1 Around SuSy 1970 E.P. Likhtman a aAll-Russian Institute of Scientific and Technical Information, Usievich Str. 20, Moscow 125315,Russia 1 e-mail: [email protected] 0 0 2 n Let me turn back to look at the past, and tell vided the basis for the science, whichin six years a you about the history of construction of the first was called the SuperSymmetry. J supersymmetricmodel. The papers I will shortly At the first stage I had to establish whether 8 1 reviewhavebeenforthemostpartratherinacces- theproposedalgebrawasuniqueorwhetherthere sible. A part of the results obtained is presented weresomealternatives. Inordertodothis,itwas 1 onlyinmyPh.D.thesis. Iwillalsomentionsome necessarytosolveasystemofequationsfortheal- v lost opportunities and delusions. gebra structure constants, which follow from the 9 I graduated from the Physical Department of Jacobi identities. I restricted myself to the set 0 2 Moscow State University obtaining the diploma of four complex spinor charges and found that 1 with honors in 1968. At that time, I was a there were four versions of such algebras: two of 0 co-author of two publications on some plausible them arenowknownas =1 and =2 super- 1 N N charge-parity breaking effects in atomic nuclei. I algebras, in two others the momentum did not 0 dreamedofsolvingmoreambitiousproblems,but commute with spinor charges (a` la de-Sitter al- / h this required becoming a postgraduate student. gebra). These results were obtained in 1968 but p However my diploma supervisor, Yuri Shirokov, were published later only in 1972, because their - p couldnotarrangemypostgraduatestudies—nei- relationtophysicswasnotimmediatelyclear. We e ther at the Physics Department of Moscow State chose the simplest ( =1) supersymmetricalge- h N University, where he was a Professor, nor at the bra for further investigations. : v MathematicsInstituteoftheAcademyofScience, Twice a week, before seminars of the Theoret- i where he was a staff member. Therefore, he rec- ical Department, I showed the result of my cal- X ommended me to Yuri Golfand who was working culations to Golfand, and we discussed them. At r a in the Theory Department of the Lebedev Phys- thattimeIusedtext-booksondifferentaspectsof ical Institute of the Academy of Sciences. applying group theory to physics, but I did not In the late 1960’s — early 70’s, the staff mem- read any new publications. Later, when I was bers of the Theoretical Department were famous preparing my Ph.D. thesis, I had to write a re- Soviet physicists such as Igor Tamm, Vitaliy view of modern literature on the subject, and I Ginzburg,and Andrey Sakharov. At that time faced difficulties again. Just recently I have read Golfandwasstillpreparinghisdoctoralthesis[1]. the historical review “Revealing the path to the A moderate height, a rapid gait and a charming superworld” by Marinov in the Golfand Memo- well-wishing smile were to level down the differ- rial Volume [2] with great interest. I don’t know encesbetweenouragesandpositions. Heshowed whether Golfand knew about Felix Berezin’s pa- meseveralcommutationandanticommutationre- pers or any other publications on the subject. lations between the operators of momentum, an- Maybe,hebelieved,thathehadalreadyinformed gularmomentumandsomespinorsandexplained meaboutallthatwasnecessaryforourwork. By to me that their consistency was verified by Ja- all his visual appearance Golfand gave me, and cobiidentities. Thesecommutationrelationspro- probably not only me, an idea that, contrary to 2 others, we worked on a very serious and impor- constructed and following general properties of tant problem. At the same time, he seemed to irreducible representations were established: have good relations with all staff members of the First, the maximum spin in any irreducible TheoreticalDepartment. He likedjokesandban- • representationdiffersfromminimumoneby ters very much, in particular about gaps in my not more than one. This conclusion was education. obtained by expanding the group operator The main problem to be solved was to relate (notthesuperfield)inTaylorserieswithre- the constructed algebra to quantum field theory. spect to Grassmann parameters of spinor Nobody knew whether such a relation existed, translations. The expansion turned out to and if yes,whether its representationswere finite have a cut-off and therefore the irreducible dimensional. I use here the word “algebra” in- representations were finite! stead of “group” for the following reason. We introduced the notion of supercoordinates and The second discovered property was that consideredthegroupwithGrassmannparameters • the number of bosonic and fermionic de- and established the supercoordinate transforma- grees of freedom in every multiplet was the tions under spinor translations. But we didn’t same, and, as a consequence, the total vac- guess to expand the superfield with respect to uum energy of bosons and fermions was theGrassmannvariablesandtoestablishtherela- equal to zero. tionbetweenthe superfieldandasetofthe usual fields forming the supermultiplet. The notion of IciteheretheLebedevPhysicalInstitutepreprint the Superfieldwasnotpresentedinthepublished #41, 1971 (see Appendix), of which the only papers [3,4], because for us it did not provide a reader was probably Misha Shifman. I wanted mechanismforconstructingasuperinteraction;it very much to publish the obtained results imme- was presented in my Ph.D. thesis only. The way diately, in 1969, but Golfand believed that they to success, as it seems to be now, was not so el- would attract no attention. Moreover, he didn’t egant and, therefore, more laborious. I began to wantto losetime onthe manuscriptpreparation. seek the representations of the algebra in terms Therefore, I began constructing an interaction of of creation and annihilation operators. the found multiplet. As long as we had to do with the algebra, the At that time the Academy of Science was re- mainproblemwastoproperlyhandlethegamma- ceiving letters from inventors with requests to matrices and not to forget to change the signs in make examination of their projects. Post grad- certain places of the Jacobi identities while deal- uate students had to make sense of them and ing with anticommutators. When we turned to to give a conclusion. I was given a project of fieldtheory,wefounditverydifficulttogetaccus- a rocket, where liquid moved inside the rocket in tomed to the fact that both bosons and fermions a closed tube: straight in one direction and in are in the same multiplet and even more difficult the zigzag fashion in the opposite direction. The to imagine how they transform into each other authoroftheprojectbelievedthatduetotherel- under spinor translation. There was no way to ativistic effects a force has to arise and push the get an answer from a crib as I used to do at my rocket forward. I could not point out to the au- exams on the Marx-Lenin philosophy. Golfand thor that this contradicted the momentum con- encouragedme: “Acat maylook ata king.”The servation law, because the author, evidently, did onlythingthatwasclearfromtheverybeginning notknowaboutit. Ihadtospendalotoftimein was that all particles in the multiplet have equal ordertofindmistakesinhisconsiderations. After masses, but this fact didn’t give optimism. this case I was afraid to find myself in a similar Atlast,in1969thesuperspinoperatorandtwo situation, in the position of luckless inventor. irreduciblerepresentationsofalgebra(chiralmul- The psychological barrier associated with the tiplet of spins zero and one half and vector mul- fermi-bose mixture was broken, but some techni- tiplet of spins zero and one half and one) were cal difficulties arose. The method of construct- 3 inginteractionswastowritedownageneralform persymmetry. Nevertheless, summarizing the re- of spinor generators of the algebra not only in sults of three years’ work I conclude that under terms of the second but also the third powers the continual support by Golfand, I had solved of the fields. Consequently, the number of un- the problem formulated by him, that is I demon- known constants in front of different combina- stratedthequantumfieldrealizationofsupersym- tions of fields was about a dozen. Today I solve metry by a specific example. about eight hundred of nonlinear equations with What stimulated Golfand to formulate the ease and have problems with the computer only problem? It is clear that it was not any spe- if the number of equations exceeds a thousand. cific experimental result such as constant speed To solve the equations in 1970 I used pen and of light or the approximate equality of the neu- the reverse side of blueprints of old drawings. tronandprotonmasses. Therewerealsonoques- It is easy to solve a problem from a text-book, tions of overcoming some internal contradictions when you know, that the answer does exist and in field theory as it took place in constructing the only thing to do is to find it. When one of general relativity or the Weinberg-Salam model. equations contradicted the others I didn’t know I think that he was inspired by numerous posi- whetherthiswastheresultofarithmeticmistakes tive results which were obtained by applying one or whether the problem had no solution at all. or another symmetry in physics during the 20th My postgraduate term went to finish and I had century. to think about my Ph.D. thesis and future job. Attheendof1971IgotajobatthePhysicsDe- Finally, when constants obtained fromone of the partment of All-Union Institute of Scientific and equationsappearedtosatisfytheothers,Itookit Technical Information and hardly found time to asamiracle. Moreover,theunknownconstantsat continue my work on fermi-bose symmetry. Gol- the fourth powersof fields in the spin translation fand’s position was even worse: he lost his job operators could be set to zero. The system was and became unemployed. solved,thefirstsuperinteractionwasconstructed. Ireallyhateddivergencesinquantumfieldthe- Nowit is knownasmassivesupersymmetric elec- ory,andthe hypothesiscame tome thatthe can- trodynamics. celation of the vacuum energies divergences for The time came to draw the results up. I was free fields, that I found, would take place for in- goingtowritedownonebigconsistentpaper,but teracting fields as well. Knowing coupling con- Golfand took another decision. Time of publica- stants it was easy to establish that the one-loop tioninourjournalswasverylarge,andhedecided mass divergences of bosonic fields in constructed towritedownashortpaperforJETPLetters[3]. model will not be quadratic, but logarithmic as Golfand cut down my manuscript without any for fermionic field. Golfand was not excited by pitytofittherequiredvolume. Deletedfragments this result because he believed it happened just were scattered around in other publications. At by chance. Maybe his intuition left him, but thesametimeIconstructedtheselfinteractionfor maybe his mind was occupied by completely dif- the vector multiplet. I succeeded in constructing ferent non-scientific problems. Considering the onlythetrilinearpartoftheinteraction,therefore actionofspinortransformsonS-matrix,Icameto the resultwaspresentedonly in my Ph.D.thesis. the conclusion, that the cancelation of the diver- The title of the thesis was so complicated, that genceswasnotaccidentalandhadtotakeplacein duringthe defense ofmy Ph.D.thesis inSeptem- higher orders. However, the divergences did not ber 1971, the scientific secretary of our institute disappear completely. What if the nonrenormal- faced difficulties in reading it aloud. The title izabletheorybecamerenormalizableone? Unfor- seemed to reflect our understanding of the prob- tunately,withmymethodsforconstructingmod- lem. We erroneously believed, that constructing els therewereno chancesto treatnonrenormaliz- supersymmetricinteractionthroughthepowerse- able models. riesexpansionofgroupgeneratorsinthecoupling In 1974 Igor Tyutin told me about the Wess- constant was related to specific features of su- ZuminopaperwhichIhavereadwithgreatinter- 4 est. The linear realization of representations ow- essary to arrange the lattice better, but maybe ing to the presence of auxiliary fields, covariant it is necessary to seek the unknown symmetry of derivatives with respect on Grassmann variables the equations of motion to use it for preliminary — it was very beautiful, and allowed one to con- analytical analysis. structthe superinteractionrathereasily. Integra- InconclusionIwouldliketothanktheorganiz- tion overGrassmannvariables,usedin the paper ers of the conference Thirty Years of Supersym- by Salam-Strathdee, finally set both types of ar- metry for giving me the opportunity to share my gumentsofsuperfieldsonequalfooting. Thepos- vision of thirty-year old events, and for financial tulates ofthis integrationwereformulatedbyFe- support. I would like to express my gratitude to lixBerezinin1965. UsingthesetechniquesIcon- Misha Shifman for comments to my notes in the structed two models with the Abelian and non- Golfand Memorial Volume [2]. These comments Abelian massive vector fields and proved their madesomeaspectsoftheSovietlifeinthe1970’s renormalizability. At the same time I also noted more palpable for the Western readers. And, the that the proof of renormalizability could be car- last but not the least, Yuri Golfand remains for riedoutinanotherway,usingthestandardmech- me notonly the manwho taughtme a trade,but anism of spontaneous gauge symmetry breaking. the man who made me love the risky business — Thequestionofanyrelationofsupersymmetryto not to be afraid to go deep in the forest by the high energy physics remained open. paths nobody walked on before. Thank you for After the publication of these two papers [5], I your attention. practically stopped my work on supersymmetry because of strong competition and the absence REFERENCES of any support. I began to seek my own unique 1. The academic hierarchyin Russia follows the topic. A small number of assumptions and the GermanratherthantheAnglo-Americanpat- simplicityofconstructions,togetherwithnontriv- tern. An approximate equivalent of Ph.D. in ial and maybe not very realistic (at first glance) the US is the so called candidate degree. The results — that was the main impression, which I highest academic degree, doctoral, is analo- gained from my collaboration with Golfand. On gous to the German Habilitation. The doc- theotherhand,myabilitytooperatethepersonal toraldissertationisusuallypresentedatama- computers allows me to try to solve problems, turestageoftheacademiccareer;onlyafrac- while I do not even know about the existence of tion of the candidate degree holders make it their solutions. I heard that to seek a black cat to the doctoral level. in a dark room is very risky business, especially 2. The Many Faces of the Superworld — Yuri initsabsence. Nevertheless,todayIamtryingto Golfand Memorial Volume, Ed. M. Shifman study numerically the model of Born-Infeld elec- (World Scientific, Singapore, 2000). tromagneticfieldinteractingwith amembraneof 3. Yu.A. Golfand and E.P. Likhtman, JETP finite size and with a string boundary, where it Lett. 13, 323 (1971) [Reprinted in Supersym- is beforehandknownthatthedivergencesofelec- metry,Ed.S.Ferrara,(North-Holland/World tricandmagneticenergiescanceleachother. But Scientific,Amsterdam–Singapore,1987)Vol. this cancelation is not the final aim. It appears 1, page 7]. that some combination of mass, electric charge 4. Yu.A. Golfand and E.P. Likhtman, in and magnetic momentum is independent on two I.E. Tamm Memorial Volume Problems of unknown dimensional constants of the model on Theoretical Physics, Eds. V.L. Ginzburg et the one hand and, on the other hand, is related al., (Nauka, Moscow 1972), page 37. [English to well-known observable value — the fine struc- transl. in Ref. 2, p. 45]. ture constant. At the present time, the solutions 5. E.P. Likhtman, JETP. Lett., 21, 109 (1975); obtained slowly converge with the growth of the JETP. Lett., 22, 57 (1975). number of the lattice cells. Maybe this points to the absence of solutions, or maybe it is nec- 5 APPENDIX At the suggestion of Misha Shifman I present below a verbatim translation of my paper written in 1970 which circulated as FIAN preprint No. 41 (1971). The final authorization for issuing this preprint was obtained on April 12, 1971. LEBEDEV PHYSICS INSTITUTE OF THE USSR ACADEMY OF SCIENCES Theory Department Preprint No. 41 Moscow 1971 IRREDUCIBLE REPRESENTATIONS OF THE EXTENSION OF THE ALGEBRA OF GENERATORS OF THE POINCARE´ GROUP BY BISPINOR GENERATORS E.P. Likhtman 1. INTRODUCTION (1) imposes rigid constraints on the form of the quantum field interaction. In constructing this In Ref. 1 a special extension of the algebra P example we used two linear irreducible represen- of the generators of the Poincar´e group was con- tations of the algebra (1). Their definition was sidered. The extension was performed by virtue not given in Ref. 1. In this work I present the of introduction of the generators of the spinorial definition of these representations of algebra (1), translations W and W¯ , α β and build other representations. The properties of these representations are investigated. [Mµν,Mσλ]− =i(δµσMνλ+δνλMµσ δ M δ M ); µλ νσ νσ µλ − − [Pµ,Pν]− =0; 2. THE SPACE OF STATES AND IN- VARIANT SUBSPACES [Mµν,Pλ]− =i(δµλPν δνλPµ); − i Inordertodetermineinwhichspacetherepre- [Mµν,W]− = [γµ,γν]W; (1a) 4 sentationsofalgebra(1)actnotethatthealgebra W¯ =W+γ0; is its subalgebra. Therefore, any representa- P [W,W¯] =γ+P ; tion of algebra (1) is also a representation of the + µ µ algebra . The spacesinwhichthese representa- [W,W]+ =0; [Pµ,W]− =0; (1b) P tionsactcoincide. However,irreduciblerepresen- 1 γ± =s±γ ; s± = (1 γ ), γ2 =1. tationsofalgebra(1)arereduciblewithrespectto µ µ 2 ± 5 5 . This reducible representationsplits in several P Thespinorialindicesareomitted,whiletheex- irreducible representations of and one and the P pressions of the type d d are to be understood same irreducible representation can be involved µ ν asd d d d d d d d hereafter. Inthesame several times. To distinguish them we will intro- 0 0 1 1 2 2 3 3 − − − workarealizationofthisalgebrawasconstructed duce the number χ of the irreducible representa- in which a Hamiltonian describes interactions of tions. quantumfields. Thisexampleshowsthatalgebra Of physical interest are those representations 6 of (1) which can be reduced to (several) repre- spin j one could obtain a vector with a nonva- 1 sentations of characterized by mass and spin. nishing projection on the state vectors with spin P therefore, the basis vector of the space in which j , j j > 1. To see that this is impossible 2 2 1 | − | we build representations of (1) can be written as note that the consecutive action of the operators follows: from algebra (1) in the most general case can be represented as a polynomial in these operators. κ,p ,j,m,χ , (2) λ Due to the (anti)commutation relations (1) the | i form of this polynomial is where κ is the mass, p is the spatial momen- λ tum, j stands for the spin, m is its projection on the z axis, and, finally, χ is the number of the Cµαν,.,....;.β;λ,...... irreducible representation of . X M ... P ... W ... W¯ ..., P µν λ α β Inthespacewiththebasisvectors(2)thereare × × × × × × × × subspaces invariantunder the action of the oper- where Cα,...;β,... are numerical coefficients. ators from the algebra in Eq. (1). According to µν,...;λ... The product of any number of operators M µν Schur’s lemma [2], in order to find invariant sub- andP doesnotchangethespinofthestate. The λ spaces it is necessary to find invariant operators number of operators W (and W¯) in each term which,bydefinition,commutewithalloperations can be equal to one or two since if it is larger of algebra (1). It is easy to observe that the op- than 2 then the product vanishes because of the eratorP2 doeshavethisproperty. Therefore,the µ anticommutationrelationinEq.(1). Bythesame space of which the basis vectors correspond to reasontheproductW W =0onlyprovidedthat α β particles of one and the same mass κ will be the 6 α = β. Such operator does not change the spin invariant subspace. The spins of the vectors of 6 of the state as well. Thus, an invariant subspace this invariant subspace cannot be all the same, can contain only the spins j, j+ 1, j+1. since the square of the spin operator is not an 2 In orderto build the irreducible representation invariant operator of algebra (1), ofalgebra(1)itisnecessarytofindthematrixel- 1 ementsoftheoperatorsfromalgebra(1)between [Γ2µ,W]− 6=0; (Γµ = 2εµνλσMνλPσ). these states. Instead, the invariant operator of algebra (1) is D2 where 3. EQUATIONS FOR THE REDUCED µ MATRIX STATES 1 P P D =Γ + (W¯γ W µ νW¯γ W), µ µ 2 µ − P2 ν The matrix element of the operator of conven- σ P2 =κ2 >0. tional translations can obviously be written as σ ′ ′ ′ ′ Otherinvariantoperatorsinalgebra(1)arelikely hκ,pλ,j,m,χ|Pµ|κ,pλ,j ,m,χi to be absent. = (δ κ2+p2 +δ p ) A priori it is not exactly known which vectors µ0 λ µλ λ q ′ (2) form the basis of the irreducible representa- × δ(pλ−pλ)δjj′δmm′δχχ′, (3) tionof(1),sincethepropertiesoftheoperatorD2 µ are not known. Besides the fact that the mass of where λ = 1,2,3. The operator M will have µν allstates is the same,one canassertthatthe dif- its conventional form as well. We will be inter- ference between the maximal and minimal spins estedin the matrixelements ofthe operatorsW α is the irreducible representation of (1) does not andW¯ , which,due to Eq.(1),arediagonalwith β exceed 1. In other words, irreducible representa- respect to κ and p . One can readily convince λ tionsof(1)containnomorethan3distinctspins. oneself that the operators s−W and W¯s+ satisfy Otherwise, by consecutively applying the opera- the trivial commutation relations. Therefore, we tors from algebra (1) to the state vectors with willlimitourselvestoinvestigatingsuchrepresen- 7 tations in which 1 where j1 j2 j′ is the 6j symbol. As a s−W =s¯+ =0. (cid:26) j3 j4 J (cid:27) result of this substitution we obtain, after per- forming the summation, Without loss of generality let us choose the rep- resentation of the γ matrices with a diagonal γ5. ( 1)2j′′+J+M(2J +1) Then the operators Wα and W¯β become two- JMXj′χ′ − component. Moreover, knowing the transforma- j j′ J 1 1 J tionlawofspinorsundertheLorentztransforma- × (cid:26) 1 1 j′ (cid:27)(cid:18) 2β α2 M (cid:19) tions,andassumingthatκ>0,letuspasstothe 2 2 − reference frame with p = p′ = 0. In this ref- j j′′ J erence frame one couldλapplyλthe Wigner-Eckart × (cid:18) m m′′ M (cid:19) − − theorem [4] according to which ( 1)m+α−j′ (2j+1)(2j′′+1)(2j′+1)2 × − κ,0,j,m,χs+Wα κ,0,j′,m′,χ′ jχf j′χ′ pj′χ′ f+ j′′χ′′ h = (cid:18) mj | α21 |jm′ ′ (cid:19)(−1)j′−im′ ×+ (cid:16)(h1)J|−|j+j′′ihjχf|+ j′|χ′ ji′χ′ f j′′χ′′ − − h | | ih | | i(cid:17) (2j+1)(2j′+1) jχf j′χ′ ; = κδjj′′δmm′′δχχ′′. (5) × h | | i κ,0,jp,m,χW¯s− κ,0,j′,m′,χ′ (An analogous formula is obtained upon substi- h | β| i = (cid:18) mj′′ β12 jm (cid:19)(−1)j−m t[Wut,ioWn]o+f=Eq0..()4)Nienxtthweeanutsiecotmhemvuatlauteisonofretlhaetio6nj − symbols [4], (2j+1)(2j′+1) jχf+ j′χ′ , (4) × p h | | i j1 j2 j3 =( 1)j1+j2+j3 where(cid:18) mj α12 jm′ ′ (cid:19)aretheWignersymbols, (cid:26) 21 j3− 12 j2+ 21(cid:27) − 1 j j′ = 1, jχ−f j′χ′ are the reduced matrix (j1+j3−j2)(j1+j2−j3+1) 2 ; | − | 2 h | | i × (cid:20)(2j +1)(2j +2)(2j )(2j +1)(cid:21) elements, (2j+1)(2j′+1)is aconvenientnor- 2 2 3 3 j j j malizationpfactor. Therepresentation(4)ensures 1 2 3 =( 1)j1+j2+j3 the correct commutation relation with the mo- (cid:26) 12 j3− 12 j2− 21(cid:27) − mentum operator and the operator of the spatial (j +j +j +1)(j +j j ) 12 1 2 3 2 3 1 rotations. In order to satisfy other commutation − . × (cid:20) (2j )(2j +1)(2j )(2j +1) (cid:21) 2 2 3 3 relationsofalgebra(1),we substitute (3)and(4) Then Eq. (5) takes the form in(1b) and exploitthe formulaeof summationof the 3j symbols in the spin projections [4], 2j′+1 jχf j′χ′ j′χ′ f+ j′′χ′′ ( 1)j′+m Xj′χ′(cid:18) 2 (cid:19)(cid:16)h | | ih | | i − Xm′ + jχf+ j′χ′ j′χ′ f j′′χ′′ j1 j2 j′ j3 j4 j′ h | | ih | | i(cid:17) × (cid:18) m1 m2 m′ (cid:19)(cid:18) m3 m4 m′ (cid:19) = κδjj′′δχχ′′; (6a) − = ( 1)2j4+J+M(2J +1) ( 1)j′ jχf j′χ′ j′χ′ f+ j′′χ′′ XJM − Xj′χ′ − (cid:16)h | | ih | | i j1 j2 j′ j3 j2 J jχf+ j′χ′ j′χ′ f j′′χ′′ × (cid:26) j3 j4 J (cid:27)(cid:18) m3 m2 M (cid:19) − h | | ih | | i(cid:17) = 0, for j =0; (6b) j j J 6 1 4 , jχf j′χ′ j′χ′ f+ j′′χ′′ × (cid:18) m1 m4 M (cid:19) h | | ih | | i 1Relaxingthisrequirem−entwouldrequire,ofnecessity,the Xj′χ′ ′′ introductionofanindefinitemetric[3]. = 0, except for j =j =0. (6c) 8 Thesearetheequationsthatwereourgoal. Their tinct spin values enter. In this case j′ in Eq. (6) ′ solution will allow us to find the explicit form of takes only one value, and the summation over j s+W and W¯s−. Note that Eq. (6) describes the is,infact,absent. Using thisfact, letus multiply representationinwhichtheinvariantoperatorD2 Eq. (6b) by ( 1)j′(2j′+1)/2 and add up the re- µ − need not necessarily be proportional to the unit sult with Eq. (6a). Then on the right-hand side operator, i.e. we get reducible representations of one will find a nonsingular matrix acting in the algebra (1), generally speaking. space of the variable χ (at fixed j, j′, and j′′). On the left-hand side the matrix will be nonsin- 4. THE NUMBER OF PARTICLES IN gularonlyifj =0. (Cf. Eq.(6c)). Therefore,the 6 THE REPRESENTATION OF ALGE- representation of algebra (1) with two spins can BRA (1)AND SOME SOLUTIONS OF containonlyspin-0andspin-1/2states. Itiseasy EQ. (6) to check that in this case the simplest solutionof the system with n = 2 (χ = 1,2) and n = 1 0 1/2 First of all, starting from Eq. (6) I will deduce (χ=1) has the form constraintsonthe numberofparticlesinthe rep- resentationofalgebra(1). Tothisendletusmul- 0 0 1 tiply Eq. (6a) by ( 1)2j(2j+1)/2 and sum over jχf j′χ′ =√κ 0 0 0 , (8) j =j′′ and χ=χ′′−. After this operation the left- h | | i 0 1 0 hand side will vanish. To see thatthis is the case it is sufficientto transposetwo factorsin the sec- where the matrix on the right-hand side acts on ondterm(whichisjustifiedsincethisisinsidethe the state trace)andtousethefactthat( 1)2j =( 1)2j′+1 − − a (see Eq. (4)). Then the second term will differ b . from the first one by the sign only. The right- c hand side of Eq. (6a) must also vanish, and we obtaintheconstraintonthenumbernofthepar- The amplitudes a and b describe the spin-0 par- ticles with spin j in the representationof algebra ticles while c describes the spin-1/2 particle. (1), In the case of the three-spin representations, j, j +1/2, and j +1, the lowest spin j can be ( 1)2j(2j+1)n =0. (7) Xj − j arbitrary. Thesimplestsolutionwithnj =1(χ= 1), n = 2 (χ = 1,2), and n = 1 (χ = 1) j+1/2 j+1 As is known [5], in relativistic quantum field can be written in a form analogous to (8), theory, the particle energy operator,being trans- 0 0 1 0 formed to the normal form, contains an infinite term which is interpreted as the vacuum energy. jχf j′χ′ =√κ 1 0 0 1 . (9) It is also known that the sign of this term is dif- h | | i 0 0 0 0 ferentforparticlessubjecttothe BoseandFermi 0 0 1 0 − statistics. According to Eq. (7), the representa- Therepresentations(8)and(9)areirreducible. tionsofalgebra(1)includeparticleswithdifferent One can readily convince oneself that this is the statistics, so that the number of the boson states case even without calculating the eigenvalues of isalwaysequaltothatofthefermionstates. From the operator D2. It is enough to observe that this it follows that the infinite positive energy of µ there exist no representations with a lesser num- the boson states is canceled by the infinite nega- ber ofparticlessatisfyingthe necessarycondition tive energy of the fermion states in any represen- (7). The question of existence/nonexistence of tation of algebra (1). other irreducible representations of (1), besides Afterthesepreliminaryremarksweproceeddi- those found here, requires further investigation. rectly to solving Eq. (6). Let us try to find rep- resentation in which only particles with two dis- 9 ↔ 5. SECOND-QUANTIZED RELATIVIS- d3x ω∗(x)∂ ∂ ω(x) γ+ TIC REPRESENTATIONS OF ALGE- − Z (cid:16) µ 0 (cid:17)× µ BRA (1) + i d3x ψ¯ (x)∂↔ γ−ψ (x) γ+ 2Z (cid:16) 1 0 µ 1 (cid:17)× µ In this section we will show how one can pass i ↔ from the representations (8), (9) acting in the + d3x ψ¯c(x)∂ γ−ψc(x) γ+ spacewiththebasisvectors(2)totherelativistic- 2Z (cid:16) 1 0 µ 1 (cid:17)× µ covariantform of these representations,acting in = d3xT (x) γ+. (11) the space of the occupation numbers. The oper- Z µ0 × µ ators of the algebra in such representations have InthiscalculationIusedtheFierzidentity,the tobeexpressibleintermsofthesecond-quantized equations of motion, and the commutation rela- free fields with equal masses but distinct spins. tions for free fields. Let us further note that the Since relativistic equation for spin-1/2 particles energy-momentumtensorT isnonsymmetricin αβ describes both particles and antiparticles, it is the case of the spinor field, generally speaking. necessary to introduce antiparticles in the repre- However, if one of the indices is zero, then sentation(7) . Then the operators of the algebra willbe expressibleintermsofnon-Hermitianfree Tµ0 =T0µ, scalar fields ϕ(x), ω(x), and a spinor field ψ (x). 1 andtheintegralontheright-handsideofEq.(11) Let us show that the operator becomes the energy-momentum tensor of the Wo =s+Wo = 1 ϕ∗(x)∂↔ s+ψ (x) fields ϕ(x), ω(x), and ψ1(x). All other relations 0 1 i Z (cid:16) in (1) can be checked in a similar manner. The ↔ actionoftheoperatorWoonthefieldoperatorsis + ω(x)∂ s+ψc(x) d3x, (10) 0 1 a linear transformationof these fields. Schemati- (cid:17) cally, one can write it as follows: (wherethe superscriptomeansthatthe operator is bilinear in the field operators while the super- ϕ(x) ψ(x) ω(x) 0, script c means the charge conjugation) satisfies 2 ω∗(x)→ ψc(x→) ϕ∗(→x) 0. (anti)commutation relations (1). For instance, → → → Let us pass now to generalization of the repre- [Wo,W¯o] =i d3xd3y sentation (9) to cover the case of the quantized + ZZ fields. We will limit ourselves to the option that ↔ ↔ ϕ∗(x)∂ γ+i∂ D(x y)∂ ϕ(y) the lowest spin is zero, while two spin 1/2 parti- × (cid:16) x0 µ xµ − y0 (cid:17) clesmaybeconsideredrelatedbytheoperationof + i d3xd3y thechargeconjugation. Thentheoperatorsofal- ZZ gebra (1) in this representationwill be expressed ↔ ↔ ω(x)∂ γ+i∂ D(x y)∂ ω∗(y) in terms of a Hermitian scalar field χ(x), Hermi- × (cid:16) x0 µ xµ − y0 (cid:17) tian vector transverse field Aµ(x), and a spinor + i d3xd3y field ψ2(x). This irreducible representation can ZZ differ from that in Eq. (10) by the mass of the ↔ ↔ ψ¯ (y)s− ∂ D(y x)∂ s+ψ (x) particlesandmustdifferbytheeigenvaluesofthe × (cid:16) 1 × y0 − x0 1 (cid:17) invariant operator Dµ2. The operator Wo in this + i d3xd3y representation has the form ZZ 1 ↔ × (cid:16)ψ¯1c(y)s−×∂↔y0 D(y↔−x)∂↔x0 s+ψ1c(x)(cid:17) Wo =s+Wo = i√2↔Z (cid:16)χ(x)∂0 s+ψ2(x) = −Z d3x(cid:16)ϕ∗(x)∂µ ∂0 ϕ(x)(cid:17)×γµ+ + Aµ ∂0 γµ+ψ2(x)(cid:17)d3x. (12) 2TheoperatorW isdefineduptoaphasefactor,seeRef.3. One can verify Eq. (12) in the same manner as 10 Eq. (10), This can be seen also from the fact that the ac- tion of the operatorsWo and W¯o on the vacuum 1 [s+Wo,W¯os−] = d3xd3y always yields zero, and + −2iZZ ψ¯ (y)s− ∂↔ D(y x)∂↔ s+ψ (x) Pµ =Tr(γµ−[Wo,W¯o]+). × (cid:16) 2 × y0 − x0 2 (cid:17) 1 ↔ Therefore, the representation (14) also possesses + 2iZZ d3xd3y(cid:16)ψ2(y)γµ+×∂y0 this property, while the vector and spinor mass- less particles can only be in two states with the 1 ↔ × (δµν + µ2∂yµ∂yν)D(y−x)∂x0 γν+ψ2(x)(cid:17) opposite chiralities. 1 ↔ χ(x)∂ ∂ χ(x) d3x γ+ 6. CONCLUSION − 2Z (cid:16) 0 µ (cid:17) × µ 1 ↔ Summarizing,wefoundseveralirreduciblerep- A (x)∂ ∂ A (x) d3x γ+γ−γ+ − 2Z (cid:16) µ 0 α ν (cid:17) × µ α ν resentations of algebra (1). In these represen- tations conventional fields are united in certain = γ+P , (13) µ µ multiplets. A question arises whether one can whereµ=0isthemassofthefieldsχ(x),A (x), identify thesemultiplets withsomeobservedpar- µ 6 andψ (x). Inthisrepresentationtheactionofthe ticles. In answering this question the main diffi- 2 operator Wo on free fields can be schematically culty lies in the fact that masses of all particles depicted as in the multiplet are identical, while spins are dif- ferent. Therefore, at present algebra (1) and its χ(x) realizationsmustbeconsideredasjustamodelof ψ2c(x) ր ց ψ2(x)→0. a Hamiltonian formulation of quantum field the- ց Aµ(x) ր ory. At intermediate stages the mass µ enters in the In conclusion I express my deep gratitude to denominator in Eq. (13). Therefore,it cannot be Yu.A. Golfand for his constant attention to my set to zero. The limit µ 0 can be realized by work and stimulating discussions. → abandoning the condition of transversalityof the vector field and passing to the diagonal pairing REFERENCES 1 1. Yu.A. Golfand and E.P. Likhtman, Pis’ma [Aµ(x),Aν(y)]− =−2δµνD(x−y). ZhETF, in press. 2. M.I. Petrashen and E.D. Trifonov, Applica- In this case the field ψ (x) becomes two- 2 tion of Group Theory in Quantum Mechanics component (s+ψ = 0). The first term in inter- 2 (Nauka, Moscow, 1967). mediatecalculationsin(13)becomesunnecessary 3. E.P. Likhtman, Kratkie Soobscheniya po and disappears; therefore,there is no need in the Fizike (FIAN Brief Communication in introduction of the scalar field χ(x). At µ = 0 Physics), in press. the operator Wo has the following form 4. A. Edmonds, Angular Momenta in Quantum Wo = s+Wo Mechanics, in the collection Deformation of 1 ↔ Atomic Nuclei (Foreign Literature Publish- = A (x)∂ γ+ψ (x) d3x. (14) i√2Z (cid:16) µ 0 µ 2 (cid:17) ers, Moscow, 1958). 5. V.B. Berestetskii, E.M. Lifshitz, and InSecs.2–4whereIinvestigatedtheproperties L.P. Pitaevsky, Relativistic Quantum Theory of the representations with nonvanishing mass, (Nauka, Moscow, 1968). it was shown that the numbers of the fermion and boson states in the representations of alge- bra (1) coincide. Therefore, the operator Po is µ automatically representable in the normal form.