentation algebra which contains allbounded operators, i.e.ifyand garetwo arbitrary states, there isanoperator ofthe representation algebra forwhich AY =¢and AY’ =if¥/isorthogonal toy. Itisclear that thecenter ofthealgebra con- tains only theunit operator and multiply thereof. Infact, ifCisinthecenteronecandecompose Cy=ay+y'sothaty’shallbeorthogonal toy.However,¥’must vanish since otherwise Cwould notcommute with theoperator which leaves yinvariant and transforms every function orthogonal toitinto 0.For similar reasons, amust hethesame forall. For irreducible representations there isnoclosed linear manifold ofstates, (excepting themanifold ofallstates) which isinvariant under allLorentz transformations. Infact, according tothe above definition, a¢’arbitrarily close toany gcan berepresented by@finite lincar combination aD(LiW +mD(La)v + +++agD(La)y- Hence, aclosed linear invariant manifold contains every state ifitcontains one, ‘This is,infact, the more customary definition forirreducible representations and theone which will beused subsequently. Itiswell known that allfinite dimensional representations are sums ofirreducible representations. ‘This is nottrue,” ingeneral, inaninfinite number ofdimensions. The second class ofrepresentations will becalled factorial. For these, the center oftherepresentation algebra still contains only multiples ofthe unit operator. Clearly, the irreducible representations are allfactorial, but not conversely. For finite dimensions, the, factorial representations may contain oneirreducible representation several times. This isalso possible inaninfinite number ofdimensions, but inaddition tothis, there are the “continuous” representations ofMurray andvonNeumann.” These arenotirreducible as there areinvariant linear manifolds ofstates. Ontheother hand, itisimpossible tocarry thedecomposition sofarastoobtain asparts only irreducible repre- sentations. Inalltheexamples known sofar, therepresentations into which these continuous representations can bedecomposed, are equivalent tothe original representation. The third class contains allpossible unitary representations. Inafinite number ofdimensions, these can bedecomposed first into factorial repre- sentations, andthese, inturn, inirreducible ones. Von Neumann’ hasshown that the first step still ispossible ininfinite dimensions. We can assume, therefore, from theoutset that wearedealing with factorial representations. Inthetheory ofrepresentations offinite dimensions, itissufficient todeter- nine only theirreducible ones, allothers arcequivalent tosums ofthese. Here, itwill benecessary todetermine allfactorial representations. Having done that, weshall know from theabove theorem ofvon Neumann, that allrepre- sentations areequivalent tofinite orinfinite sums offactorial representations. Itwill beone ofthefesults ofthedetailed investigation that theinhomo-geneousLorentzgrouphasno“continuous” representations, allrepresentations UNITARY REPRESENTATIONS OF LORENTZ GROUP 155 can bedecomposed into irreducible ones. Thus the work ofMajorana and Dirac appears tobejustified from this point ofview aposteriori. D.Classification ofunitary representations from thepoint ofview of infinitesimal operators ‘The existence ofaninfinitesimal operator ofacontinuous one parametric (cyclic, abelian) unitary group haxbeen shown byStone."' Heproved that the operators ofsuch agroup canbewritten asexp(iHt) where Hisa(bounded or unbounded) hermitean operator and¢isthegroup parameter. However, the Lorentz group has many one parametric subgroups, and the corresponding infinitesimal operators H,, H:,--- are all unbounded. For every H; an everywhere dense setoffunctions ¢ean befound such that Hg can bedefined. Itisnot clear, however, that aneverywhere dense setcan befound, toall members ofwhich every 1can beapplied. Infact, itisnot clear that one such ¢can befound. Indeed, itmay beinteresting toremark that foranirreducible representation theexistence ofonefunction ¢towhich allinfinitesimal operators canbeapplied, entails theexistence ofaneverywhere dense setofsuch functions. This again hastheconsequence that onecanoperate with infinitesimal operators toalarge extent intheusual way. Proor: LetQ(t) beaoneparametric subgroup such that QQ’) =Q(t+t’) Iftheinfinitesimal operator ofallsubgroups canbeapplied to¢,the (6) lim(QW —De exists. Itfollows, then, that the infinitesimal operators ean beapplied toRy also where Risanarbitrary operator oftherepresentation: Since R“Q(t) Ris also aone parametrie subgroup lim((R" QR —Dg=limR7-(QW —DRe ‘=o rary also exists and hence also (Risunitary) lim(QW —DRe. Every infinitesimal operator canbeapplied toRyifthey alleanbeapplied toy, and the same holds for sums ofthe kind () alte +anlage ++++ +ankue. ‘These form, however, aneverywhere dense setoffunctions iftherepresentation isirreducible. Iftherepresentation isnotirreducible, one can consider thesetNoofsuch wave functions towhich every infinitesimal operator canbeapplied. This setis MM. H.Stone, Proc, Nat. Acad. 16, 173, 1990, Ann, ofMath. $3,643, 1992, also J.v. Neumann, ibid, $3,567, 1992. 156 E,WIGNER clearly linear and, according totheprevious paragraph, invariant under the operations ofthegroup (i.e. contains every Rgifitcontains ). ‘The same holds fortheclosed setNVgenerated byNoand also ofthesetPoffunctions which areperpendicular toallfunctions ofNV. Infact, ify,isperpendicular to allgnofN,itisperpendicular alsotoallRg, and, fortheunitary character of R,theRg, isperpendicular toallgs,i.e.isalso contained inthesetP. Wecandecompose thus, byaunitary transformation, every unitary repre- sentation intoa“normal” anda“pathological” part. Fortheformer, there is aneverywhere dense setoffunctions, towhich allinfinitesimal operators canbe applied. ‘There isnosingle wave functions towhich allinfinitesimal operators ofa“pathological” representation could beapplied. According toMurray andvonNeumann, iftheoriginal representation was factorial, allrepresentations into which it-ean bedecomposed willbefactorial also. Thus every representation isequivalent toasum offactorial repre- sentations, part ofwhich is“normal,” theother part “pathological.” Itwillturn outagain that theinhomogeneous Lorentz group hasnopath- ological representations. Thus this assumption ofMajorana and Dirac also willbejustified aposteriori. Every unitary representation oftheinhomogenous Lorentz group canbedecomposed intonormal irreducible representations. It should bestated, however, that therepresentations inwhich theunit operator corresponds toevery translation have notbeen determined todate (cf.also section 3,end). Hence, theabove statements arenotproved forthese repre- sentations, which are,however, more truly representations ofthehomogeneous Lorentz group, than oftheinhomogeneous group. While allthese points may beofinterest. tothemathematician only, thenew representation oftheLorentz group which willbedescribed insection 7may interest thephysicist also. Itdescribes aparticle with acontinuous spin. Acknowledgement. Thesubject ofthispaper wassuggested tomeascarly as 1928 byP.A.M.Dirac who realised even atthat date theconnection ofrepre- sentations with quantum mechanical equations. Iamgreatly indebted tohim alsoformany fruitful conversations about thissubject, especially during the years 1934/35, theoutgrowth ofwhich thepresent paper is. Tam indebted alsotoJ.v.Neumann forhishelp andfriendly advice. 3,Summary oFENsuING Szcrions Section 4willbedevoted tothedefinition oftheinhomogeneous Lorentz group and the theory ofcharacteristic values and characteristic vectors ofa homogeneous (ordinary) Lorentz transformation. The discussion will follow very closely thecorresponding, well-known theory ofthegroup ofmotions in ordinary space and thetheory ofcharacteristic values oforthogonal trans- formations.” Itwillcontain only astraightforward generalization ofthe methods usually applied inthose discussions. *Cf. e.g. B,Wigner,Le.ChapterIII,0.VeblenandJ.W.Young,PrajectiveGeometry, Boston1917, Vol. 2,especially Chapter VII UNITARY REPRESENTATIONS OF LORENTZ GROUP 187 InSection 5,itwillbeproved that onecandetermine thephysically meaning less constants intheD(L) insuch away that instead of(8a) themore special equation @ D(Li)D(L2) =+D(LsL2) will bevalid. ‘This means that instead ofarepresentation uptoafactor, we can consider representations uptothesign. For thecase that either LyorLz isapure translation, Dirac” hasgiven aproof of(7)using infinitesimal operators. Aconsideration very similar tohiscanbecarried out, however, also using only finite transformations. For representations with afinite number ofdimensions (corresponding toan only finite number oflinearly independent states), (7)could beproved also if both L;and L;arehomogeneous Lorentz transformations, byastraightforward application ofthemethod ofWeyl andSchreier."* However, theLorentz group hasnofinite dimensional representation (apart from thetrivial oneinwhich the unit operation corresponds toevery L). ‘Thus themethod ofWeyl and Schreier cannot beapplied. Itsfirst step istonormalize theindeterminate constants in every matrix D(L) insuch away that the determinant ofD(L) becomes 1 Nodeterminant can bedefined forgeneral unitary operators. ‘The method tobeemployed here willbetodecompose every LLinto aproduct, oftwo involutionsL=MNwithM*=N*=1.ThenD(M)andD(N)willbe normalized sothat their squares become unity and D(L) =D(M)D(N) set Itwill bepossible, then, toprove (7)without going back tothetopology ofthe group. Sections 6,7,and 8will contain the determination oftherepresentations. ‘The pure translations form aninvariant subgroup ofthewhole inhomogeneous Lorentz group andFrobenius’ method" willbeapplied inSection 6tobuild uptherepresentations ofthewhole group outofrepresentations ofthesubgroup, bymeans ofa“little group.” InSection 6,itwillbeshown onthebasis ofanas yetunpublished work” ofJ.v.Neumann that there isacharacteristic (in- variant) setof“momentum vectors” forevery irreducible representation. ‘The irreducible representations oftheLorentz group will bedivided into four classes. ‘The momentum vectors of the 1stclass are time-like, 2ndclass arenull-vectors, butnotalltheir components will bezero, 3rdclass vanish (i.e., alltheir components will bezero), 4thclass arespace-like. Only thefirst two cases will beconsidered inSection 7,although thelast ease PA. M.Dirac, mimeographed notes oflectures delivered atPrinceton University, 1934/35, page 5a.1H.Weyl,Mathem.Zeits.23,271;24,328,877,789,1925;0.Schreier,Abhandl.Mathem. Seminar Hamburg, 4,15,1926;5,28,1927. 4G. Frobenius, Sitz. d.Kon. Preuss. Akad. p.501, 1898, I.Schur, ibid, p.164, 1906;F.Seitz,Ann.ofMath.$7,17,1936 158 B,WIGNER may bethemost interesting from themathematical point ofview. Ihope to return toitinanother paper. Ididnotsucceed sofaringiving acomplete discussion ofthe3rdclass. (All these restrictions appear intheprevious treatments also.) InSection 7,weshall find again allknown representations oftheinhomo- geneous Lorentz group (i.e., allknown Lorentz invariant equations) and two new sets. Sections 5,6,7willdealwith the“restricted Lorentz. group” only, i.e.Lorentz transformations with determinant 1which donot reverse thedirection ofthe time axis. Insection 8,therepresentations oftheextended Lorentz group will beconsidered, thetransformations ofwhich arenotsubject tothese conditions, 4.Description orTHe INHoMocENEoUs Lorentz Grove A. Aninhomogeneous Lorentz transformation L=(a,A)istheproduct ofa translation byareal vector a 8) m= rta (i=1.2,3,4) and ahomogeneous Lorentz transformation Awith real coefficients (9) r=Dedar. ‘The translation shall beperformed after the homogeneous transformation. ‘The coefficients ofthehomogeneous transformation satisfy three conditions: (1)TheyarerealandAleavestheindefinite quadratic form—2}—x}—2}+24invariant: (10) APA’ =F where theprime denotes theinterchange ofrows and columns and Fisthe diagonal matrix with thediagonal elements —1,—1,—1,+1.—(2) Thedeter- minant|Au|=1and—(@)Ay>0. Weshall denote theLorentz-hermitean product oftwoveetors xandyby (ay (2,9) =-2in —aly —atu +sty. (The star denotes theconjugate imaginary.) If{z,x]<0thevector xis called space-like, if{z,z}=0,itisanullvector, if{2,2}>0,itiscalled time- like. Arealtime-like vector liesinthepositive light cone ifx,>0;itliesinthe negative light cone ifx1<0.Two vectors zand yarecalled orthogonal if {z,y}=0. Onaccount ofitslinear character ahomogeneous Lorentz transformation is completely. definedifAvisgivenforfourlinearlyindependent vectorsv”, 2)9. 0,oo. From (11)and(10)itfollows that {»,w}={Av,Aw)forevery pairofvectors v,w.This willbesatisfied forevery pair ifitissatisfied forallpairs o,» UNITARY REPRESENTATIONS OF LORENTZ GROUP 159 offour linearly independent vectors. The reality condition issatisfied if (Av®)* =A(o*) holds forfoursuch vectors. ‘The scalar product oftwo vectors xand yispositive ifboth lieinthepositive light cone orboth inthenegative light cone. Itisnegative ifone liesinthe positive, the other inthe negative light cone. Since both zand yaretime-like ln >laP +a tials lal >lal tly +y. Hence, by Schwarz’s inequality |2fye| >|ztyi +z1ys+ziya| andthesignofthescalar product oftwo real time-like vectors isdetermined bythe product oftheir time components. Atime-like vectoristransformed byaLorentztransformation intoatime-likevector. Furthermore, onaccount ofthecondition Aw>0,thevector »® with thecomponents 0,0,0,1remains inthepositive light cone, since thefourth component ofAv”isAg. Ifvisanother vector"* inthepositive light cone fo, o} >0andhence also {Av, Av} >0andAv" isinthepositive light cone also. ‘The third condition for aLorentz transformation can beformulated also asthe requirement, that every vector in(oron) thepositive light cone shall remain in(or,respectively, on)thepositive light cone. ‘This formulation ofthe third condition shows that the third condition holds fortheproduct oftwo homogeneous Lorents transformations ifitholds forboth factors. The same isevident for the first two conditions. From AFA’ =Foneobtains bymultiplying with A“from theleftand AT =(A)' from theright F=A'F(AY’ sothat thereciprocal ofahomo- geneous Lorentz transformation isagain such atransformation. ‘The homo- geneous Lorentz transformations form agroup, therefore. One easily calculates that theproduct oftwo inhomogeneous Lorentz trans- formations (b,M)and (c,N)isagain aninhomogeneous Lorentz transformation (a,A) (a2) (b,M)(c, N)=(a,A) where (12a) Aa=DOMgNas a=6+DMacy, or,somewhat shorter (12b) A=MN; a=b+Me. B.Theory ofcharacteristic values and characteristic vectors ofahomogeneous Lorentz transformation Linear homogeneous transformations are most simply described by their characteristic values and vectors. Before doing this for the homogeneous . Lorentz group, however, weshall need two rules about orthogonal vectors. ‘Wherever aconfusion between vectors and vector components appears tobepossible,upperindiceswillbeusedfordistinguishing differentveetorsandlowerindicesfordenoting the components ofavector. 160 ‘EB,WIGNER (1]If{v,w}=0andfv,v}>0,then(w,w}<0;if{v,w}=0,fv,v}=0, thenwiseither space-like, orparallel to(either {w,w]<0, orw=co). Proor: (13) oto =ofuy +vluy +vfs. BySchwar2’s inequality, then (14) fo fae SClon+LoefF+fsPC|wrF+|ae|?+|osP) For|vg?>|[+|ve|?+|vsPitfollowsthat|wef?<|wsfF+|ws+|ws If|m |’=|u[ +|| +|)? thesecond inequality stillfollows ifthein- equality signholds in(14). ‘Theequality signcanhold only, however, ifthe first three components ofthevectors vandwareproportional. Then, on account of(13)andboth being nullvectors, thefourth components areinthe same ratio also. [2]Iffourvectors v,v,v®,varemutually orthogonal andlinearly inde- pendent, oneofthem istime-like, three arespace-like. Proor: Itfollows fromtheprevious paragraph thatonlyoneoffourmutually orthogonal, linearly independent vectors canbetime-like oranull vector. It remains tobeshown therefore onlythatoneofthem istime-like. Since they arelinearly independent, itispossible toexpress bythem anytime-like vector oe. av” Wsp>ao, ‘Thescalarproductoftheleftsideofthisequation withitselfispositiveandtherefore {Faw £cue}>o7 or (15) Dias fo, o) >0 andone{v“,«|must bepositive. Four mutually orthogonal vectors arenot necessarily linearly independent, because anullvector isperpendicular toitself, ‘Thelinear independence follows, however, ifnone ofthefour isanullvector. Wegoover now tothecharacteristic values \ofA. These make thedeter- minant|A—1|ofthematrixA—Alvanish. [3]Ifdisacharacteristic value, \*,X*and*~' arecharacteristic values also. Poor: For\*thisfollows from thefactthatAisreal. Furthermore, from |.4—M1|=0also| A’—A1|=0follows, andthismultiplied bythedeter- minants ofAFandF™gives [AF|-[a'—ot|-| Fl=|arart —yal=[1-4] =0, sothat2isacharacteristic value also, 4]Thecharacteristic vectors vyandv,belonging totwocharacteristic valuesdxand Asareorthogonal ifMd. #1. UNITARY REPRESENTATIONS OFLORENTZ GROUP 161 Proor: on,va}=(Avy,Avg}=[Arosdata]=ArAatoi,02} ‘Thusif{ry,v2}#0,7d=1. 15]Ifthemodulus ofacharacteristic value is|X|#1,thecorresponding characteristic vector visanull vector and itself real and positive. From |v,v}={Av, Av]=|[[o, 0}the{»,»}=0follows immediately for JA|#1.Ifwere complex, A*would beacharacteristic value also. The characteristic vectors ofXand X*would betwo different null vectors and, becauseof[4],orthogonal toeachother.Thisisimpossible onaccountof[1].‘Thus 2isrealandvarealnullvector. Then, onaccount ofthethird condition forahomogeneous Lorentz transformation, \must bepositive. 6]Thecharacteristic value dofacharacteristic vector voflength nullisrealand positive. IfAwere notreal, *would beacharacteristic value also. ‘The corresponding characteristic vector v*would bedifferent from v,anull vector also, and per- pendicular to»onaccount of(4]. This isimpossible because of[1]. [7]Thecharacteristic vector vofacomplex characteristic value d(themodulus of which is1onaccount of{5})isspace-like: |v,v}<0. Proor: \*isacharacteristic value also, thecorresponding characteristic vector isv*. Since (A*)*A =0?#1,[o*,v}=0.Since they aredifferent, at least oneisspace-like. Onaccount of|v,»}=[o*,v*}both arespace-like. If allfour characteristic values were complex andthecorresponding characteristic vectors linearly independent (which istrueexcept ifAhaselementary divisors) weshould have fourspace-like, mutually orthogonal vectors. ‘This isimpossible, onaccount of[2]. Hence [8]There isnotmore than onepair ofconjugate complex characteristic values, ifAhasnoelementary divisors. Similarly, under thesame condition, there is ‘notmore than onepair d,X”'ofcharacteristic values whose modulus isdifferent from 1.Otherwise their characteristic vectors would beorthogonal, which ‘they cannot be,being null vectors. Forhomogeneous Lorentz transformations which donothave elementary divisors, thefollowing possibilities remain: (a)There isapair ofcomplex characteristic values, their modulus is1,on account of[5] (16) MaMa [N=|el=1, andalsoapairofcharacteristic values Xs,4s,themodulus ofwhich isnot1 ‘These must bereal and positive: (16a) M=M MEN DO. ‘The characteristic vectors oftheconjugate complex characteristic values are conjugate complex, perpendicular toeach other andspace-like sothatthey can be normalized to —1 an not; foo)=fn,et]=0 for, am}=for,m} =—1 162 E,WIGNER those oftherealcharacteristic values arerealnuulvectors, their scalar product ean benormalized to1 wae mae fa}= (17a)\ ! {vo} ={v0} =0. Finally, theformer pairofcharacteristic vectors isperpendicular tothelatter kind (7b) for,es}=for,0}=for,0s}=fon,ve}=0. Itwillturn outthat alltheother cases inwhich Ahasnoelementary divisor arespecial cases of(a). A c.Sa Fra.1.Position ofthecharacteristic valuesforthegeneral casea)inthecomplex plane. Inease b),Xvand\«coincide andareequal 1;incase c),X:and 2coincide and areeither +lor—1, Incase d)both pairsds=4=landA;=ds~+1coincide. (b)There isapair ofcomplex characteristic values 4, =di'=M, AixOT,[Ar]=[Ae]=1.Nopairwith||#1,however. ‘Thenonaccountof[8],stillXy=AFwhich gives with|\y|=1,4s=&1.Since theproduct Add =1,onaccount ofthesecond condition forhomogeneous Lorentz transformations, also y=dy=+1.‘The double characteristic value 1has twolinearly independent characteristic vectors vsand vwhich canbeassumed tobeperpendicular toeach other, {v,v4]=0.According to[2],oneofthe four characteristic vectors must betime-like andsince those of\.anddyare space-like, thetime-like onemust belong to:t1. This must bepositive, thereforeXs=dy=1.Outofthetime-likeandspace-likevectors{vs,v5}=—1 and{%,v4}=1,onecanbuildtwonullvectors%+%and%—%.Doing this, ease (b)becomes thespecial case of(a)inwhich thereal positive char- acteristic values become equal 4s=Xz"=1. (c)Allcharacteristic values arereal; there ishowever onepair Xs=XJ, UNITARY REPRESENTATIONS OF LORENTZ GROUP 163 de=Av!themodulus ofwhich isnotunity. ‘Then {v%,#6}={%,%)=0 and s>Oand one can conclude for\:and Xz,asbefore fordsand A,that dy= dz=+1. This again isaspecial ease of(a); here thetwo characteristic values ofmodulus 1become equal. (a) Allcharacteristic values arereal and ofmodulus 1. Ifallofthem are+1, wehave theunit matrix which clearly canbeconsidered asaspecial case of(a). ‘The other case isky=4=—1, As=by=+1. The characteristic vectors of dsand A;must bespace-like, onaccount ofthethird condition forahomogeneous Lorentz transformation; they can beassumed tobeorthogonal and normalized to—1. This isthen aspecial case of(b)and hence of(a)also. The cases (a), (b), (©), (@)areillustrated inFig. 1. ‘The cases remain tobeconsidered inwhien Ahas anelementary divisor. We set therefore (18) Made =Deve} Mee =ete +0 Itfollowsfrom[5]thateither|A,|=1,oF{ve,v4}=0.Wehave{v,w.}= (Ave, Awe} =|. [vewe}+(ee,24].Fromthisequation (19) {vem} =0 follows for|\,|=1,sothat(19)holdsinanycase. Itfollows thenfrom [6]that A,isreal, positive and »,,w,canbeassumed tobereal also. The last equation now becomes {x,w.]=X2{v-,we}sothateither4,=1or{vs,we)=0. Finally, we have fw.,we}=[Aewe,Aewe}=AE[we,we)+BWelwe, vo}+(ve,ve). ‘This equation now shows that (198) (ws,v6}=0 evenif4=1.From(19),(19a)itfollowsthatw,isspace-like andcanbenormalized to (19b) {we, we}=—1. Inserting (19a) into thepreceding equation wefinally obtain (19) = 1 [9]IfA,hasanelementary divisor, allitscharacteristic roots are1. From (19¢) weseethat theroot oftheelementary divisor is1and this isat least. adouble root. IfAhadapair ofcharacteristic values ds#1,ds=Xi’, thecorresponding characteristic vectors »,and »;would beorthogonal to»,andtherefore space-like. Onaccountof{5],then|x|=|x|=1and(x,2}=0. Furthermore,from{w,,0%}={Aetwoe,Asi}=Arfwe,i}+Arte,1}andfrom {v.,1}=Oalso{w,,1}=Ofollows. Thusallthefourvectorsv,v2,%.,w.would bemutually orthogonal. This isexcluded by(2]and (19). 164 B,WIGNER ‘Two cases areconceivable now. Either thefourfold characteristic root has only onecharacteristic vector, orthere isinaddition tov,(atleast) another characteristic vector v. Intheformer case four linearly independent vectors 4%,,W,.ZeZecould befound such that. Ade = Nave =te+06 Mees etu Ate =ete However [0,2] ={Aeve, Aote] ={ve,xe]+[ve24]from which {ve,ze]=0 follows. On the other hand bwey te]=[Aeros Ate] =[WeyZe}+Le,we}+fe,Ze}+fe,wel, ‘Thisgiveswith(19a)and(19b){ve,ze]=1sothatthiseasemustbeexcluded.(e)There isthus avector »;sothatinaddition to(18) (18a) Aw =0 holds. From {we,0]={Aewe,Adi}=[we,m1}+{ve,vi}follows (19d) fe,m1}=0. ‘Theequations (18),(18a) willremain unchanged ifweaddtow.and»;amultiple ofv.. Wecanachieve inthiswaythat thefourth components ofboth w,and »%vanish. Furthermore, »canbenormalized to—1andadded tow,also with anarbitrary coefficient, tomake itorthogonal to. Hence, wecanassume that (19e) mu=wae=0;inn} =—1; fume,nm}=0, Wecanfinally define thenullvector z.tobeorthogonal tow,and;andhave a sealar product 1with v (ast) teer2ed=beywe)=fen)=0;feo}=1 ‘Then thenullvectors v,and2,represent themomenta oftwolight beams in opposite directions. IfwesetA,z: =av,+bw.+cz+dvtheconditions {2,0} =[Aze, Ao] give, ifwesetfor»thevectorsv.,we,v1theconditions ©=1;b=¢;2ac—b*—d' =0;d=0.Hence Ate=VeNatty=we+ (20) Aa =r Mee =Fe+we+dee ALorenta transformation with anelementary divisor canbebest characterized bythenullvector v,which isinvariant under itandthespace part ofwhich forms with thetwoother vectors w,and»;three mutually orthogonal vectors in ordinary space. The twovectors w,and»arenormalized, »;isinvariant under «while thevector v,isadded tow,upon application ofA.. The result ofthe application ofA,toavector which islinearly independent ofv,,w,and0is, aswesaw,already determined bytheexpressions forAwe, AdeandA.. ‘TheA.(y) which have theinvariant null vector veandalso w.(and hence also UNITARY REPRESENTATIONS OF LORENTZ GROUP 165 ‘m)incommon and differ only byadding tow,different multiples ye.ofve, form aeyclic group with y=0,theunit transformation asunity: Ada)Ady’)=Ady+7’). The Lorentz transformation M(a) which leaves vand w,invariant but re- places v,byav,(and z,bya‘z,)hastheproperty oftransforming A.(y) into M(q) Ac(y)M(a)! =Adley). (+) Anexample ofA,(y) and M(a) is 100t)|| jo4 Y 7 Ady) = 2 a0mytat ont|10 y be +h) 200 tO)\ ol 0 o| M(a)= 1 ~y|00Hata’) Ka-a')) 100Me-a@') Ha+a’)) ‘These Lorentz transformations play animportant réle inthe representations with space like momentum vectors. ‘Abehavior like (+) isimpossible forfinite unitary matrices because the characteristic values ofM(a)'A.(y)M(a) and A.(7) arethesame—those of Ad(ya) =Ad(q)" thea®powers ofthose ofAc(v). ‘This shows very simply that the Lorentz group has notrue unitary representation inafinite number of dimensions. C.Decomposition ofahomogeneous Lorentz transformation into rotations and ‘anacceleration inagiven direction ‘The homogeneous Lorentz transformation is,from the point ofview ofthe physicist, atransformation toauniformly moving codrdinate system, theorigin ofwhich coincided at¢=0withtheoriginofthefirstcodrdinate system. One can, therefore, first perform arotation which brings the direction ofmotion of the second system into agiven direction—say the direction ofthe third axis— and impart itavelocity inthis direction, which will bring ittorest. After this, thetwo codrdinate systems can differ only inarotation. This means that every homogeneous Lorentz transformation can bedecomposed inthe following way” (21) A= RZS *Cleg. L.Silberstein, TheTheoryofRelativity,London1924,p.142. 166 E,WIGNER whereRand§arepurerotations, (ie.Ric=Ru=Su=Su=Oforix4 andRy=Su=1,alsoR’=R',S’=S™)andZisanacceleration inthedirection ofthethird axis, ic. j1 000 Jo100} 2Jooad |lo0dba} witha—b*=1,a>6>0.Thedecomposition (21)isclearly notunique. Itwillbeshown, however, that Zisuniquely determined, ic.thesame inevery decomposition oftheform (21). Inorder toprove thismathematically, wechose Rsothat inR*A =Ithe first two components inthe fourth column I.=In=0become zero: R™ shall bring thevector with thecomponents Aw, Ax, Asinto thethird axis. ‘Then wetake Iy=(Aix+Ale+Ad)! andIq=Auforbandatoform Z; theysatisfytheequation Ii,—Ij,=1.Hence,thefirstthreecomponents ofthefourth column ofJ=ZT=Z"RAwillbecomezeroandJu=1,becauseof Jia—Sis—Sh—Sic=1.Furthermore, thefirstthree components ofthe fourth rowofJwillvanish also, onaccount ofJi,—Ji,—Js—J=1ice. J=S=Z"R" Aisapure rotation. This proves thepossibility ofthede- composition (21). ‘Thetrace ofAA’=RZ*R™ isequal tothetrace ofZ’,i.e.equal to2a"+26+2=4a=4b*+4whichshowsthattheaandbofZareuniquely deter-mined. Inparticulara=1,6=OandZtheunitmatrixifAA’=lie.Aa pure rotation. Itiseasy toshow now that thegroup space ofthehomogeneous ‘Lorents transformations isonly doubly connected. Ifacontinuous series A(t) of homogeneous Lorentz transformations isgiven, which isunity both for¢=0 and for¢=1,weeandecompose itaccording to(21) (21a) AW =ROZOSW. Itsalsoclearfromtheforegoing,that.R(f)canbeassumedtobecontinuous in¢,exceptforvaluesof¢,forwhichAw=An=Ax=0,Le.forwhichAisapure rotation. Similarly, Z(¢) will becontinuous in¢and this will hold even where A(t)ispure rotation. Finally, S=Z-'R™A willbecontinuous also, except where A(t) isapurerotation. Let usconsider now the series ofLorentz transformations (21b) AO =ROZO'SO where thebofZ(0)' isstimes thebofZ(t). Bydecreasing sfrom 1to0we continuously deform thesetAx(!) =A(t)ofLorentz transformations intoasetof rotations Aa(!) =R()S(t). Both thebeginning Ag(0) =1andtheendA,(1) =1 ofthesetremain theunit matrix and thesets A,(t) remain continuous in¢for UNITARY REPRESENTATIONS OF LORENTZ GROUP 167 allvalues ofs.This last fact isevident forsuch ¢forwhich A(f) isnot arota- tion: forsuch ¢allfactors of(21b) arecontinuous. But itistrue also for& forwhich A(t) isarotation, and forwhich, hence Z(t) =Land A,(ls) =A(t) = A(t). AsZ(t) iseverywhere continuous, there will beaneighborhood oft inwhich Z(t) and hence also Z(})' isarbitrarily close totheunit matrix. In thisneighborhood A(t) =A(). S()™'Z()*Z()" S(t)isarbitrarily close to A(O; and, iftheneighborhood issmall enough, this isarbitrarily close to Ao) =Ad(l). ‘Thus (21b) replaces thecontinuous setA(t) ofLorentz transformations bya continuous setofrotations. Since these form anonly doubly connected mani- fold, themanifold ofLorentz transformations can notbemore than doubly connected. The existence ofatwo valued representation” shows that itis actually doubly and notsimply connected. ‘Wecanform anewgroup" from theLorentz group, theelements ofwhich are theelements oftheLorentz group, together with away A(t), connecting A(1) =Awith theunity A(0) =E.However, twoways which canbecontinuously deformed into each other are not considered different. The product ofthe element “Awith theway A(t)” with theelement “Iwith theway I(f)” isthe element AIwith theway which goes from Ealong A(t) toAand hence along AI()) toAI. Clearly, theLorentz group isisomorphie with thisgroup andtwo elements (corresponding tothetwoessentially different ways toA)ofthisgroup correspond tooneelement oftheLorentz group. Itiswellknown,” that this group isholomorphic with thegroup ofunimodular complex two dimensional transformations, Every continuous representation oftheLorentz group “uptothesign” isa singlevalued, continuous representation ofthisgroup. ‘Thetransformation which corresponds to“Awith theway A(f)” isthat d(A) which isobtained bygoing over from d(Z) =d(A(0)) =1continuously along d(A(t)) tod(A(1)) =d(A). D.The homogeneous Lorentz group issimple Itwillbeshown, first, that aninvariant subgroup ofthehomogeneous Lorentz group contains arotation (i. atransformation which leaves 2invariant).— Wecanwrite anarbitrary element oftheinvariant subgroup intheform RZS of(21).Fromitspresence intheinvariant subgroup followsthatofS-RZS-S*=SRZ =TZ. IfX,istherotation byxabout thefirstaxis, X,ZX, =Z* andX,7ZX,"' =X,TX,X.2X, =X,TX,Z" iscontained intheinvariant subgroup alsoandthusthetransform ofthiswith Z,ie.Z'X.TX, also. The product ofthiswith TZisTX.TX,which leaves invariant. IfTX,TX, =1 wecan take TY,TY,. Ifthis istheunity also, TX,7X, =TY,TY, and T commutes with X,Y, ,ie.isarotation about thethird axis. Inthiscase the 8CE.H.Weyl, GruppentheorieundQuantenmechanik, Ist.ed.Leipzig1928,pages110-114, 2nded.Leipzig1931,pages130-133.Itmaybeinteresting toremarkthatessentially thesame isomorphism hasbeen recognized already byL.Silberstein, lc.pages 148-157. 168 BE,WIGNER space like (complex) characteristic vectors ofTZlieintheplane ofthefirst two codrdinate axes. Transforming TZbyanacceleration inthedirection ofthe first coordinate axis weobtain anew element oftheinvariant subgroup for which thespace like characteristic vector will have anot vanishing fourth component. Taking this forRZS wecantransform itwith Sagain toobtain a new SRZ =TZ. However, since Sleaves x,invariant, thefourth component ofthespace like characteristic vectors ofthis TZwill not vanish and wecan obtain from itbytheprocedure just described arotation which must becon- tained intheinvariant subgroup. Itremains tobeshown that aninvariant subgroup which contains arotation, contains thewhole homogeneous Lorentz group. Since thethree-dimensional rotation group issimple, allrotations must becontained intheinvariant sub- group. Thus therotation byxaround thefirst axis X,,and also itstransform with Zand also @X0".X, =2-X.0"X, =2 iscontained intheinvariant subgroup. However, thegeneral acceleration in the direction ofthe third axis can bewritten inthis form. Asallrotations are contained inthe invariant subgroup also, (21) shows that this holds forall elements ofthehomogeneous Lorentz group. Itfollows from this that thehomogeneous Lorentz group hasapart from the representation with unit matrices only true representations. Itfollows then from theremark attheend ofpart B,that these have allinfinite dimensions. ‘This holds even forthetwo-valued representations towhich weshall beledin Section 5equ. (52D), asthegroup elements towhich thepositive ornegative unit matrix corresponds must form aninvariant subgroup also, and because the argument attheend ofpart Bholds fortwo-valued representations also. One easily sees furthermore from theequations (52B), (52C) that itholds forthe inhomogeneous Lorentz group equally well. 5,Repucrion orReeresentations Ur to4Factor ToTwo-VaLvep REPRESENTATIONS ‘The reduction willbeeffected bygiving each unitary transformation, which is defined bythephysical content ofthetheory and theconsideration ofreference only uptoafactor ofmodulus unity, a“phase,” which willleave only thesign ofthe representation operators undetermined. The unitary operator cor- responding tothetranslation awillbedenoted by(a), that tothehomogeneous Lorentz transformation Abyd(A). Tothe general inhomogeneous Lorentz transformation then D(a, A)=T(a)d(A) will correspond, Instead ofthe relations (12), weshall usethefollowing ones. (22B) T(a)T(b) =(a, b)T(a +b) (2c) (A)T(a) =w(A, a)T(Aa)d( A) (22D) d(A)d(Z) =w(A, Da(AD. UNITARY REPRESENTATIONS OFLORENTZ GROUP 169 ‘The warenumbers ofmodulus 1.They enter because themultiplication rules (12) hold fortherepresentatives only uptoafactor. Otherwise, the relations (22) areconsequences of(12) and canintheir return replace (12) Weshall replace theT(a), d(A) by9(a)T(a) and9(A)d(A) respectively, for which equations similar to(22)hold, however with (22) (a,b)=1; (Aa)=1;(A,1)=41 AL Itisnecessary, first, toshow that theundetermined factors intherepresenta tionD(L) canbeassumed insuch away that thew(a,b),0(A, a),w(A, I)become —apart from regions oflower dimensionality—continuous functions oftheir arguments. This isaconsequence ofthecontinuous character oftherepresenta- tion and shall bediscussed first. (a)From thepoint ofview ofthephysicist, thenatural definition ofthe continuity ofarepresentation uptoafactor isasfollows. ‘Theneighborhood 3 ofaLorentz transformation Lo=(b,I)shall contain allthetransformations L=(a,A)forwhich |a,—b,|<Sand|Aw—Iu|<4.Therepresentationuptoafactor D(L) iscontinuous ifthere istoevery positive number¢,every normalized wave function gandevery Lorents transformation Losuch aneigh- borhood 6ofLethat forevery Lofthis neighborhood onecanfind an@of modulus 1(the@depending onLandg)such that (ug, u,)<¢where (23) uy=(D(Lo)—2D(L))e. Letusnow take @point Lointhegroup space andfindanormalized wave function gforwhich |(y,D(Lo)y) |>1/6. There always exists a¢with this property, if|(y,D(Le)y)| <1/6then ¥=ap+BD(Lo)e withsuitably chosen and 6willbenormalized and|(y,D(Ls)¥) |>1/6. Weconsider then such ‘aneighborhood %ofLsforallLofwhich |(,D(L)e) |>1/12. Itiswell known" that thewhole group space canbecovered with such neighborhoods. Wewant toshow nowthattheD(L)g canbemultiplied with such phase factors (depending onL)ofmodulus unity thatitbecomes strongly continuous inthe region 2. Weshall chose thatphase factor sothat(y,D(L)y) becomes realandpositive. Denoting then (23) (DL)—D(L))e =Us, the(Uy, Uy)canbemade arbitrarily small byletting Lapproach sufficiently near toLy,ifL;isin9.Indeed, onaccount ofthecontinuity, asdefinedabove,thereisan@=e*suchthat(u,u)<€ifLissufficiently neartoLy where u=(D(Li) —e*D(L))e. 1This condition isthe“separability” ofthegroup. Cf.e.g.A.Haar, Ann. ofMath., 34, 147, 1933. 170 E, WIGNER ‘Taking theabsolute value ofthescalar product ofuwith goneobtains |(@,D(La)e)—cos«(y,D(L)¢)—isinx(e,D(L)e)|=|(ew)|SVe, because ofSchwartz's inequality. Ifonly y/e <1/12, thexmust besmaller than x/2 because theabsolute value iscertainly greater than thereal part, and both (y,D(L;)¢) and (y,D(L)g) arereal and greater than 1/12 Asthe absolute value isalso greater than the imaginary part, we sink <12. On the other hand, Uy,=ut (e*—1)D(L)g, and thus (Uy,Uo!S(uywh+le=1)SVe+Bin4/2 (Uy, Uy) S625 «. (b)Itshall beshown next that ifD(L)¢ isstrongly continuous inaregion and D(L) iscontinuous inthesense defined atthe beginning ofthis section, then D(L)y with anarbitrary yis(strongly) continuous inthat region also. We shall see, hence, that the D(L), with any normalization which makes a D(L)e strongly continuous, iscontinuous inthe ordinary sense: There isto everyLy,andeveryya8sothat(Uy,Uy)<«where Uy=(Dia) =DY) ifLisintheneighborhood 6ofL,. Itissufficient toshow thecontinuity ofD(L)y where yisorthogonal to». Indeed, every ¥’can bedecomposed into two terms, ¥/=ay+Bytheone of which isparallel, the other perpendicular to¢.Since D(L)¢ iscontinuous, according tosupposition, D(L)y’ =aD(L)p +AD(L)¥ willbecontinuous also if, D(L)¢ iscontinuous. The continuity oftherepresentation uptoafactor requires that itispossible toachieve that (uy,uy) <€and (Uysg ,Uyeg) <€where (23a) uy=(D(Ls)—2,DL))¥, (23b) Use=(DULi)~QrpD(L))(Y +@), with suitably chosen 0's. According tothe foregoing, xtalso ispossible to choose Land Lrsoclose that (Uy, Uy) <& Subtracting (23’) and(23a) from (23b) andapplying D(L)~' onboth sides gives, (2p=Mee+(l=Mpegle=DL)Huyeg—uy—Uy) ‘The scalar product ofthe right side with itself isless than 9¢. Hence both |—%,| <3and|1 —%,,| <3¢or|1—94)<6¢. Becauseof Uy=uy—(1—9)D(L)¥, the(Uy, Uy) <(uy, uy)!+|)—94|andthus (Uy, Uy) <496. UNITARY REPRESENTATIONS OF LORENTZ GROUP 171 ‘This completes theproof ofthetheorem stated under (b). Italso shows that notonly thecontinuity ofD(L)y hasbeen achieved intheneighborhood ofLe bythenormalization used in(a)but also that ofD(L)y with every ¥,i.e., the continuity ofD(L) Itisclear also that every finite part ofthe group space can becovered by finite number ofneighborhoods inwhich D(L) can bemade continuous. Itis easy toscethat thewof(22) will bealso continuous inthese neighborhoods so that isispossible tomake them continuous, apart from regions oflower dimen- sionality than their variables have. Inthefollowing only the fact will beused that they can bemade continuous intheneighborhood ofany a,band . B. (a)Wewant toshow next that all7(a) commute. From (22B) wehave (24) T(a)T(b)T(a)* =c(a,b)T(b) where (a, b)=w(a, b)/w(b, a) and hence (24a) (a,b)=c(b,a). ‘Transforming (24) with 7(a’) one obtains T(a’)T(a)T(b)T(a)*T(a’)' =e(a,b)T(a')T()T(a’)* or ala’, a)T(a’ +a)T(B)a(a’, a)*T(a' +a) =e(a,B)e(a', 6)T'(6) or (25) e(a, b)e(a’, b)=c(a+a’,b) Itfollows” from (25) andthepartial continuity ofe(a,b)that (26) c(a,t)=exp(2xXa.f(0)) and, since thisisequal to(b,a) =exp(—2ni D>befe(a)) (27) 2S00)+bafela))=n(a,b) where n(q,6)isaninteger. Setting in(27)forbthevector e”’the\component ofwhich is1,alltheothers zeroandforf.(e™) =—fa Ala) =n(a,e™) +Dasa, and putting this back into (27) weobtain (28) Jfalard, +bay)+acn(b,e)+ben(a,e)=n(a,6). 1G. Hamel, Math, Ann. 60,460, 1905, quoted from H.Hahn, Theorie derreellen Funk- tionen. Berlin 1921, pages 581-583. 172 E.WIGNER Assuming forthecomponents ofaand bsuch values which aretranscendental both with respect toeach other and thefa(which arefixed numbers), onesees that (28) cannot hold except ifthecoefficient ofevery onevanishes (29) Jathe=0;nlb,e) =0, sothat (26) becomes (0) cla,8)=exp(213faas.). Itisnecessary now toconsider theexistence ofanoperator d(A)satisfying (22C). ‘Transforming thisequation withthesimilar equation containing 6instead ofa d(A)T(@)a(A)"d(A)T(a)a(A)"d(A)TW)"day =(A, )T(Ab)o(A, a)T(Aa)o(A, bY'T(AB)* =o(A, @)e(Ab, Aa)T(Aa), while thefirst lineisclearly d(A)c(b, a)T(a)d(A) =..o(A, ae(b, a)T(Aa) whence (31) e(b, a)=(Ab, Aa) holds forevery Lorentz transformation A. Combined with (30) this gives &(faa, =Lhrdecdmads) =n'(a,b), where n’(a, b)isagain aninteger. Asthis equation holds forevery a,b Jo=LSnbedas f=ASA must hold also, for every Lorentz transformation. However, the only form invariant under allLorentz transformations are multiples ofthe Fof(10). Actually, because of(29), fmust vanish and c(a, b)=1,allthe operators corresponding totranslations commute (32) T(a)T(b) =T()T(a). Itiswell toremember that itwas necessary forobtaining this result touse the existence ofd(A) satisfying (22C). (b) Equation (32) isclearly independent ofthe normalization ofthe T(a). Ifwecould fixthetranslation operators infour linearly independent directions e,ee, e"sothat foreach ofthese directions (33) T(ae®))T(be) =T((a +bye) bevalid forevery pair ofnumbers a,6,then the normalization (88a)Thare+age+aye“+age)=Tae)Tae)Taye)T(ae) and (32) would ensure thegeneral validity of (34) T(a)T@) =Tl+d). UNITARY REPRESENTATIONS OF LORENTZ GROUP 173 Asthefourlinearly independent directions ¢"”,-.-,ec”weshalltakefour null vectors. Ifeisanull veetor, there is,according tosection 3,ahomogeneous Lorentz transformation” A,such that Ae=2c. ‘We normalize 7(e) sothat (35) a(A)T@d(A)? =TO) This isclearly independent ofthenormalization ofd(A.). Wefurther normalize forall(positive and negative) integers n (85a) AA)"T(e)d(A)™ =T(2"e). Itfollows from this equation also that (36) T(2%e)* =d(A,)"T(e)"d(A.)-* =d(A,)"d(A)T()d(A.)""d( A,"=T2"**e). This allows ustonormalize forevery positive integer k (5b) T(k-2"e) =Te)" insuch away that thenormalization remains thesame ifwereplace kby2" andnbyn+m. This ensures, together with (36), thevalidity of T(ve)T(ue) =T(%+w)e) (36a) si d(A,)T(ve)d(A.)* =T(2ve) foralldyadic fractions »and p. Itmust beshown that ifn,2,95, ---isasequence ofdyadic fractions, converging to0,limT(ve)=1.FromT(a)-T(0) =o(a,0)T(a)itfollowsthatT(0) isaconstant. According tothe theorem ofpart (A)(b), the T(ve), if multiplied byproper constants 2,willconverge to1,ie.,bychoosing anarbi- trary ¢,itispossible tomake both (1—2%T(ve))e =wand (1—2T(ve)) d(A.)"e =wu’arbitrarily small, bymaking »small. Applying d(A,) tothe second expression, oneobtains, for(36a), that (1—9T(2ve))e =d(A.)u’ is also small. Ontheother hand, applying (ve) tothefirst expression one sees that (T(ve) —%7'2ve))p =T(ve)u approaches zero also. Hence, thedifference ofthese two quantities (1—T(ve))e goes tozero, i.e.T(viee converges tay ifr, v2,5,~~isasequence ofdyadic fractions approaching 0. Now v1,v2,»s, «++beasequence ofdyadic fractions converging toanarbi- trary number a.Itwillbeshown then that 7(v.e) converges toamultiple of T(ae) and this multiple ofT(ae) willbethenormalized T(ae). Again, itfollows from thecontinuity that there aresuch 9;that 2%7'(we)e converges toT(ae)e. The 0;'T(v,e)‘2:7(v.e)gwillconvergetoy,therefore,asbothiandjtendto infinity. However, according totheprevious paragraph, T((v; —»,)e)e tends togand thus 05'Q; tends to1.Itfollows that 9;'converges toadefinite number 2.Hence9%".27(vie)e converges to2T(ae)y whichwillbedenoted,henceforth, byT(ae). Forthe(ae), normalized inthis way, (33) will hold, 41The index¢denotesherethevector¢forwhichA,¢=2¢;thisA,hasnoelementary divisor. aa E,WIGNEH sinceifwi,we.as.---aredyadicfractions converging tob,weobtain,withthehelp (36a) Tae)T(bee=limTm+wee=T(a+dodo. ‘This argument notonly shows that itispossible tonormalize theT(ae) and hence by(33a) the T(a) sothat (34) holds forthem but, inaddition tothis, that these T(a) willbecontinuous intheordinary sense. c. Itisclear that (34) will remain valid ifone replaces 7(a) byexp (2xi{a, e})T(a) where cisanarbitrary vector. This remaining freedom inthenormalization of (a) will beused toeliminate thew(A, a)from (22C). ‘Transforming (22C) d(A)T(a)d(A)* =w(A, a)T(Aa) with d(M) oneobtains ontheleftside o(M, A)d(MA)T(a)o(M, \)* d(MA)* =o(MA, a)T(M Aa) while theright side becomes w(A, a)w(M, Aa)T(MAa). Hence (37) (MA, a)=w(M, Aa)o(A,a). Ontheother hand, theproduet oftwo equations (22C) with thesame Abut with aand 6respectively, instead ofayields with thehelp of(34) w(A, a)wo(A, b)=w(A(a +b)). Hence: (A,a)=exp(2rifa,f(A)}), where f(A) isavector which can depend onA. Inserting this back into (37) one obtains fa,((MA)} =[Aa, $M} +fa,f(A) +n, ta,(MA) —AM) ~f(A)} =n, where nisaninteger which must vanish since itisalinear function ofa. Hence (38) SOA) =ATY(M) +f(A). Ifwecan show that themost general solution oftheequation is (39) HA) =(A =Ino, where visavector independent ofA,thew(A, a)will become w(A, a)=exp (2ri}(\ —1)a,v}).‘Then(A,a)in(22C)willdisappear ifwereplace T(a)by exp (2rifa, })7(a). UNITARY REPRESENTATIONS OF LORENTZ GROUP 175 The proof that (39) isaconsequence of(38) issomewhat laborious. One ean first consider the following homogeneous Lorentz transformations a 0 0S co 0 —s 0 oOaq10 0Cy0S| Xa, = 5 Y(ae,y2) = uM=Towg0fond*a0 S00G 0S06 (40) .|a&00 -sa00 Zax, 3) == lo oa 0 0SsCy where ¢=c0sa4;8;=sinag;C;=Chys; 8;=Shy:. All the X(a, 7)com- mute. Letuschoose, therefore, twoangles &,71forwhich 1—X(a1, 71)" hasareciprocal. Itfollows then from (38) Xa, f(Xler, 10) +[(X(a, 1)=Xr 1)H(X(a,Y))+SX(ar 1) 41) orf(X(@,y)) =(l= Xa) "T'U —X@, ¥)"Y(Xla, n)) S(X(a, 1)=(1=X(a, 7)ox, where vxisindependent ofa,y. Similar equations hold forthef(¥(a, y))and S(Za,1)). Letusdenote nowX(x,0) =X;¥(x,0) =¥;Z(x,0) =Z.These anticommute inthefollowing sense with thetransformations (40): (42) YX(a,y¥ =ZX(a,1)Z =X(a, 7)". From (38) one easily calculates S(YX(a,y)¥)=(¥X(a,7)+VAY)+YIX(@, 1), or,because of(41) and (42), after some trivial transformations (43) (1=X(a,7))(1 —¥)(ex —wy)=0. Asa,ycan betaken arbitrarily, the first factor can bedropped. This leaves (1=¥)(ox —o)=0,orthat thefirst and third coniponents ofvxand vyare equal. One similarly concludes, however, that (1—X)(or —vx)=Oandthus thatthefirstthreecomponents ofvx,vyandalsoofveareequal.For 7:=72=71=0the transformations (40) are the generators ofall rotations, i.e.allLorents transformations Rnotaffecting thefourth codrdinate. Asthe4-4matrix element ofthese transformations is1,theexpression (1—Ro isindependent ofthefourth component of»and(1—R™')ox =(1—R')or = (1—R)ez. Itfollows from (38)that iff(R) =(1—Rox andf(S) = (1—Sox, then (SR) =(1—R*S“)ox. Thus f(R) =(1—Roe is valid with the same oxfor allrotations. Now S(X(a, YR) =RL —X(a,y) ox +(l= Rox =(1=(X(@ DR) or- 176 E.WIGNER: One easily concludes from (38) that thef(Z) corresponding totheunit operation vanishes andf(A"!)=—Af(A). Hencef(R"'X(a, y)')=(1—X(a,7)R)vx 5 and one concludes further that forallLorentz transformations A=RX(a, y)S, (89) holds with »=—vx ifRand Sarerotations. However, every homogene- ousLorentz transformation can bebrought into this form (Section 4C). ‘This completes theproof of(39) and thusofw(A,a)=1 D. ‘The quantities w(a, 6)and w(A, a)forwhich ithasjust been shown that they canbeassumed tobe1,areindependent from thenormalization ofd(A). We canaffixtherefore anarbitrary factor ofmodulus 1toallthed(A),without interfering with the normalizations sofar accomplished. Inconsequence hereof, the ensuing discussion will besimply adiscussion ofthe normalization | “& Fic. 2 oftheoperators forthehomogeneous Lorentz group and theresult tobeobtained will bevalid forthat group also. Partly because the representations uptoafactor ofthe three dimensional rotation group may beinteresting inthemselves, but more particularly because theprocedure tobefollowed forthe Lorentz group can beespecially simply demonstrated forthis group, thethree dimensional rotation group shall betaken upfirst. Itiswell known that thenormalization cannot becarried sofarthatw(A, I)=1 in(22D) and there arewell known representations forwhich w(A, I)=1. We shall allow this ambiguity therefore from theoutset. One can observe, first, that theoperator corresponding totheunity ofthe group isaconstant. This follows simply from d(A)d(B) =w(A, B)d(A). ‘The square ofanoperator corresponding toaninvolution isaconstant, therefore. UNITARY REPRESENTATIONS OF LORENTZ GROUP 77 ‘The operator corresponding totherotation about theaxis ebytheangle x, normalized sothat itssquare beactually 1,will bedenoted byé; =1.The @are—apart from thesign—uniquely defined. Arotation Rabout vbytheangle aistheproduct oftwo rotations by* about ¢:and ¢where ¢;and ¢,are perpendicular tovand ¢,arises from e:by rotation about vwith a/2. Choosing forevery vanarbitrary eperpendicular to 2,wecan normalize, therefore (44) d(R)=+82 Now d(R) commutes with every d(S) ifSisalso arotation about v.‘This is proved inequations (24)-(30). ‘The fuin(30) must vanish onaccount of(29). a Fro. 3 (Also, both &and Scan bearbitrarily accurately represented aspowers ofa very small rotation about »). Hence, transforming (44) byd(S) one obtains (44a) a(R) =£d(S)za(S)*-a( Spend). Now d(S)éd(S)* corresponds toarotation by about anaxis, perpendicular tovand enclosing anangle @with e,,where 8istheangle ofrotation ofS. Since thesquare ofd(S)éd(S)-* isalso1,(44a) issimply another wayofwriting d(R) =&% asaproduct oftwo &and wesee that the normalization (44) is independent ofthechoice oftheaxis e(Cf. Fig. 2). Forcomputing d(R)d(T) wecandraw theplanes perpendicular totheaxes of rotation ofRand T'and useford(R) =éé such adevelopment that theaxis ec ofthe second involution coincide with the intersection line ofthe above-men- tioned planes, while ford(T) =éér wechoose the first involution tobearota- tion about this intersection line (Fig. 3). ‘Then, the product (45) d(R)U(T) =énbetode =ener 178 E,WIGNER will automatically have the normalization corresponding to(44). ‘This shows that theoperators normalized in(44) give arepresentation uptothesign. For the Lorentz group, the proof can beperformed along the same line, only the underlying geometrical facts are less obvious. Let AbeaLorentz trans- formation without elementary divisors with thecharacteristic values ¢, 7, €%,¢®aadthecharacteristic vectors 1,02=vt.v3,14,a8described insection 4B. We want tomake \=MN with M? =N?=. For AN =M,wehave ANAN =Land thus AVA =N. SettingNv,=22auve,weobtain$VAv,= DLraudws =Laur. Because ofthelinear independence ofthevgthis Amounts toAten =atx: allaearezero, except those forwhich Ade =1.As innone oftheeases (a), (b), (¢),(d)ofsection 4BisA;orXsreciprocal toone of thelasttwo A,thevectors »and tywill betransformed byNVinto alinear com- bination of»;and vagain, and thesame holds fort3and v. ‘This means that N can beconsidered astheproduct oftwo transformations N=N.N,, thefirst inthev0»plane, thesecond inthevs»,plane. (Instead of»,0,plane one really should say0;+v2,iy—tvsplane, aso; and %arecomplex themselves. ‘This willbemeant always byviveplane, ete.). ‘The same holds forMalso. Both N,and N,must satisfy thefirst and third condition forLorentz trans- formations (cf.4A) and both determinants must beeither 1,or—1. Further- more, thesquare ofboth ofthem must beunity. Ifboth determinants were +1, theN;had tobeunity itself, while NV,could betheunity orarotation byrinthev0»plane. Thus 0%,v2,¥s,v¢would be characteristic vectors ofNitself. Ifhoth determinants are —1(this will turn out tobethecase), Nyisare- flection onaline intheoyeplane and N,areflection inthev0plane, inter- changing v5and v. Inthis ease 0,vs,03,%would not allbecharacteristic vectors of N. Ifvy,v2,vs,v4arecharacteristic vectors ofN,they arecharacteristic vectors ofM=ANalso. ‘Then both Mand Nwould becither unity, orarotation by+ inthenwplane. Ifboth ofthem were rotations intheoy"plane, their product .\ would betheunity which wewant toexclude forthepresent. Wecanexclude theremaining cases inwhich thedeterminants ofN,and N,are+1byfurther stipulating that neither MnorNshall betheunity inthedecomposition A= MN. Hence Nistheproduct ofareflection intheve:plane (46a) Ns=s; Ns=—s where s,ands,aretwoperpendicular realvectors intheviv:plane (46) seaeln ten; 8=ile —Dy), and ofareflection inthev0,plane (46) Nhat; Ny=—b, UNITARY REPRESENTATIONS OF LORENTZ GROUP 179 where again t,,t,arerealvectors inthevs»,plane, perpendicular toeach other, t,being space-like, t,time-like: (46d) heen tern; G=en— en. Thus Nbecomes arotation byxinthepurely space like sl,plane. ‘The M can becalculated from M=AN Ms; =ANs, =As,=ey, +omy =4°%(8,—is.)+478)+is)=cosQy-s)+sin2y-8, (466)Ms,=sinQy-8;=08ae,-_Mi,=ANE=AML=ey+oy =4G +4)+$M —4)=Ch2x-th +Sh2x4, Mt,=—Sh2x-t,—Ch2x-4,. ‘Thus Malso becomes aproduct oftworeflections, oneintheow=#48,the other inthets=ft,plane. This completes thedecomposition ofAintotwo involutions. One oftheinvolutions can betaken tobearotation byxinan arbitrary space likeplane, intersecting both thenw:and thevs»planes, asthe freedom inchoosing »and yallows ustofixthe lines s,and ¢,arbitrarily in those planes. The involution characterized by(46) will becalled N,, hence- forth. The other involution Misthen asimilar rotation, inaplane, however, which iscompletely determined once thes,t,plane isfixed. Itwill bedenoted byM,, (itis,infact My =Nosy nx). One sees thecomplete analogy tothe three dimensional ease ifone remembers that yand xare the half angles of rotation. ‘The d(M) and d(N) sonormalized that their squares be1shall bedenoted by 4\(M,,) and di(N,,). Wemust show that thenormalization for (47) aA) =+4,(M,,)di(N,,) isindependent ofyand x.Forthis purpose, wetransform (47a) d(A) =d(Ma)di(Noo) with d(A:) where A,has the same characteristic vectors asAbut different characteristic values, namely e”,e”, &ande™. Since A.\MwAr' =M,, and AiNwAr' =N,,wehave d(A;)di(Mw)d(A:)~* =wdi(M,,) where w=+1,as the squares ofboth sides are 1.Hence, (47a) becomes iftransformed with d(A,) just (a7) d(A)d(A)d(Ay)* =£d(M,,)d(N.,). ‘Thenormalization (47)wouldbeclearlyindependent of»anduifd(A,)com- muted with d(A). Again, theargument contained inequations (24)-(30) can beapplied and shows that (48) d(Ay)d(A)d( As)=exp(2mif(2yu —2x»))a(A) 180 E,WIGNER holdsforevery7,x,»,4.However, theexponential in(48)mustbe1ify=0;v=2n/n; x=}nusince inthiscase A=Af. Thus exp(—4s'ifu) =1for every andf=Oand theleftside of(47b) canbereplaced byd(A); thenormali- zation in(47) isindependent of»and u. Inorder tohave theanalogue of(45), wemust show that, having two Lorentz transformations A=M,,N,, and I=PayQog wecan chose »,uand a,8so that N,, =Pagie.that theplane ofrotation s,ofN,,coincide with theplane ofrotation ofPas. Asthe latter plane can bemade toanarbitrary space- like plane intersecting both thewiw and thewst0, planes (where wi, ts, wa, w are thecharacteristic vectors ofJ),wemust show theexistence ofaspace like plane, intersecting allfour planes nv, vv, wit, wx. Both the first and thesecond pair ofplanes areorthogonal. One can show* that ifAand Jhave nocommon null vector ascharacteristic #Wefirst suppose theexistence ofareal plane pintersecting allfour planes vit, exes,wy;,wns.Ifpintersects vv:theplaneqperpendicular topwillintersecttheplanevoyperpendicular tovie:.Indeed,thelinewhichisperpendicular tobothpandvies(thereis‘suchalineaspandneintersect) iscontained inbothgandoes.Thisshowsthatifthereisaplaneintersecting llfourplanes,theplaneperpendicular tothiswillhavethispropertyAlso, 2)x fa, Fig. 4gives aprojection ofalllines into the2:21 plane. One sees that there are, in general, twointersecting planes, only inexceptional cases isthere only one. Iftheplanep—theexistenceofwhichwesupposeforthetimebeing—containstime-likevector,4willbespace-like (Section4B,[1]).Bothinthiseaseandifpcontainsonlyspace-like vectors, thetheorem inthetextisvalid. ‘There isalastpossibility, that pis tangent tothelight cone, i.e.contains only space like vectors and anull vector v.‘The space-like vectors ofpareallorthogonal tov,otherwise pwould contain time-like vectors also. Inthis case theplane g,perpendicular topwill contain »also. ‘The line inwhich wsintersects pisspace-like andorthogonal tothevector inwhich eyintersects p.‘The latter intersection must coincide with »,therefore, asnoother vector ofpisorthogonal to UNITARY REPRESENTATIONS OF LORENTZ GROUP 181 vector, there arealways two planes, perpendicular toeach other which intersect four such planes. One ofthese isalways space like. Itispossible toassume, therefore, that both N,, and Pag aretherotation byxinthis plane. Thus ao) (Ad)=sdi(Mra)di(N,)d(Pas)di(Qas)=di(M,,)d(Qas), and d(A)d(I) hasthenormalization corresponding totheproduct oftwo involu- tions, neither ofwhich isunity. This is,however, also the normalization adopted ford(AI). Hence (49a) a(ad() =#d(Al) holds ifA,Jand AJare Lorentz transformations corresponding toone ofthe cases (a), (b), (c)or(d)ofsection 4Band ifAand Jhave nocommon character- istic null vector. Inaddition tothis (49a) holds also, assuming d(E) =+1, ifany ofthe transformations A,I,AJisunity, orifboth characteristic null vectors ofAandJareequal,as inthiscasetheplanesv3»,andtwandalso vivand wyw» coincide and there aremany space like planes intersecting all. IfAand Jhave one common characteristic null vector, v5=us,theothers, y.and ‘uwrespectively, being different, one can use anaritifice toprove (49a) which will beused inlater parts ofthis section extensively. One can find aLorentz transformation Jsothat none ofthepairs I—J;A—IJ;AIJ —J“hasa common characteristic null vector. This will betrue, e.g. ifthe characteristic null vectors ofJare %and another null vector, different from rm,wand the characteristic vectors ofAZ. ‘Then (49a) will hold foralltheabove pairs and d(A)d(I) =£d(A)d(I)d(J)d(J") =2dA)dTJ)d(J) =4d(AIJ)d(J) =+4(AD). any spave-like vector init. Hence, vis the intersection ofpand #» and iseither »oFvy« ‘One can conclude inthe same way that»coincideswitheitherwsoFw,alsoandweseethat ifpistangent tothelight cone thetwo transformationsAandJhaveacommonnullvector ascharacteristic vector. ‘Thus the theorem inthe text iscorrect ifweean show the exiet-fenceofanarbitraryrealplanepintersecting allfourplanesoes,vse,wes».Letusdrawacodrdinatesysteminourfourdimensional space,the2:2,planeofwhichis the vits plane, the 2sand 2.axes having the directions ofthe vectors ry—veand 0,+0%, respectively. The three dimensional manifold Mfcharacterized byz.=1intersects all planes inaline, theesplane intheline atinfinity ofthe2.23 plane, theextplane inthezs axis. The intersection ofMwith thew,w, and wsws planes willbe lines inMwith directions perpendicular toeach other. They will have acommon normal through theorigin ofM, intersecting itatreciprocal distances. ‘This follows from their orthogonality inthe four ‘dimensional space. ‘Applane intersecting ov,and 4»will bealine parallel to2:2; through the2,axis. Ifwe ‘draw such lines through allpoints oftheline corresponding towes, the direction ofthis linewillturnby+ifwegofromoneendofthislinetotheother. Similarly, thelinesgoingthrough theline corresponding toww, will turn byxintheopposite direction. Thus the firat set oflines will have atleast one line incommon with the second aet and this line will correspond toareal plane intersecting allfour planes ei, rs, wee, ws... This com- pletes theproof ofthetheorem referred tointhetext. 182 B.WIGNER This completes theproof of(49a) foralleases inwhich A,Jand Afhave no elementary divisors. Itisevident also that weeanreplace inthenormalization (47)thedybyd.Onealsoconcludes easily thatd(M)* isinthesame representa- tion either +1forallinvolutions M,or—1forevery involution. The former ‘ones will give real representations, the latter ones representations uptothe sign. IfAhasan elementary divisor, itcanbeexpressed inthev,wv,,2s,»scheme asthe matrix (Cf. equ. (20)) 11 40} aeob to\ ajoo10! 10001) and can bewritten, inthesame scheme, astheproduct oftwo Lorentz trans- formations with thesquare 1 {1 -14O}l100OF ‘0 -110|/0-100} Ae=MoNo =| | i"=lo 01oO|/o01oF jo ©0-1/[0 001) Wecannormalize therefore d(A.) =:d(Mo)d(No). IfAeabewritten asthe product oftwo other involutions also A,=M,N; thecorresponding normaliza- tion will beidentical with theoriginal one. Inorder toprove this, letuscon- sider aLorentz transformation Jsuch that neither ofthe Lorentz transforma- tions'J,NoJ,NiJ,AeJ=MoNoJ=MsNiJhaveanelementary divisor. Sincethenumber offree parameters isonly 4inease (e),while 6forcase (a),this is always possible. Then, for(45a) d(Mo)d(No)d(J) =+d(Mo)d(NoJ) =+d(MoNoJ) =+d(M\NiJ) =+d(Mi)d(NJ) =+d(Mi)d(Ni)d(J) and thus d(M,)d(No) =+:d(M,)d(N}). This shows also that even ifAIisincase(¢),w(A,I)=1,since(49)leadstothecorrectnormalization.IfA=MN has ahelementary divisor, Inot, d(A)d(J) still will have the normalization corresponding totheproduct oftwo involutions. One can find again aJsuch that neither ofthetransformations J,J’, IJ,NIJ, MNIJ, have anelementary divisor. Then a(Myd(N)d(T) =dM) a(N)aCaCI)a)* =£d(Ma(N)aI Dad) =d(M)a(N TIA) =d(AJ)a(J™), The last product has, however, thenormalization corresponding totwo involu- tions, aswasshown in(49a), since neither AIJ, norJisincase (e). UNITARY REPRESENTATIONS OF LORENTZ GROUP 183 Lastly, wemust consider theease when both Aand Jmay have anelementary divisor. Inthisease, weneed aJsuch that neither ofJ,J~', IJhave one. Then, because ofthegeneralization of(49a) just proved, inwhich the first factor isin,case (e) a(A)d(D) =d(Adda) =zaAya) aS) =£d(AlJ)d(J™) which hastheright normalization, This completes theproof of (50) w(A,1)=+1 forallpossible cases, and thenormalization ofallD(L) ofarepresentation oftheinhomogeneous Lorentzgroupuptoafactor,iscarriedoutinsuchawaythatthe normalized operators give arepresentation uptothesign. Itiseven carriedsofarthatinthefirsttwoofequations (22)#=1canbeset.Weshallconsider henceforth systems ofoperators satisfying (7), or,more specifically, (22B) and (22C) with o(a, b)=w(A,a)=Land(22D)witho(A,I)=£1. E Lastly, itshall beshown that therenormalization notonly didnotspoil the partly continuous character oftherepresentation, attained atthefirst normali- zation inpart (A) ofthis section, but that the same holds now everywhere, intheordinary sense forT(a) and, apart from theambiguity ofsign, also for d(A). For T(a) this was proved inpart (B)(b) ofthis section, ford() itmeans thattoeveryAi,eand¢thereissucha5thatoneofthetwoquantities (61)(d(Ax) ¥d(A))e, (d(A1) ¥d(A))e) <€ ifAis intheneighborhood 6ofA,. ‘The inequality (51) isequivalent to (Sia) (GFd(Ad))e, (1Fd(Ao))e) <6 where Ay=Ar'A now canbeassumed tobeintheneighborhood oftheunity. ‘Thus, thecontinuity ofd(A) atA=Eentails thecontinuity everywhere.” Infact, itwould besufficient toshow that thed(X), d(Y) and d(Z) correspond- ingtothe transformations (40) converge to-t1, asa,yapproach 0,since one can write every transformation inthe neighborhood ofthe unit element asa product A=Z(0,vs)¥(0,v2)X(0, 11)X(a1,0)¥(a2,0)Z(as, 0)andtheparam-eters ar, ---,72will convergeto0asAconvergesto1.However,weshall carry out the proof foranarbitrary Awithout anelementary divisor. Ford(A), equations (46) show that asAapproaches E(ie.,asyandxapproach zero) both Mw and Nw approach the same involution, which weshall callK. Let usnow consider awave function ¥=¢+di(K)¢ or,ifthis vanishes y= e—a&(K)y. We have di(K)y =+¥. IfAissufficiently near tounity, %J,vonNeumann, Sitz.d.kén.Preuss.Akad.p.76,1927. 184 E. WIGNER di(Noo)¥ will besufficiently near toQdi(K)¥ =-9¥ and allwehave toshow isthat @approaches +1. ‘The same thing will hold fordi(Mu). Indeed from d(Noo¥ —%¥=witfollows byapplying di(Noo) onboth sides ¥—Oy= (di(Noo) +2)u. As(u,u)goes tozero, @must goto£1, and consequently, also di(Nw)¥ goes to¥orto—¥. Applying di(Mu) tothis, one sees that \(Mco)di(Noo)¥ =d(A)¥ goestoEYasAgoestounity,‘Theargumentgivenin (A)(b) shows that this holds not only for¥but forevery other function also, i.e.d(A) converges to+1 =d(E) asAapproaches £. ‘Thus d(A) iscontinuous intheneighborhood of£and hence everywhere. According tothelast remark inpart 4,theoperators -td(A) form asingle valued representation ofthe group ofcomplex unimodular two dimensional matrices C. Let usdenote the homogeneous Lorentz transformation which corresponds intheisomorphism toCbyC.Our task ofsolving theequs. (22) hasbeen reduced tofinding allsingle valued unitary representation ofthe group with theelements (a,C]=(a,1][0,C],themultiplication rule ofwhich is {a,Ci](0,Cal=[a+Cib,C:C3]._ Fortherepresentations ofthisgroup Da, C]= T(a)d{C] wehad T(a)T() =T(a +b) (52a) a(C\T(a) =T(Ca)a{c} d(C]d{Cs]=a{CxCs). Itwould bemore natural, perhaps, from themathematical point ofview, touse henceforth this new notation fortherepresentations and lettheddepend onthe Crather than ontheCorA.However,inordertoberemindedonthegeometri- calsignificance ofthegroup elements, itappeared tometobebetter tokeep the oldnotation. Instead oftheequations (22B), (22C), (22D) wehave, then (52B) T(a)T() =T(a +b) (52C) d(A)T(a) =T(Aa) d(A) (52D) d(A) d() =£a(an). 6.Repuction ofTHE REPRESENTATIONS OFTHE INHOMOGENEOUS LORENTZ Group ToRerresentations or4“Lirruz Grove” ‘This section, unlike theother ones, will often make use ofmethods, which though commonly accepted inphysics, must befurther justified from arigorous mathematical point ofview. This has been done, inthe meanwhile, byJ. von Neumann inanasyetunpublished article and Iammuch indebted tohim forhiscodperation inthis respect and forhisreadiness incommunicating his results tome. Areference tohispaper" willbemade whenever hiswork is necessary formaking inexact considerations ofthis section rigorous. %J,vonNeumann, Ann.ofMath.toappearshortly. UNITARY REPRESENTATIONS OF LORENTZ GROUP 185 A. Since thetranslation operators allcommute, itispossible™ tointroduce such a coordinate system inHilbert space that the wave functions (p, {)contain momentum variables ps,ps,ps,puand adiscrete variable {sothat (53) Tlaelp, t)="el,2). pwill stand forthefour variables p,,ps,ps,Pe Ofcourse, thefact that theLorentzian scalar product enters intheexponent, rather than theordinary, isentirely arbitrary and could bechanged bychanging thesignsofps,Pr,Ps- The unitary sealar product oftwo wave functions isnotyetcompletely defined bytherequirements sofarmade onthecoordinate system. Itcan beasum- mation over ¢and anarbitrary Stieltjes integral over thecomponents ofp: (54) wo=d JVp,1)"o(D,1)dflp,£). ‘The importance ofintroducing aweight factor, depending onp,forthe scalar product liesnotsomuch inthepossibility ofgiving finite but different weights to different regions inpspace. Such aweight distribution g(p, ¢)always could be absorbed into thewave functions, replacing allg(p, $)byVg(p, 1)-¢(p, $). The necessity ofintroducing thef(p, ¢)liesrather inthepossibility ofsome regions ofphaving zero weight while, ontheother hand, atother places points may have finite weights. Onaccount ofthe definite metric inHilbert space, the integral J&¥(p, §)over anyregion r,foranyf,iseither positive, orzero, since itisthe ‘scalar product ofthat function with itself, which is1intheregion rofpand thevaluefofthediscrete variable, zero otherwise. Letusnow define theoperators (65) P(e, 1)=oA",5). ‘Thisequation defines thefunction P(A)¢, which is,atthepoint p,f,asgreat as thefunction gatthepoint A“'p, ¢.Theoperator P(A) isnotnecessarily ‘uni- tary, onaccount oftheweight factor in(54). Wecaneasily calculate PCA)T(ael, 9)=Tla)e( Ap, 1)=oP 6(A*D, 2), T(Aa)P(Ae(p, $)="1P(AoLp, 8)=(Ap, 0), sothat, for {A“'p, a]={p,Aa}, wehave (66) P(A)T(a) =T(Aa)P(A). ‘This, together with (52C), shows that d(A)P(A)* =Q(A) commutes with all T(a) and, therefore, with themultiplication with every function ofp,since the exponentials form acomplete setoffunctions ofpi,pps, Pe. Thus (87) (A)=Q(A)P(A), 186 E,WIGNER where Q() isanoperator inthespace ofthe¢alone" which eandepend, how- ever,ontheparticular valueofpintheunderlying space: (57a) QAAeP, =Tw, A)rwe(P, 1): Here, Q(p, A)ry are the components ofanordinary (finite orinfinite) matrix, depending onpand A. From (57), weobtain de, 9)=XQ, MryPelP, 0) (87b) * 4=LAP, Mela, ). [Astheexponentials form acomplete setoffunctions, wecanapproximate the operation ofmultiplication with any function ofpr, pr, Pr, Pebyalinear combination (58) Se=XeaTasdy. Ifwechoose f(p) tobesuch afunction that (58a) S(p)=Ap) the operation ofmultiplication with f(p) will commute with alloperations of the group. Itcommutes evidently with the T(a) and the Q(p, A),and on account of(56) and (58), (58a) also with P(A). ‘Thus theoperation of(58) belongs tothecentrum ofthealgebra ofour representation. Since, however, wweassume that therepresentation isfactorial (cf.2),thecentrum contains only multiples ofthe unity and (58b) S(p)e(p, 8)=cep,$). This canbetrue only ifgisdifferent from zero only forsuch momenta pwhich can beobtained from each other byhomogeneous Lorentz transformations, because f(p) needs tobeequal tof(p’) only iftherais aAwhich brings them into each other. Itwill besufficient, henceforth, toconsider only such representations, the wave funetions ofwhich vanish except forsuch momenta which can beobtained from one byhomogeneous Lorentz transformations. One can restriet, then, thedefinition domain ofthe¢tothese momenta. ‘These representations can now naturally bedivided into the four classes ‘enumerated insection 3,and two elasses contain two subclasses. ‘There will be representations, thewave functions ofwhich aredefined forsuch pthat () {pp} =P >0 (3) p=0 @Inpl=P=0;p~0 (4)[pp]=P<0. ‘The classes 1and 2contain two sub-classes each. Inthe positive subclasses P,and 0,the time components ofallmomenta are py>0,inthe negative UNITARY REPRESENTATIONS OF LORENTZ GROUP 187 subclasses P_and 0_the fourth components ofthe momenta are negative Class 3willbedenoted by0». IfPisnegative, ithasnoindex. From thecondition that d(A) shall beaunitary operator, itispossible to infer“ that onecanintroduce acodrdinate system inHilbert space insuch a way that e) [oo =[aon ifQ(p, A)zy #0forthepofthedomain r,Otherwise, risanarbitrary domain inthespace ofpi, m2, ps, peand Aristhedomain which contains Apifrcon- tains p. Equation (59) holds for all5,»,except for such pairs for which QP, A)ry =0. Itispossible, hence, todecompose the original representation insuch away that (59) holds within every reduced part. Neither T(a) nor (A) can have matrix elements between such »and forwhich (59) does not hold. Inthe third class ofrepresentations, the variable pcan bedropped entirely, and T(a)o(s) =¢(¢), ie., allwave functions areinvariant under theoperations ofthe invariant subgroup, formed bythe translations. ‘The equation T(a)¢(¢) =¢(¢) isaninvariant characterization ofthe representations ofthe third class, ie., acharacterization which isnot affected byasimilarity transformation. Hence, thereduced parts ofarepresentation ofclass 3also belong tothis class. Since no wave function ofthe other classes can remain invariant under all translations, norepresentation ofthethird class ean becontained inany repre- sentation ofone ofthe other classes. Inthe other classes, the variability domain ofpremains three dimensional, Itispossible, therefore, tointroduce instead ofpx,ps,ps,pethree independent variables. Intheeases 1and 2with which weshall beconcerned most, ps,ps,Pscanbekept forthese three variables. Onaccount of(59), theStieltjes integral canbereplaced byanordinary integral” over these variables, theweight factor being|pe"=(P+pi+pi+pi)? cm) wet=Eff[ven%el0.0Invr*arsdnedn. Infact, with theweight factor |py|theweight ofthedomain rie., W,= fJf|p«\"*dpadpadpsisequaltotheweightofthedomain W,asrequired” by(69). Having thescalar product fixed inthis way, P(A) becomes aunitary ‘operator and, hence, Q(A) will beunitary also. ‘Wewant togive next acharacterization oftherepresentations with agiven P, which isindependent ofthecodrdinate system inHilbert space. Itfollows from %The invariance ofintegrals ofthe character of(598) isfrequently made use ofin relativity theory. One can prove itbycalculating the Jacobian ofthe transformation Pim Aupi tSapa tAap: +(P+ pit pit pp @=1,2,3) which comes outtobe(P+ pi+pi+pDMP +p? +p?+ PP)-%. Eau, (60a) willnotbe ‘used inlater parts ofthis paper. 188 E.WIGNER (53), that inarepresentation with agiven Pthewave functions Yi, va, «+> which aredifferent from zero only inafinite domain ofp,form aneverywhere dense set,toallelements ofwhich theinfinitesimal operators oftranslation can beapplied arbitrarily often HimA-*(T(he)—1)"¥=limh(E"=1)" (0)ray mo =pel, where ¢will beaunit vector inthedirection ofacodrdinate axis oroppositely directed toit. Hence forallmembers yofthis everywhere dense set (61)timz(LT (hex)—27(hex)+I=(pitpitpi—pv=Py, where exisaunitvector in(oropposite) thek"*codrdinate axisandthe+is+ fork=4,and—fork=1,2,3.Ontheother hand, thereisno¢forwhich (61a) lim2A*(T(@hes) —27(hes)+De ifitexists, would bedifferent from —Py. Suppose thelimit in(61a) exists and is—Pe +¢’. Letuschoose then anormalized ¥,from theabove set,such that (Y,e')=8with 5>0and anhsothat theexpression after the lim sign in (61a) assumes thevalue —Pp+y’+uwith (u,u)<6/3and also theexpression after thelimsign in(61), with oppositely directed ebecomes —Py+u’with (u',w)<6/3. ‘Then, onaccount oftheunitary character ofT(a) and because ofT(-a) =T(a)* (2+WT(hes)—27(her)+De), =(=2W*(T(—2ha) ~27(—ha) +Ihre), or —Pe) +e) +Hu) =—PW, 0)+(u's9), which isclearly impossible. ‘Thus ifthelimin(61a) exists, itis—Pyand this constitutes acharacterization ofthe representation which isindependent ofsimilarity transformations. Since, according totheforegoing, itisalways possible tofind wave functions forarepresentation, towhich (61a) can beapplied, every reduced part of representation with agiven Pmust have this same Pand norepresentation with one Pcan becontained inarepresentation with another P. The same argument can beapplied evidently tothepositive and negative sub-classes of class 1and 2. UNITARY REPRESENTATIONS OF LORENTZ GROUP 189 B. Every automorphism L—L°ofthegroup allows ustoconstruet from one representation D(L) another representation (62) D%(L) =D(L?). ‘This principle will allow ustorestriet ourselves, forrepresentations with finite, positive ornegative P,toone value ofPwhich can betaken respectively, tobe +1, and —1. Itwillalso allow incases 1and 2toconstruct therepresentations otthenegative sub-classes out ofrepresentations ofthe positive sub-classes. ‘Thefirstautomorphism isa°=aa,A°=A.Evidently Equs.(12)areinvari-ant under this transformation. Ifweset, however, Tale =Tale; aAe =d(A)e, then theoccurring p Tale =Taay =ele =ery, will bethepoccurring fortheunprimed representation, multiplied bya. This allows, with areal positive a,toconstruct allrepresentations with allpossible numerical values ofP,from allrepresentation with one numerical value ofP. Ifwetake anegative, therepresentations ofthenegative sub-classes areobtained from therepresentations ofthepositive sub-class. Incase P=0,evidently allrepresentations goover into themselves bythe transformation (62). Incase P=0,and P=0_itwill turn out that for positive a,(62) carries every representation into anequivalent one. c Onaccount of(53) and (56), (57), theequs. (52B) and (52C) areautomatically satisfied and theQ(p, A); must bedetermined by(52D). This gives (63)EQ,AeQAP, DeoeIAp,8)=&ELQP,ADroe(TAD,0). Since this must hold forevery ¢,one would conclude (63a) LAW,AQP, Doo=+QP,ADro- Actually, this conclusion isnot justified, since two wave functions must be considered tobeequal even ifthey aredifferent onasetofmeasure zero. Thus ‘one cannot conclude, without further consideration, that the two sides of(63a) areequal atevery point p.Ontheother hand,” thevalue ofQ(p, A)rycanbechangedonasetofmeasurezeroandonecanmakeitcontinuous intheneighbor-hood ofevery point, iftherepresentation iscontinuous. This allows then, to justify (68a). Itfollows from (63a) that Q(p, 1)ry =Sry- 190 E,WIGNER Letuschoose" nowabasic pparbitrarily, Wecanconsider then thesubgroup ofallhomogeneous Lorentz transformations which leave this peunchanged. Forallelements d,«ofthis“little group,” wehave Pe,WeQ(Po,Dee=+QPoMeo (64). gq) =+a0), where g(A)isthematrix q(A)r =Q(Po, \)ry- Because oftheunitary character of Q(A), theQ(ps, A)ry isunitary matrix and q(A) isunitary also. Ifweconsider, according tothelast paragraph ofSection 5,thegroup formed out ofthe translations and unimodular two-dimensional matrices, rather than Lorentz transformations, the +sign in(64) canbereplaced bya+sign. In thiscase, \and¢areunimodular two-dimensional matrices andthelittle group isformed bythose matrices, thecorresponding Lorentz transformations i,<to which leave ppunchanged ipo=ip=Po. Adopting this interpretation of(64), one can also see, conversely, that the representation q(X) ofthelittle group, together with the class and Pofthe representation ofthewhole group, determines thelatter representation, apart from asimilarity transformation. Inorder toprove this, letusdefine forevery patwo-dimensional unimodular matrix a(p) insuch away that thecorrespond- ingLorentz transformation (65) &(p)po =P brings ppinto p.The a(p) can bequite arbitrary except ofbeing analmost everywhere continuous function ofp,especially continuous for p=psand (ps) =1.Then, wecan set A(a(p) eo,8)=9(P,8), (66)° d(a(p))e(p, 3)=e(po, $)- ‘This isequivalent tosetting in(58) (66a) 1,a(p))=1 and can beachieved byasimilarity transformation which replaces o(p, $)by LyQo, a(p))rye(p, 2).Asthematrix Q(po, a(p)~) isunitary, thisis Unitary transformation. Itdoes notaffect, furthermore, (53)since itcontainsp only asaparameter. Assuming this transformation tobecarried out, (66) will bevalid and will define, together with thed(A), alltheremaining Q(p, A)uniquely. Infact, calculating d(A)g(p, §),wecandecompose Ainto three factors (67) A= a(p). ap) 'Aa(A%p). (Ap) UNITARY REPRESENTATIONS OF LORENTZ GROUP 191 ‘Thesecondfactor8=a(p)"'Aa(Aé'p) belongs intothelittlegroup:ap)"Aa(X*p)po =a(n) "AK"p=a(n)"p=po.Wecanwrite,therefore ("p =py aA)e(p, 8)=d(a(p))d(8)d(a(p’)) ‘eC,$) (67a) =dB)d(a(p’))'e(m ,$) =Laedlale'y elo»,2)=Xa(Bsve(', ») ‘This shows that allrepresentations ofthe whole inhomogeneous Lorentz group ‘areequivalent which have the same Pand thesame representation ofthe little group. Further than this, the same holds even ifthe representations ofthe little group are not the same forthe two representations but only equivalent toeach other. Letusassume qi(A) =sqx(A)s. Then byreplacing o(p, t)by XD.s(F,n)e(p, 2)weobtain anewform oftherepresentation forwhich (53)still holds but q:(8) forthe little group isreplaced byqi(8). Then, bythe trans- formation just described (Eq. (66), wecan bring d(A) forboth into the form (67a). The equivalence oftwo representations ofthe little group must be defined astheexistence ofaunitary transformation which transforms them into each other. (Only unitary transformations areused forthewhole group, also). Onthe other hand, ifthe representations ofthe whole group areequivalent, therepresentations ofthelittle group areequivalent also: the representation ofthe whole group determines the representation ofthe little group uptoa similarity transformation uniquely. ‘The representation ofthe little group was defined asthe set ofmatrices Q(po, d)ry iftherepresentation issotransformed that (53) and (66a) hold. Having twoequivalent representations DandSDS” =D°forboth ofwhich (53) and (66a) holds, theunitary transformation Sbringing thefirst into thesecond must leave alldisplacement operators invariant. Hence, itmust have theform (57a), ice.,operate onthe¢onlyanddepend onponlyasonaparameter. (68) Sele,t)=XSP)rvelp, 0). Denoting thematrix Qforthetwo representations byQand Q®,the condition SD(A) =D°(A)S gives that (68a) LSM, Me=LEM, MSAD) holds, forevery A,foralmost every p. Setting A=a(p:) wecan letpapproach prinsuch away that (68a) remains valid. Since Qisacontinuous function ofp bothQ@,A)andQ°(p,A)willapproach theirlimitingvalue1_{tfollowsthatthere isnodomain inwhich (69) S(p:) =S(a(p:)*p) =S(po) wouldnothold,i.c.,that(69)holdsforalmosteveryp:..Sinceallourequationsmust hold only foralmost every p,theS(p);, canbeassumed tobeindependent 192 E, WIGNER ofpand(68a) then tohold forevery palso. Itthen follows that therepresenta- tions ofthelittle group inDand D°aretransformed into each other bySry« ‘The definition ofthelittle group involved anarbitrarily chosen momentum vector py. Itisclear, however, that the little groups corresponding totwo different momentum vectors ppand pareholomorphic. Infact they can be transformed into each other bya(p): IfAisanelement ofthelittle group leavingpinvariant thena(p)"‘Aa(p) =6isanelement ofthelittlegroupwhichleaves poinvariant. Wecanseefurthermore from (67a) that ifAisinthelittle group corresponding top,ie.Ap=pthen therepresentation matrix q(8) of the little group ofpo,corresponding to8,isidenticalwiththerepresentation matrix ofthelittle group ofp,corresponding toA=a(p)Ba(p)*. ‘Thus when characterizing arepresentation ofthewhole inhomogeneous Lorentz group byP and therepresentation ofthelittle group, itisnotnecessary tosay which pois leftinvariant bythelittle group. D. Lastly weshall determine theconstitution ofthelittle group inthe different cases, 1,. Incase 1,wecan take forppthe vector with the components 0,0,0,1. ‘The little group which leaves this invariant obviously contains allrotations inthespace ofthefirst three coirdinates. This holds forthelittle group ofall representations ofthe nrst class.0,IncaseOy,thelittlegroupisthewholehomogeneous Lorentzgroup.1, Incase P=—1thep,canbeassumed tohave thecomponents 1,0,0,0. ‘Thelittle group then containsall transformations which leave theform —2}—z} +xiinvariant, i.e.,isthe2+1dimensional homogeneous Lorentz group. The same holds forallrepresentations with P<0. 0,.The determination ofthe little group for P=0,issomewhat more complicated. Itcan bedone, however, rather simply, forthe group ofuni- modular two dimensional matrices. The Lorentz transformation corresponding tothematrix\°alwithad—be=1bringsthevectorwiththecomponents 21,22,22, 2,intothevector with thecomponents z{,73,x5,21,where” abl\ixetas atimeflatct)xitahxitinh «)|||:tma+ioe|-|272 aaah ed) lini x—x/ loratl lait xii) ‘The condition that anull-vector py,saywith thecomponents 0,0,1,1beinvari- antiseasilyfoundtobe|@|’=1,c=0.Hencethemostgeneralelement ofthelittle group canbewritten jee”Getaye) m™ore | UNITARY REPRESENTATIONS OF LORENTZ GROUP 193 with realz,y,6and 0$B< 4x. The general element (71) canbewritten as t(z, y)6(8) where 1x+iy|| om0| (71a) tz,y)=| bb 6@=| |.lo 1 fo ef] The multiplication rules forthese are (7b) Ha,ye) =Me+ ty), (71e) 8(B)t(z, y)=t(zcosB+ysinB,—xsinB+ycos6)&(B), (ia) 3(8)5(8’) =(8+6’). One could restrict thevariability domain of6in4(8) from 0to2x. As3(2x) commutes with allelements ofthe little group, itwill beaconstant and from 8(2n)* =(4x) =1itcanbe&(2r) =+1. Hence (8+2x)=+8(8) and inserting a++intoequation (714) onecould restrict 8to0 <B<2x. ‘These equations areanalogous totheequations (52)-(52D) andshow that the little group is,inthiscase, isomorphic with theinhomogeneous rotation group of two dimensions, i.e.thetwo dimensional Euclidean group. Itmay bementioned that theLorentz transformations corresponding to U(z,y)have elementary divisors, andconstitute alltransformations ofclass e) in4B,forwhich v,=py. The transformations 4(8)canbeconsidered tobe rotations intheordinary three dimensional space, about thedirection ofthe spacepartofthevectorpy._Itispossible, then,toproveequations (71)alsodirectly. 7,Tae Representations oFTHE Lrrtie Grours ‘A.Representations ofthethree dimensional rotation group byunitary transformations. ‘Therepresentations ofthethree dimensional rotation group inaspace with a finite member ofdimensions arewell known. ‘There isoneirreducible representa tion with thedimensions 1,2,3,4,---each, therepresentations with anodd number ofdimensions aresingle valued, those with aneven number ofdimen- sions aretwo-valued. These representations willbedenoted byD‘?(R) where thedimension is2j+1.‘Thus forsingle valued representations jisaninteger, fordouble valued representations ahalf integer. Every finite dimensional representation can bedecomposed into these irreducible representations. Consequently those representations oftheLorentz group with positive Pin which therepresentation ofthelittle group—as defined by(64)—has afinite number ofdimensions, canbedecomposed into such representations inwhich therepresentation ofthelittle group isoneofthewell known irreducible repre- sentations oftherotation group. This result willhold forallrepresentations oftheinhomogeneous Lorentz group with positive P,since weshall show that even theinfinite dimensional representations ofthe rotation group can be decomposed into thesame, finite, irreducible representations. 194 B,WIGNER Inthefollowing, itismore appropriate toconsider the subgroup ofthe two dimensional unimodular group which corresponds torotations, than therotation group itself, aswecan restrict ourselves tosingle valued representations inthis cease (cf. equations (52)). From (70), one easily sees! that the condition for {Ibi) . *[e‘Itoleavethevectorwiththecomponents 0,0,0,1invariant isthatit shall beunitary. Itis,therefore, the two dimensional unimodular unitary group therepresentations ofwhich weshall consider, instead ofthe representa- tions ofthe rotation group. Let usintroduce adiscrete codrdinate system inthe representation space anddenotethecoefficients oftheunitaryrepresentation byg(R).whereRisa‘two dimensional unitary transformation. ‘The condition for the unitary character oftherepresentation q(R) gives (72) LaRyaWee=duiLeg(RV(Ben=be, (72a) TlaWaP=1; Lla®@al=1 ‘This show also that |q(R) |<1and theq(R), aretherefore, asfunctions ofR, square integrable: SlaBea Par exists if[...dRisthewell known invariant integral ingroup space. Since this isfinite fortherotation group (ortheunimodular unitary group), iteanbe normalized to1. We then have (73) LslaWalar =zSla(Bal'ak =1. The(2;+1)'D (R)u form,” acomplete setofnormalized orthogonal functions for R. We set co) aR=FOLDER). Weshall calculate nowtheintegral over group space oftheproduct ofD?(R)it and (7) ARS)on =Za(Rna(Sm The sum ontheright converges uniformly, asfor(72a) « * * 1 re +Ela@aadul ¢(Elawat Eiat’) =(Flats) canbemade arbitrarily small bychoosing anN,independent ofR,making the last expression small. Hence, (75) can beintegrated term byterm and gives (76) [D°(RRS)ud=XfDPR)La(Raa(S)rik. *H.Weyl and F.Peter, Math, Annal. 97,787, 1927. UNITARY REPRESENTATIONS OF LOKENTZ GROUP 195 Substituting ED?(RS)inDS)miforD(R)x:oneobtains (77) LDS af DRS)inRS)apdR=FgSeJD?(RVR). Intheinvariant integral ontheleftof(77), Rcanbesubstituted forRSand weobtain, for(74) and theunitary character (78) LD" SnCiin =ESraCi- Multiplying (78)byD“(S)t,, theintegration ontheright sidecanbecarried ‘outterm byterm again, since thesum over \converges uniformly * « * 17s qiEleraon| s(Liene Einst) s(Eicm)- ‘This canbemade arbitrarily small, aseven 2)Dy(2j+1)"|Cit:*converges, for(74) and (72a). The integration of(78) yields thus (79) LCC =snduCie. From q(R)q(E) =9(R)follows q(#)=1andthenq(R™) =q(k)* =q(k)'. ‘This, with thesimilar equation forD’’(R) gives TCHRYu=Rea=aK (so «DoQR)sreDiC?=Len DW =FCHDPR Ya, or (81) Ci=Chr. Ontheother hand q(B)a =3ayields (82) 2Cis=ba ‘These formulas suffice forthereduetion ofq(R). Let uschoose forevery finite irreducible representation D”anindex k,sayk=0.Wedefine then, in theoriginal space oftherepresentation q(R) vectors »“””with thecomponents Cita,Cit,City+++. ‘The vectors v*” fordifferent jorlareorthogonal, thescalar product ofthose with thesame jandLisindependent of!.Thisfollowsfrom(79)and(81) (83), 0) =LOMCH=LOMOsire=dardChhe. ‘The v“*” forallx,j,1,form acomplete setofvectors. Inorder toshow this, itissufficient toform, forevery »,alinear combination from them, the»com- ponent ofwhich is1,allother components 0.‘This linear combination is (st) LChae. 196 E,WIGNER: Infact, theXcomponent of(84)is,onaccount of(79)and(82) (85) Lejueen =EC =on. ot z However, twovwiththesamejand1butdifferent firstindices «arenot orthogonal. Weeanchoose forevery jan1,say1=0andgothroughthevectors vi,9%”,...and,following Schmidt’s method,orthogonalize andnormalize them.Thevectors obtained inthiswayshall bedenoted by ‘Then, since according to(83)thescalar produets (v””, »°*”) donotdepend on1, the vectors (862) w=Fav” willbemutually orthogonal and normalized also and thevectors w"”” forall n,j,Lwillform acomplete setoforthonormal vectors. ‘The same holds forthe setoftheconjugate complex vectors w'""”*. Using these vectors ascodr- dinate axes fortheoriginal representation q(R), weshall findthatq(R) iscom- pletely reduced. The»component ofthevector (R)v**”* obtained byapplying aR) ono” is (s7) Lah.) =LyWa-Che. The right sideisuniformly convergent. Hence, itsproduct with (2k+1) D(R)!, canbeintegrated term byterm giving (88) Lf2h+ID(R)i.q().,ChiedR =LCitaChin =buj5aCGhe. ‘Thus we have for almost all R (88a) Lg(R(o"), =ECD" Ba=LDR ulo),, or (s8b) (Rv =XD), Since both sides aresupposed tobestrongly continuous functions ofR,(8b) holds forevery R.In(86a), forevery n,thesummation must becarried out only over afinite number of\.Wecanwrite therefore immediately (80) Rye" =SDR aw, ‘This proves that theoriginal representation decomposes inthecodrdinate system ‘ofthewintowellknown finite irreducible representations D'”(R). Since thew form acomplete orthonormal setofvectors, thetransition corresponds toa unitary transformation: UNITARY REPRESENTATIONS OF LORENTZ GROUP 197 ‘This completes theproof ofthecomplete reducibility ofall(finite andinfinite dimensional) representations ofthe rotation group orunimodular unitary group. Itisclear also that thesame consideration applies forallclosed groups, i.e.,whenever theinvariant integral fdRconverges. ‘The result fortheinhomogeneous Lorentz group is:Forevery positive numeri- calvalue ofP,therepresentations ofthelittle group eanbe,inanirreducible representation, onlytheD®,D,D'”,...,bothforP,andforP_.Allthese. representations have been found already byMajorana and byDirac and for positive Pthere arenone inaddition tothese B.Representations ofthetwodimensional Euclidean group ‘This group, aspointed outinSection 6,hasagreat similarity with theinhomo- geneous Lorentz group. Itispossible, again”, tointroduce “momenta”, i.e. variables &,»and vinstead ofthe¢insuch away that (90) Uz,velo,&1,v)=(pe, &mv) Similarly, onecandefine again operators R(B) (91) R(B)e(Po ,0,»)=e(Po, &,9,v), where £=Ecos8—nsin6, (ota) A a!=Esin8+7cos6. ‘Then 4(8)R(B)* =S(B) willcommute, onaccount of(71e), with ¢(z,y)and again contain ,»asparameter only. ‘The equation corresponding to(57a) is (92) 5@)e(Pe, Em,)=LSBuelPr,ty,)- One caninfer from (90) and(92) again that thevariability domain of&,7canbe restricted insuch awaythatallpairs ,»arise from onepairgp,»»byarotation, according (91a). Wehave, therefore two essentially different cases: a) F+d=240 b) P+P=2=0, ie Fan=0. ‘Thepositive definite metric inthe£,nspace excludes theother possibilities of section 6which were made possible bytheLorentzian metric forthemomenta, necessitated by (55). Case b)canbesettled very easily. The “little group” is,inthis case, the group ofrotations inaplane and weareinterested inoneand twovalued irreducible representations. ‘These areallonedimensional (¢") (98) S(8)= where sisinteger orhalfinteger. ‘These representations were alsoallfound by Majorana andbyDirac. Fors=0wehave simply theequation Oy=0, 198 B,WIGNER fors=£3Dirac’s electron equation without mass, fors=-t1Maxwell's electromagnetic equations, ete Incase a)thelittle group consists only ofthe unit matrix and the matrix i.an|ofthetwodimensional unimodular group.‘Thisgrouphastwo irreducible representations, as(1)and (—1) cancorrespond totheabove two dimensional matrix ofthelittle group. This gives two new representations of thewhole inhomogeneous Lorentz group, corresponding toevery numerical value ofZ,Both these setsbelong toclass 0,andtwosimilar newsetsbelong toclass 0. Thefinalresultisthusasfollows:Therepresentations P..;ofthefirstsubclassP,,canbecharacterized bythetwonumbers Pandj. From these Pispositive, otherwise arbitrary, while jisaninteger orahalf integer, positive, orzero. ‘The same holds forthesubclass P_. ‘There arethree kinds ofrepresentations ofthesubclass 0... ‘Those ofthefirst kind 0,,canbecharacterized byanumber s,which canbeeither aninteger orahalf integer, positive, negative orzero. ‘Those ofthesecond kind0,() aresingle valued andcanbecharacterized byan arbitrary positive number Z,those ofthethird kind 0) aredouble-valued and also canbecharacterized byapositive Z. ‘The same holds forthesubclass 0_. The representations oftheother classes (0band Pwith P<0)have not been determined. 8,Representations oFTHe Extenpep Lorentz Grou A Asmost wave equations are invariant under awider group than the one investigated intheprevious sections, andasitisvery probable that thelaws of physics areallinvariant under thiswider group, itseems appropriate toinvesti- gate now how theresults oftheprevious sections willbemodified ifwegoover from the“restricted Lorentz group” defined insection 4A,totheextended Lorentz group. This extended Lorentz group contains inaddition tothe translations allthehomogeneous transformations Xsatisfying (10) (10) XFX' =F while thehomogeneous transformations ofsection 4Awere restricted bytwo more conditions. From (10’) itfollows that thedeterminant ofXcanbe-+1or =1only. Ifits—1, thedeterminant ofX;=X¥F is+1. Ifthefour-four clement ofX;isnegative, thatofX:=—X; ispositive. Itisclear, therefore, that ifXisumatrix oftheextended Lorentz group, oneofthematrices X, XF, —X, —XP isintherestricted Lorentz group. ForF*=1,conversely, all homogeneous transformations oftheextended Lorentz group canbeobtained from thehomogeneous transformations oftherestricted group bymultiplication with one ofthe matrices (4) 1,F,-1, -F. UNITARY REPRESENTATIONS OF LORENTZ GROUP 199 ‘The group elements corresponding tothese transformations will bedenoted by E,F,I,IF. The restricted group contains those elements ofthe extended group which can bereached continuously from theunity. Itfollows that the transformation ofanelement Loftherestricted group byF,I,orIFgives again ‘anclement ofthe restricted group. This is,therefore, aninvariant subgroup ‘ofthe extended Lorentz group. Inorder tofind the representations ofthe extended Lorentz group, weshall useagain Frobenius’ method." We shall denote the operators corresponding inarepresentation tothe homogeneous transformations (94) byd(E) =1,d(F), d(1),d(IF). For deriving theequations (52) itwas necessary only toassume theexistence ofthetrans- formations oftherestricted group, itwas not necessary toassume that these are theonly transformations. ‘These equations will hold, therefore, forelements ofthe restricted group, inrepresentations ofthe extended group also. We normalize the indeterminate factors ind(F) and d(I) sothat their squares become unity. ‘Then wehave d(F)d(I) =wd(I)d(F) ord(Z) =wd(F)d(I)d(F). Squaring this, oneobtains w*=£1. Weeanset,therefore (0s) (IF)=d()a(F)=d(F)a)a(FyY =dd) =1; dF) =£1. Finally, from (96) d(PYD(Ly)d(F) =w(L1)D(FLiF) weobtain, multiplying this with thesimilar equation forLa o(Ia)o(In) =w(Lale) which, gives w(L) =1astheinhomogeneous Lorentz group (orthegroup used in(52B)-(52D)) hastheonly one dimensional representation bytheunity (1). Inthis way, weobtain (96a) d(F)D(L)d(F) =D(FLF), (96b) d(I)D(L)d(I) =DUILI), (96¢) (IF)D(L)a(IF)* =DUIFLFD), B. Given arepresentation oftheextended Lorentz group, one can perform the transformations described insection 6A, byconsidering the elements ofthe restricted group only. Weshall consider here only such representations ofthe ‘extended group, forwhich, after having introduced themomenta, allrepresenta- tions oftherestricted group areeither inclass 1or2,ie. P20but not 0» Following then theprocedure ofsection 6,onccanfind asetofwave functions forwhich theoperators D(L) oftherestricted group have oneoftheforms, given insection 6asirreducible representations. Weshall proceed, next tofind the operator d(F). Forthewave functions belonging toanirreducible D(L) ofthe 200 E,WIGNER: restricted group, we-can introduce acomplete setoforthonormal functions VilP, 5),val, 9),+++. We then have (97) D(L)WB, £1)=XD(L)u¥e(@, $). ‘Theinfinite matrices D(L),« defined in(97)areunitary andformarepresentation which isequivalent totherepresentation bytheoperators D(L). ‘The D(L), d(F) are, ofcourse, operators, buttheD(L), arecomponents ofamatrix,ie.numbers. Wecannowformthewavefunctions d(F)¥1,d(F)¥,d(FWWs,«- andapply D(L) tothese. For(96a) and (97) wehave D(L)d(F)}e =d(F)D(PLFWW (97a)=dF) LDELP), =LDFLF),d(F Wy, The matrices D°(L)y. =D(FLF)segivearepresentation oftherestrictedgroup (FLF isanelement oftherestricted group, wehave anewrepresentation byan automorphism, asdiscussed insection 6B). Weshall findoutwhether D°(L) is equivalent D(L) ornot. ‘The translation operation inD?is (98) T°(a)=d(F)T(a)d(F) =T(Fa) which, together with (53) shows that D°hasthesame PasD(L) itself. In fact, writing (99) Urelp,2)=oFp,2) onehasUj!=Uyandoneeasily calculates U:T%a)Us =T(a). Similarly for Uya(A)U; one has Urd(A)Uie(p, $)=UrdPAF)Ui ¢(p,3) (09a) =d(PAP)Use(Fp, $)=LDQ(Fp, PAF) Uie(FAp,0) =LQ, PAP)eneA,9). This means that thesimilarity transformation with U,brings T°(a) into T(a) andd°(A) intoQ(Fp; FAF)P(A). Thus therepresentation ofthe“little group” inUd%(A)Us is GO) =Q(Fpo, PAF), Forthislatter matrix, oneobtains from (67a) 0)=QFpe,FAP)=g(a(Fps)"PAPalF; (100) Fd)=QFpo )=a(a(Fpo) (Fpo)) =40°) where d°isobtained from-d bytransforming itwith Fa(Fpp). The representations D°(L) and D(L) areequivalent iftherepresentation UNITARY REPRESENTATIONS OF LORENTZ GROUP 201 (2) isequivalent tothe representation which codrdinates q(X*) to4.The(Fp)isatransformation oftherestricted groupwhichbringsppintoa(F'ps)p> =Fpe. (Cf. (65).) This transformation is,ofcourse, not uniquely determined butifa(po) isone, themost general can bewritten asa(Fpo)t, where tp»=Po isinthelittle group. Forg(¢‘a(Fpo)”' Aa(Fpo)s) =g(0)"'g(a(Fpo)Aa(Fp»)) q(®), the freedom inthe choice ofa(Fpe) only amounts toasimilarity trans- formation of9°() and naturally does notchange theequivalence ornon equiva- lence ofq°(A) with (A). For thecase P,, wecan choose pointhedirection ofthe fourth axis, withcomponents 0,0,0,1.‘ThenFp»=pyanda(Fps)=1.Thelittlegroupisthegroup ofrotations inordinary space and FAF =d.Hence q°(X) =q(A) and D°A) isequivalent toD(A) inthis case. The same holds fortherepresenta~ tions ofclass P_. For 0,wecan assume that p>has the components 0,0,1,1.Then the components ofFps are0,0,1, 1.For a(Fpx) wecan take arotation byx about thesecond axis and Fa(Fps) will be diagonal matrix with diagonal elements 1,—1, 1,1,i, areflection ofthesecond axis. ‘Thus if)isthetrans- formation in(70), ”=a(Fpo)”'F\Fa(Fps) isthetransformation forwhich aon w(tnomae_(;+aai*). itimmm titit,meoy This is,however, clearly X=A*. Thus theoperators ofq°(A) areobtained from theoperators q(A) by(ef. (71a)) #2,»)=Ue,— (iota) (z,y)=Uz,—y) 88) =(-8). Fortherepresentations 0,,with discrete s,theq°(A) andq(A)areclearly inequiva- lentas8°(8) =(e“*) and4(8) =(e“*), except fors=0,when they areequiva- lent. Fortherepresentations 0,(=), 04(E), theq°(A) and (A)areequivalent, both inthesingle valued and thedouble valued case, asthesubstitution »>— transforms them into each other. The same holds forrepresentations ofthe class 0. IfD\(L) and D(L) areequivalent (102) U"D(L)U =D(L), thesquare ofUcommutes with allD(L). As@consequence ofthis, U*must beaconstant matrix. Otherwise, onecould form, inwell known manner,” anidempotent which isafunction ofU*andthus commutes with D(L) also. ‘Such anidempotent would lead toareduction oftherepresentation D(L) ofthe restricted group. As constant isfree inU,weean set (1028) ar +J.vonNeumann, Ann.ofMath.$8.191,1981:ref.2.0.89. 202 ‘E.WIGNER c. Returning nowtoequation (97a), ifD(L) =D(FLF) andD(L) areequivalent (P>0or0,,0_with continuous =ors=0)there isaunitary matrix U,,, corresponding toU’,such that ZDELP) Uy=XLCuDLer (102b) * * Da =be Let usnow consider the functions (103) wht Ld Applying D(L) tothese D(L)e, =DLW, +XUeD(L\UP Wy =Dil +XDUypd(F)D(FLFYY, (103a). =LD)ods+XLUndPDFLP ute =LDLw (be+ZLUndlP.) =XDLwen. Similarly (Pye =dF, +XLUnde (103) . . ; =LU (e+Lad.) =ZLUwes ‘Thus thewave functions ¢transform according totherepresentation inwhich D(L),, corresponds toLandU,,tod(F). Thesameholdsforthewavefunctions (104) =—LUade, except thatinthiscase (—U,,) corresponds tod(F). They,andd(F)¥, canbe expressed bytheyandy’.Iftheyandd(F)y were linearly independent, the gand¢’willbelinearly independent also. Ifthed(F)¥ were linear combinations ofthey,either the¢ortheg’will vanish. Ifweimagine aunitary representation ofthegroupformedbytheLandFL intheforminwhichitiscompletely reduced outasarepresentation ofthegroup ofrestricted transformations L,theabove procedure will lead toareduction ofthatpartoftherepresentation ofthegroup oftheLandFL,forwhich D(L) and D(FLF) areequivalent. IfD(L), and D°L),, areinequivalent, the¥%and d(F)¥, =¥!areor- thogonal. This isagain ageneralization ofthesimilar rule forfinite unitary UNITARY REPRESENTATIONS OF LORENTZ GROUP 203 representations.* One eanscethisinthefollowing way: Denoting Mu = (We, ¥2)one has Mu=(We,$1)=(DL, DIL!) =EDL).DWL)» Marj x M=D(L)'MD*L). Hence (105) D(L)M =MDL); M'D(L) =D°(L)M'. From these, oneeasily infers that MM" commutes with D(L), and M'M com- mutes with D°(L). Hence both areconstant matrices, and ifneither ofthem is zero, Mand M"are,apart from aconstant, unitary. Thus D(L) would be equivalent D°(L) which iscontrary tosupposition. Hence MM' =0,M=0 and the¥areorthogonal tothed(F)y =y’. Together, they give arepresenta- tion ofthegroup formed bytherestricted Lorentz group and F. Ifthey donot form acomplete set, thereduction can becontinued asbefore. One sees, thus, that introducing theoperation F“doubles” thenumber of dimensions oftheirreducible representations inwhich thelittle group wasthetwo dimensional rotation group, while itdoes not increase the underlying linear manifold intheother cases. This isanalogous towhat happens, ifoneadjoins thereflection operation totherotation groups themselves.” D. ‘The operations d(I) canbedetermined inthesame manner asthed(F) were found. Acomplete setoforthonormal functions corresponding toanirreducible representation ofthegroup formed bytheJand FLshall bedenoted by¥:, vs,+++. For this, weshall assume (97) again, although theD(L) contained therein isnow not necessarily irreducible forthe restricted group alone but contains, incase of0,,or0_,and finite s,bothsand—s. Weshallset, furthermore (106) AUP=LdPueds- We can form then thefunctions d(I)yi ,d(D)¥2, ‘The consideration, con- tained in(97a) shows that these transform according toD(ILI),,. forthetrans- formation Loftherestricted group: (106a) DL)aDy. =XDULDudDe- Choosing forLapure translation, aconsideration analogous tothat performed in(98) shows that thesetofmomenta intherepresentation L—+D(ILI) hasthe opposite sign tothesetofmomenta intherepresentation D(L). Ifthelatter *Cf.e.g. E.Wigner, ref. 4,Chapter XIL»I.Schur,Sits.d.kén,Preuss.Akad,pages189,297,1924 204 B,WIGNER belongs toapositive subclass, theformer belongs tothecorresponding negative subclass and conversely. ‘Thus theadjunction ofthetransformation Ialways leads toa“doubling” ofthenumber ofstates, thestates of“negative energy” are attached tothe system ofpossible states. One can describe allstates y1, ve,--+, dD, d(Dy2, +++byintroducing momenta pi, pz, Ps, peand restrieting thevariability domain ofpbythecondition {p,p]=Palone without stipulating adefinite sign forps. Aswesaw before, thed(I)y1, d(J)¥2, areorthogonal totheoriginal setof wave functions yi, ¥2,---. ‘The result ofthe application ofthe operations D(L)andd(F)totheyx,Ys.«+»(Le.,therepresentation ofthegroupformedbythe L,FL) was given inpart C. The D(L)d()¥. are given in(106a). On account ofthenormalization ofd(I) wecan set (106b) (DdDe=ve Ford(F)d(Dy. wehave two possibilities, according tothetwo possibilities in (95).Wecaneitherset (107) d(P)d(D. =ad, =Ld(P)ud DY5 or (107) d(P)-dDe=dDdFbe=~ZLdPeedDY,« Strictly speaking, wethus obtain two different representations. ‘The system of states satisfying (107) could bedistinguished from thesystem ofstates forwhich (107a) isvalid, however, only ifwecould really perform the transition toanew codrdinate system bythetransformation I.Asthis is,inreality, impossible, the representations distinguished by(107) and (107a) are not different inthe same sense asthepreviously described representations aredifferent. Tam much indebted tothe Wisconsin Alumni Research Foundation for their aidenabling metocomplete this research. Mapisow, Wrs.