KRQ-76-61
Z. PERJÉS
PERSPECTIVES OF PENROSE THEORY IN PARTICLE PHYSICS
Hungarian academy of Sciences CENTRAL RESEARCH INSTITUTE FOR PHYSICS BUDAPEST KFKI-76-61
PERSPECTIVES OF PENROSE THEORY IN PARTICLE PHYSICS
Z. Perjés High Energy Physics Department Central Research Institute for Physics Budapest 114, Hungary
Submitted to Reports on Mathematical Physics Presented at the Symposium on Methods of Differential Geometry in Physics and Mechanics /Invited talk/, Warsaw 1976
ISBN 963 371 183 5 ABSTRACT Existing results and some conjectures in the flat-space twistor approach to fundamental particles are reviewed. A concise introduction is given into the twistor description of dynamical systems with re*t-mass /both classical and quantum/. The Hamiltonian structure inherent to the angular momentum twistor is analyzed. The following discussion outlines the properties of n-twistor systems, the Penrose classification of particles, the l'10'SU(3) group and the problem of its twistor representations. Finally, speculative arguments are propounded as to the possible bearings of hadrcnic quark models to twbstor theory.
АННОТАЦИЯ
Рассматриваются результаты и предположения относительно фундамен тальных частиц, сделанные согласно теории твисторов. Дается краткое ознаком ление с классическим и квантовым описанием динамических систем, обладающих массой покоя. Изучается каноническая структура, вызванная тиистором импульс ного момента . Описываются теори; n-твисторных систем, классификация частиц, разработанная Пенроэом, и проблемы твисторного изображения группы l(l°)su(3). В заключение обсуждаются возможные взаимосвязи между кварковыми моделями адроиов и теорией твисторов.
KIVONAT AttekintjUk a tvisztorok elméletében a fundamentális részecskékre vonatkozó eredményeket és sejtéseket. Rövid bevezetést nyújtunk a nyugalmi tömeggel rendelkező dinamikai rendszerek klasszikus és kvantumos tvisztor- leirásába. Megvizsgáljuk az impulzusmomentum-tvlsztor által indukált kano nikus szerkezetet. Körvonalazzuk továbbá az n-tvisztorrendszerek elméletét, a részecskék Penrose-féle osztályozását és az I One of the strongest pillars of present-day particle physics is quantum field theory. Its successes depend, in a significant extent, on the flexible use of momentum and coordinate pictures it allows in various physical situations. A further extension of this versatility by including twistors as dynamical variables is an appealing thought. Our aim can be, in the first instance, to transform standard theory into a twistor picture but disregard all possible complications due to quantum gravity effects. Even if this restricted programme will not resolve many of the persisting problems of the theory, it has the merit that it is viable, by the existing theoretical aids. More than just that, as will be shovn here, the flat- space twistor theory offers hew and meaningful explanations for particle properties. The provision made in this approach is, basically, the specific way one introduces rest-mass into the theory with the aid of an infinity twistor I . [i should have said, instead, how сonformal invariance is reduced to Poincaré-invariance. But such group-theoretical notions must be rendered of little importance in a second-generation theory where curved /twistor/ spaces [Г| carry the description of events.]] In section 2 we survey the formalism which encompasses the relativistic description of mas sive particles. Next, in section 3, the beginnings of a Hamiltonian treat ment of massive particle scattering follow. The Hamiltonian structure of twistor space provides for us the quantization rules. The algebra of observables for n-twistor quantum states is develop ed in Sec. 4. A classification scheme of the states /Sec. 5/ in terms of representations of the Internal twistor symmetry groups is suggested by Penrose [19] . in Sec. 6 we take up the detailed investigation of three- twistor systems. Their 14-parameter internal group is [7'J I(l0)SU(3), a semi- direct product of SU(3) rotations with translations in the three-complex- dimensional space С . We give the appropriate solution of the state-label ing problem and align available mathematical armour for obtaining the /discrete/ twistor representations. The ensuing 3-twistor classification of quantum status, likely to га!зе considerable discussion in tho future, I.as bo« obtained after this lecture was delivered, jointly with G.A.J. Sparling, and it will be published elsewhere [_42] . Some more speculative arguments, concerning a possible interpreta tion of hadronic structure in twistor terms, are surveyed in Sec. 7. 2. PARTICLES WITH REST-MASS The kinematical data of a particle, or more generally, of a material system in special relativity are appropriately comprised in the ten com ponents of the total four-momentum [_2] P, and angular momentum N . These tensors are linked up by Poincaré transformations. Let us exploit the skew-symmetry property Ma b - - Mba for rep- AB resenting» thth ab _ AB A'B' . -Л'В' AB M /2.1/ M = и С + и с Lorentz rotations and space-time translations combine into SU(2, 2) ma trices [\\ when they are applied upon the array of components AB -2lp И- /2.2/ '»? By its transformation properties [*_"]» Л is a symmetric |_l- twistor and is called the angular momentum twistor. A symmetric Й- twistor can have 20 independent component» and wo expect constraint equations to be satisfied by Ai<»1*H1 '. Accordingly, we find the liermiHan relation i(A"ßI I'1" A ) - O. /2.3/ BY RY We can check the validity of /2.3/ by using the component forms of the in finity twistor AB О о ,«e /2.4/ Л'В' 0 °J ы- Another condition on л" results from requiring that for any future-pointing null vector ,n, the inequality - 3 - Pi > О /2.5/ must hold. This expresses that the energy is non-negative. An arbitrary twistor Ü with components (U ) = ('J , -iz од) can be defined in terns of which the null vector is written la =• о 5 * Then inequality /2.5/ takes the twistor form ",.А°Чв0* i °- /2.6/ The description of translation-invariant kinematical data easily follows from this formalism. Consider the Pauli-bubanski *-ector Kbcd («bC«'d^d"bC)- /2.7/ This symmetric definition of S ensures both that S. is Hermitian and that it satisfies S P /2.8/ a o. In definition /2.7/, no reference is made to the value of the rest-mass m which is given by the angular moment im twistor; det |Außl - Л2 /2.9/ where A=im2, /2.10/ Only whon the particle lias a non-vanishing rest-mass can we define its spin vector as Sa/m. However, it is always possible to give /2.7/ a twistor form, /2.11/ The spin twistor К ' Мая the components in our fr.ime, "o 6} 8 I /2.12/ .;sA'B °J where, of course, Sa - S . it is singular and niipotent in tha senses that det and S ßS„' = 0. /2.13/ - 4 - Hence if we insist on the twistor notation for expressing the Magnitude S » -S s of the Pftuli-Lubanski vector, we auat use Penrose's equation Qs] S PS ° - S PS ° - Í S2 T TPÖ 17 14/ « SB 8 « 3 aß1 * /2-14/ Until now the discussion was kept quite general so as to accomodate both massive and zero-mass states. Next we recall that a massless particle can either be characterized by a one-index twistor Z or, also, by its angular momentum twistor quadratic in Za, A»«r 2 *<*1*>Чу . /2.15/ /A parenthesized zero subscript reminds us that /2.15/ holds for a massless particle./ No doubt Z° provides a more direct description of the particle than does the angular momentum twistor. What we need is a similar decomposi tion of A in the generic case. ThTh Aae = 2 zia jßhi* /2Л6/ к у where к » 1,2_,... ,n and summation over к is understood when considering components [2] . For each value of k, there is a twistor Z? describing a zero-mass constituent of the particle. ~ Equation /2.16/ generalizes the result that any time-like vector may be thought of as a sum of two or more null vectors. The future-pointing momentum vector P is decomposed Pa " VV + PAPA' I2'17' whence the mass yields the inequality 2 A 2 m - 2JirAp j > 0. /2.16/ Thus 12.П1 defines a basis (Ä. , p.) in the spin space. Consider now the canonical decomposition of an arbitrary angular momentum spinor uAB - a(ABB) /2.19/ We can express the principal spinor B. in terms of the basis, 1 - 5 - Ö = a Ьр /2.20/ A *A * Л and find AB .'A-B) . (A-B) v /2.21/ ц = a^i •• ír '+ bei P . For the choice /2.19/ of the origin, two twistors Za and Wa are d< fined by their components (?*) - <*ЛтТд.) (W,() = (ьЛод,) . -»fi Decompositions /2.17/ and /2.21/ entail for components of A , ,aB (,1 e, (а В,1 ~2 Z i 4 + 2 „ 1 'й1 . /2.22. Y YJ from iwistor invariance it then follows that /2.22/ is a decomposition of a Liberal angular momentum twistor into two constituents. Even in the simplest massless case there is an arbitrariness /the [•h.-isf of z'J, cf. /2.15// in this description. The invariance transformations •Hi composed of 11 twistors, turn out to be [7j /2.23/ Zk Uk \ZS. iro X V' •iht't'i.' tin; ( nxri ) matrices U and Л arc restricted by и и - l : /2.24/ /a T superscript is used for matrix transposition/. Whether such a description of the massive particle is purely a mathematical game or it uncovers physical structure is an important question. We might try to answer it by considering thf; behavior of constituent twistors in interactions. Tho idea is that the outcome of the processes may depend on the twistor strucutro of tht? interacting particles. Here we encounter the difficulty that the transition to Penrose picture [ 3. DERIVATION OF SCATTERING HAMILTONlANS Calculations carried out so far in this field all describe scatter ing on weak shock waves. The first problem scrutinized by Penrose £loD is the interaction of a zero-mass particle with a plane-fronted wave, a limiting case of the Brinkmann-Robinson space-times which have the line element dt2 - 2(du+R(v,t.Udv)dv - 2d£ ul . /3.1/ In the shock-wave limit, R(v,e,£) - e(v)0r(O + r(C)] . /3.2/ In the classical treatment, the particle is described by a null geodesic and its twistor constants of motion before and after the scattering are Za and Za+iza , respectively. Integration of the geodesic equation yields 3H(Za,Zj /3'3/ 6K " A 7Г-2- where the real-valued Hamiltonian is determined by the field function r(4). In Eqs. /3.3/, the canonically conjugate twistor variables are also complex conjugates. This is a unique feature of the twistor picture which forecasts an eventual intermingling of space-time and probability contiпил QllQ. The validity of scattering equations is restricted, originally, to the subspace of null twistors. One may allow for Za to range over the whole of twistor space but the resulting equations /3.3/ cannot be easily inter preted. The spinning zero-mass particle hhat the non-null twistor za de scribes has no unique world line. However, we can study the consequences of this generalization by looking into problems of massive scattering [12]. We may us« a decomposition of the massive particle into supporting twistors. Then, we have to make a choice how the individual twistors should scatter on the wave. We shall assume that they behave in the interaction as if they were free "inside" the particle. Accordingly, we take Hpcl. " "<*i'V + "•+H(Z D'V /3'4/ and ez? - -i *"**• * ко эн (Я ш i —B£i£ /3.5/ - 7 - Remarkably enough, this simple assumption, together with analytic continuation .n the twistor space, leads to the law of geodesic motion for л spin-zero test particle and to Papapetrou's equations when spin is allowed. To make the statement more precise, we define the complex center-of-mass line of the particle by Za(T) - Ха(т) + i m"2 Sa, x tC /3.6/ ./here Xa(c) gives the real center of mass for real values of the proper time T . The complex world line ol the particle will generally meet the wave at a complex point Za(r ) and the constituents are arranged to pass through this same point. He expand the field quantities in power series of the magnitude -f the imaginary part of Z (i ) . Then we assume for a test о particle that we c&n drop all terms of higher than first order. It may seem that these are too restrictive examples to draw any general conclusions from them, flowever, the scattering on any gravitational wave can be viewed as a succession of infinitesimal scatterings on weak s.hock waves. In this sense, the shock wave calculations probe the structure of the gravitational interaction. These ideas have been used also in a study of electromagnetic interactions [12] but the question how the electric charge is precisely coded into constituents was, at thi?« stage, carefully evaded. The particle i 55 assumed to taVe part in electromagnetic interactions via its "electric dipole twistor" С , the electromagnetic analog of angular momentum twistor. In terms of spinor components, AB .A •2 id 3 B' M - /3.7/ tf Here j is the electric current represented by the particle, and the elec- r~ AB tromagnetic dipole spinor (_13 J~\ d contains the electric and magnetic moments. In an infinitesimal electromagnetic scattering the dipole twistor changes by the amount ÍCaS = -i Гн, Caß] . /3.8/ Gonerically, the Hamiltonian H is expected to depend on the electromagnetic potential and the dipole momentum twistor. For a plane-fronted wave [12] with null generator В passing through the point of »cattering, H is a function H (',,£) where - 8 Similarly, from the principle of equivalence it follows that gravi tational interactions do not depend on the internal structure of particles. The description of gravitational scattering should not rely on a preferred choice of supporting twistors. Therefore a formulation must exist, where H is a functional H(A ,A „), analogously to the above description of electro magnetic scattering. In addition, one has to show that internal-symmetry transformations /2.23/ preserve the Hamiltonian structure of equations /3.5/. In effect, it must be shown that transformations /2.23/ are canonical. The canonically conjugate variables in Eqs. /3.5/ are PK - -i 2к 73.IP/ q*« Z* Consider the function S а S(g,g), 5 " -*<*а - Í Aab **ty ü+| P = ff . <1 ="! . /3.12/ ээ ЭР Equations /3.12/ yield the internal-symmetry transformations /2.23/ which are therefore canonical transformations. 4. N-TWISTOR QUANTUM STATES When the constituent twistors Z. describe a quantized system, they obey commutation rules [*»fl 5S h /4.1/ as it follows from Eqs. /3.5/. Physical states are suitably represented by functions of several twistor variables, a space on which Zy act as linear operators. The transition to coordinate picture is carried out by taking certain contour integrals of these twistor functions [б]. But information is lost in the procedure and it may be conjectured that the twistor picture provides a fuller /or perhaps exhaustive/ description of the state. - 9 - In standard relativistic quantum field theory, the Hilbert space oi state functions accomodates unitary representations of the Poincaré group. This gives rise to a classification of states. A state belonging to a re ducible representation is usually considered compound while elementary particles are classified into irreducible /unitary/ representations [l4,16J. In the Penrose picture, the Poincaré group [1Ъ\ is accompanied by the group of internal transformations /2.23/. The composition law for the latter group may be written 3 /4.2/ (y2. 42> The infinitesimal operators of the group /4.2/ provide internal quantum numbers which further characterize the state. This is the basis of the clas sification of elementary particles in the theory. The word "elementary" can well be taken to mean "irreducible". "o find the infinitesimal operators, we adjoin to Eq. /2.23/ its llermitian conjugate, л e- [».-*; ° 5 У ьс* S ß О Ö .5 6 e %£ '4.3/ b a fe B> c e Hence we may wite \j.f\ В*? - Z° Z* a a a 9 ß A i г z ab Xa6 Za Zb /4.4/ d«b ж jeß га zb a 8 Operators B— form a Hermitian matrix, /4.5/ and operators d . satisfy /4.6/ We shall call dafe the partial mass operators. The Lie algebra commutators are - lo - gad /4.7/ [V •§] - «f «ao - «Í-ьс d d - О . [ sb - cd] - o. [dÄ. a*] Generally, operators /4.4/ are not all independent. To show this, we introduce a new notation [б»I'D where twistor indices are represented by "valence lines", Za ** Za ' * 2 ~ ?§ /4,8/ and index contraction is achieved by connecting the lines z. Щ *"* z»\ z~ l*-9l Skewing of indicest zl zb - zl 7ъ " \ h I*'10i The graphical representation of fundamental twisters goes as follows, á« ..,| x«e „ LJ e«f»Y« ~ Mil x ai ~ П eaey6 ~ ТТП • /4.11/ There are certain identities satisfied by the fundamental twisters. ТЬеяе identities give rise to relations among operators /4.4/. The e twistor satisfies j I I 1 | | j | ГГП * ' ' I г /4Л2а/ fiom which it follows (J_jJ~^n , j_j__|. . /4.12b/ The infinity twistor is self-dual, U - \ I 1 П , м-"«/ - 11 - and it has the property that it is simple, -fcj LJ " 0 . /4.13b/ All contractions of infinity twistor valence lines yield zero since 1 I I - О . /4.13c/ We insert identity /4.13a/ and its conjugate in /4.13c/ and use Eq. /4.12b/ to obtain -РЯ- /4.14 Next we transvec- identities /4.13b/ and /4.14/ with constituent twistors Zy. We roust have a sufficient number n of independent constituent twistors to get nonzero results on the left hand sides. For ц > 3, identity /4.14/yields constraints involving B? , d . and d— . For n > 4, we obtain from /4.13b/ that the skew quantity d . is simple. We can see this at once in terms of twistor components: 'ab •X - *2 /4.IS/ It should be noted that in Eqs. /4.4/ all independent twistor in variants are included. New invariants cannot be produced without the use of e twistors. Because of identity /4.13b/, any invariant containing a connect ed network of e's can be broken up into invariants with no more than one e in each. Consider next L1JJ П • I I I I- | I I l_i "ifHJ"- /4Л6/ Again we transvec •- with constituent twistors and obtain the relationship zaz°zczd zazbzc zd UJJ П -J-H-f-f /4.17/ Zze ZfZ . VZ Z.r I 1 Hence we see that invariants containing a single e factor are dependent on other invariants which do not contain any e factors. - 12 - 5. PENROSE CLASSIFICATION OF STATES The properties of Multiple twistor systems show a remarkable di- versit/. A detailed case study of some of these, systems will help to en lighten certain ideas on their possible relevance in particle physics. We may pass the simplest case of a one-twistor with the remark that its "internal" transformations Z » e Z /5.1/ /with 9 real/ form the one-parameter group U(l). The infinitesimal oper ator, as an observable, is the helicity of the massless particle that the one-twistor represents Q.9J. Considering now bitwistors /n » 2_/, the corresponding group turns out to be isomorphic with the semi-simple group [43] OÍ3) • Б(2). Here E(2) is the group of the Euclidean motions in the 2-plane. Its two com muting translations are provided by the skew matrix g of /2.23/ /now a 2x2 matrix/. The complex phase of у gives rise to the rotations in the plane. Let us introduce the bitwistor notation /5.2/ where we accept the practice of suppressing indices as common in particle physics. The infinitesimal operators of the 0(3) factor group become [l7,19j Sj - * Oj* /5.3/ with at as the three Pauli matrices. For E(2) we have the'partial mass' operators П „ « d - Ф * - i a - * f /5.4/ i\ aß г\ , и and the'phase' operator /5.5/ The two Casimirians of the group are - 13 - m2 - 2 d a , j(j+l) - »i »i • /5-6/ The internal operators s. have a striking structure familiar írom the non-relativistic theory of spin. We may project out the components of the Pauli-Lubanski operator /2.7/ in the rest frame of the particle defin ed by the composing twistors. These projections are just operators /5.3/. In this way the internal structure and space-time spin of a bitwistor par ticle turn cut to be related Г20). Having identified operators /5.3/ with the rest-frame components of the spin, we discover that Eq. /5.3/ does not contain the infinity twistor, hence the spin of a bitwistor particle is conformally invariantI Me may con jecture that this higher invariance property will present itself in the be havior of form factors. In effect the claim is that Dirac particles with form factor g • 2 are composed of bitwistors. Thereby one is led to a model of leptons which are Dirac particles. Penrose has attempted to classify [J2lJ the spin - 5 leptons in the 3=1/2 unitary irreducible representations /irreps/ of the bitwistor group 0(3) a E(2). Because of the non-compactness of this group, the irrep contains an infinite number of states, whereas the number of /light/ leptons and their antiparticles is eight. Penrose's solution /the "lepton cube"/ is to truncate the j - 1/2 irrep by introducing an auxiliary condition which selects the observed states. It ir a prediction of his model that heavy leptons /if existing/ have spins higher than 1/2. Alternatively, the possibility is open that one can dispense with the unitarity condition imposed on the representa tions. We now turn to the similarly interesting case of three-twietors. Their Aa -preserving transformations /2.23/ constitute a 15-parameter group. ft ft In the 24-dimensional three-twistor space the independent components of A define quartics which are transitivity surfaces of the group. Since these surfaces are 14-dimensional, the group is doubly transitive in them. It can be proved [7] that for n * j, a 14-parameter subgroup of /2.23/ exists which acts simply transitively on these surfaces. The subgroup (у,Д) is ob tained by imposing у е S'J(3). In three dimensions, an equivalent parametrlzation (y,t) for the group element (Ц>Д) becomes possible, where t84e— Abc ' '5'7' Composition rule /4.2/ takes the form - 14 - In /5.8/ one recognizes [7,22] the group structure of inhomogeneous linear transformations in the complex Euclidean 3-space, A be The invariants of /5.9/ are dg dq and the Levi-Civita symbol e . These considerations show that the 14-parameter "minimal" group which carries a 3-twlstor decomposition of a given particle into all pos sible 3-twi.stor decompositions is isomorphic to the inhomogeneous SU(3) group /5.9/. There are many possible inhomogenizations of a given semi-simple group. Rosen [23] has proved that the inequivalent irreducible representa tions of a semi-simple Lie algebra correspond in a one-to-one fashion to its all possible different /i.e., not isomorphic/ inhomogenizations. His result must be applied with care to group inhomogenizations. Here the in- homogeneous infinitesimal operators belong to some representation of the semi-simplé subgroup. When this representation is complex, its conjugate must also occur in the inhomogenization. An example is the fundamental rep resentation (X,p) « (1,0) of SU(3) /we are using highest-weight labels [24] for the representation/. In the inhomogenization, the corresponding transla tion operator occurs together with its conjugate [(A,p)- (0,1)]. We may denote the various inhomogeneous extensions of semi-simple groups according to the following patterns I(A> G . /5.10/ Here G is the symbol of the semi-simple subgroup and (A) contains the labels of the irreducible representation. In the notation /5.10/ the minimal three-twistor /5.8/ is I( 'su(3). /However, for the fundamental inhomogeni zation of a group, we may conveniently drop the representation labels./ It may be of interest to note in this context that particle physicists had studied [25,26,27] another inhomogeneous SU(3) group, the I<13)SU(3) as a candidate of strong interaction symmetries. This 16-parameter group includes translations in the octet space. The 3-twistor group is the smallest internal-symmetry group for which the constraints derived in Sec. 4 hold among Infinitesimal operators. This property of the transformation group is to be related to the existenco of a transitive subgroup. We rewrite the general expression /4.4/ for the infinitesimal operators as 15 *S - Í e~ *bc BJ> « Г Z* /5.11/ be a a a 3a " 1 eabc * where now a.b.... take the values 1,31 and 3_. Identity /4.14/ may be written d* Bb a. - О . /5.12/ а о That is, the infinitesimal operators are not algebraically independent. We can form the linear combinations a . 1 ,a _r •Í - - 4 + i 6i Bl /5.13/ satisfying (e§) - £ , в5 - о . /5.14/ Operators 0r , d- and d possess 14 algebraically independent В а omponui.ts. From commutation rules /4.1/ we obtain [Bl' 1] " 6I sl - 6| »I [da, ßg = 5ä db _ 1 6b di |>3,d&J-o-[d*faj ß a 6 5 /5.15/ [V fi ~1 b - 7 Í ft What we have found is the Lie algebra of the minimal 3-twistor transforma tion group /5.8/. Here we have a vehicle for classifying 3-twistor states and we are hopeful that the SU(3) hadron multiplets will find their place in the scheme. However, w-? encounter the difficulty that a theory of V SU(3) has yet to be developed. 6. THE !(10,SUC3) GROUP (10), By its semi-direct product structure, the I SU(3) is one of the scantily studied non-semisimple Lie groups. Here we survey the existing re sults that may help constructing the representation. Equipped with the Lie algebra commutators, Eqs. /5.15/, we first determine the number of Casimirians. According to a theorem by Beltrametti --> -> - 16 - and Blast [28] , for an arbitrary r-parameter Lie group the number v of Casimirians is v « r - rank e^ a. /6.1/ where c., are the structure constants and a. the group parameters. The *J * i- к Т symbol "rank" is a shorthand for the maximal rank of the matrix c, . a. as a function of the parameters. The recipe is not very convenient for us since it involves rank of a 14x14 matrix. Ne shall follow, instead, a constructive method. It will be helpful to use an SU(3) blob notation: e| _ ф . d2 „ \ e« „ | da „ \ eabc „ LJJ p /6p2/ The Casimirians must be SU(3) scalers since otherwise they would not commute with the SU(3) generators. Me shall now produce the independent SU(3) in variants. Plrst, the invariants which contain only ß's. We recall a theorem by Okubo [29] graphically expressed as $W« 3 <>\£ + (* $g) -2)^(| ф\^) -^\g)) J /6.3/ All chain products of more than two ф factors can be expressed by shorter chains. Hence the independent ф invariants are с я C2 and C3 are Casimirians of the SU(3) subgroup but do not commute with d translations. From the translations we can construct /6.5/ 2 A«m /2 is a Casimirian, and has positive eigenvalues. The remaining chains are ВД /b.bal > which serves to define the "baryon number operator" B, and - }7 - X /6.6b/ •!h" spin square operator [3cJ J - - и" SS* has the expression J2 = i B2 - |в + i C - Л"1 Thf- llormitian operator В is no Casimirian since [d*, в] - • § da, faa, в| - - § aa . /6.8/ .1' is a Casimirian. The invariants containing e's remain to be constructed. Because of the identities J=p{ - +-H- ; LiJT1 * ^ ' /6'9/ ito more than a single e can occur in the invariant. The only possibility is •щ /6.10/ The operator X is not Hermitian and it is not a Casimirian either, since, for example, [X,B] = -2X . /6.11/ From identities /6.9/ it follows that XX is a dependent quantity. The role of operator X seems to be quite puzzling and it has no counterpart in the fairly well developed theory [31J of E(3). In order to label uniquely the states of the representation space, a complete set of commuting Hermitian operators has to be found. Here again there is a general theorem by G. Bacah £32} . Racah's theorem says that the number i of labeling operators can be expressed I » r/2 + v /6.12/ where r denotes the number of group parameters and v the number of Casimirians. However, it is not quite clear from Racah's argument how general the theorem is. Certainly it holds for semi-simple groups. But there are other examples /one of which is [33]/the group E(3) where tne number of labeling operators varies with the basis in the representation space. - 18 - For our case Rac.th's theorem yields the number I * 8. The method of induced representations [34]] is applicable because I SU(3) has a semi- direct product structure. The induced representations are labeled indeed by 8 parameters [3"fj. However, they are not appropriate for us. The reason for this is thai, the observables Br * Z,Z- and ß» = Z_Z- become Euler operators 11a £ f a when they act on twistor functions. A twistor function has to be homogeneous [б] to describe a free particle. Hence Br ' 8= and В must be diagonal. One could also argue that hadrons are «igenstates of the hypercharge Y « ß| + ß| /6.13/ and isospin projection I2 - ^(Bj - B§) - /6.14/ The induced representations are not diagonal in and I . However, the following eight independent commuting Hermitian operators can be found: 2 2 2 J , m . В, С2, С3, У, I , Iz /6.15/ 2 12 where I « I»'Iz+1' + ^2 "l " Tne su*3* Casimirians are conveniently-re placed by the highest-weight labels [36,37] {\ ,v) C2 " I (Х2+и2+Хи+ЗА+Зц) C3 - | (p-XM2X+u+3) To sum up, the basis vectors for the representation spaces will be labeled |a> - |j,m; B, 3-twistor quantum states are given by the range of quantum numbers in /6.17/ It follows that the representations of the I SU(3) group in an SU(3) basis are to be determined For unitary representations, the infinitesimal operators defined by the real group parameters must be hermitian. However, the translations in the complex 3-space С belong to complex representations of the SU(3) group. Hence the components of the SU(3) tensor operators d- /representation (X,u) " (1,0)/ are not Hermitian in a unitary representation. Tneir Hermitian conjugates d& form /up to phase factors/ the conjugate tensor operator. Having noted that, we will not restrict the following discussion to unitary representations. - 19 - The boundaries of a given representation /J,m/ am located at №rws of the matrix elements of "step operators" $r- (a^b), d- and d"a . Operators ß| with a * b do not connect different SU(3) multiplets. The o.ssontial information is contained in the matrix elements [38] of d- and 1 . One can use the Wigner-Eckart theorem to get rid of the SU(3) projec- CI fion .{uantum numbers Y, T and I : ' b|d-|a > = < aj(lo)-|b >< b\\á\ a >. /6.18/ liere a;(lO)-|b> is an SU(3) Clebsch-Gordan coefficient and •bj|cl||a> is the reduced matrix element which is independent of the SU(3) projection quantum numbers. Algebraic expressions and recursion formulas for SU(3) i:lebs.-h-Gordan coefficients wev given by Hecht [39] . Quark coupling coef- iicrients occurring in the matrix elements of d- and d have been worked out by Asherova and Smirnov [40] . There are three non-vanishing reduced matrix .lements in /6.18/ for a given |a> as fallows [41} from the Clebsch-Gordan series of the :1U.-Ct product ( Ю) в (>U). PUt "2 - £S^Sf-i<»-!• <»+i.u)iidn..(4P)>i2 1)2 2 -ЛТЙЖЙ7Й-1<В-!» Points of the ((X,u),B) space comprise SU(3) multiplets. The rep- 2 2 2 resentations aro given in this space by the zeroes of U , D and S and the corresponding quantities defined by the adjoint operators d . The three reduced matrix elements /6.19/ may be determined from the three operator equations /6.5/, /6.6a/ and /6.7/. Enclosing these among states, we obtain aldidkí^|a> * ВЛ /6.20b/ On the left hand sides, we insert complete sets of states and apply the Wigner-Eckart theorem. Then, by use of the orthogonality properties of the Clebsch-Gordan coefficients we obtain from /6.20a/ - 20 - U2 + D2 + S2 = 1 . /6.21/ The rest of the left hand sides in /6.20/ has the overall structure Ф = Tr <п|т£ dk d. | a> /6.22/ where T. is a Hermitian tensor operator and we are summing /Tr/ over the SU(3) projection indices of state |a>. Evaluation of /6.22/ yields [44] ф * ~ I dim(b>u{a,(ll),b, (Ю); a , (10); p} | U(l,2,3,4;5,6;p) - I <1;2j5> <5;4 | 3> .<2;4|6> <1;6|3> ,, . /6.24/ all SU(3) pi p:>* pz* pi0 projection indices Here p, are multiplicity indices. There exist explicit results [39,47~] for some of the coefficients /6.24/ in the literature, but these do not, unfortunately, include the ones we require in /6.23/. The results -if these computations have to be relegated to a subsequent publication [4 2j. 7. SPECULATIONS Whereas it is fairly easy to obtain the continuous representations of i SU(3/ by the method of induced reprerentations, we may infer from the last section that the discrete representations pose a more difficult problem [4Ű. i wish to stress that if is not even clear if we are to con struct the unitary representations 4n applications to hadrons. For we can not expect that Wigner's unitarity theorem Г49] holds herei Whatever the case turns out to be, the resolution will throw new light on our ideas about quarks. Shortly after the concept of quarks was introduced into strong-in teraction physics, it has been argued that quarks would not be observed be cause they are too heavy [50]. As a consequence, quark bindings ought to be strong to explain the resulting masses. Here we encounter a clear-cut con tradiction with results of deep-inelastic scattering experiments indicating that quarks behave as if they were "free" inside hadrons fsij. Physicists were thereby led to a concept of "imprisonment" of quarks and even the idea of "hadronic bags" has been suggested. - 21 - It is not my purpose to criticire these theories on some deeper philosophical grounds, in particular since even their advocators admit [52] the rudimentary nature of the models. Let us try to glean, instead, another but consistent picture from the experimental facts, within the very frame work of twistor theory. We venture to say that quarks are the twistors themselves and find the corollaries of our assumption on some long-standing ijuestions [5зЗ of the theory. a/ Why free qruarks have not been observed? We obtain a subtle answer here; in a sense, "quarks' can indeed be seen: these are the zero- mass particles. When imprisoned, the twistors as independent variables of the state function carry the information about "quark structure'. And the scattering calculations of Sec. 3 show that twistor constituents do, behave in /at least some/ interactions as though they were free! b/ Are hadrons extended? Three twistors have three "partial" center of masses. This structure is already extended in space-time in the normal sense, yet a twistor structure can entail a wilder sort of extendedness. Sparling f54] has pointed out that it may not always be possible to project out a local quantum field from the twistor state function. c/ Quark statistics: The old solution is color [55]. No need arises for introducing color here since the twistor variables into which quarks are coded are not physical particles. d/ Relativization of quark model; Now relativistic e/ Charm and <*/J: Is there one more quark? SU(4)? It may seem that we have plenty of room for them here, by introducing state functions depend ing on an arbitrary number of twistors. On the other hand, Penrose |56j finds that already four-twistor state functions are subject to a set of restrictive conditions. Moreover, we obtained in Sec. 4 algebraic relations among internal auantum numbers, the number of which grows fast when increasing the number of constituents. Both of these observations would seem to lessen the chances that twistor functions of too many variables can represent new fundamental particle states. f/ Dual behavior of resonances 8 Recent investigations by Hughston I 57] show that a one-to-one correspondence between duality diagrams and twistor graphs for massive particle interactions exists. 22 - REFERENCES [1] R. Perrose, GRG Journal 7, 3). /19161 [2] We are adopting the abstract index notation /R. Penrose, in Battelle Recontres, Eds. CM. DeWitt and J.A. Wheeler; Benjamin, 1968/: indices, generally, exhibit the structure of the tensorial entities but cannot take numerical values. However, we shall use matrix brackets /instead of introducing fancy index font types/ for denoting components. Accord ingly, we take for the components of the metric: (g.J-diag/1,-1,-1,-1/. Space-time indices, inside component brackets, range through the values 0,1,2 and 3. Similarly, Greek twistor indice. can be chosen from 0,1,2, or 3. Twistor internal labels a,b,... may range through positive integer values. Spinors are indexed by capital Latin fonts, A,B,... and A*,B',... [3] R. Penrose, J. Math. Phys. 8, 345 /1967/ [4] With our attention restrained here to the Poincare group [5] R. Penrose, personal communication [6] R. Penrose and M.A.H. MacCallum, Physics Reports 6C, 241 /1973/ [7] Z. Perjés, Phys. Rev. 11D, 2031 /1975/ [8] Why, indeed, has not been formerly named so his beautiful quantum picture? [93 The blame is, in part, on the reluctance of many particle physics ex perts to deviate from the beaten tracks of their favorite formalisms- [10] R. Penrose, Int. J. Theor. Phys. 1, 61 /1968/ [113 R. Penrose, in Magic without Magic: John Archibald Wheeler, a Collection of Essays in Honor of his 60th Birthday, Ed.J.R. Klauder,-Freeman, San Francisco, 1972 [12] P. Tod and Z. Perjés, preprint KFKI-76-12 [13] E.T. Newman and J. Winicour, J. Math. Phys. 15, 113 /1974/ [14] S.S. Schweber: An Introduction to Relativistic Quantum Field Theory, Row, Peterson and Co., 1961 [15] We are considering the flat-space version of twistor theory [16] E.P. Wigner: Group Theory and its Application to the Quantum Mechanics of Atomic spectra,Academic Press, 1959 [17] We ought to symmetrize the factors in the expressions B- for infinites- 9 imal operators of /4,2/ but this has no significance for the forthcoming discussion [18] It is no surprise that a new algebraic structure invokes a new formalism. Einstein's index notation fits well with tensor manipulations in a metric; 21 - space. However, ir. the twistor space there is no metric and the funda mental skew twistors frequently produce ugly expressions in the familiar index formalism. К . Penrose, in Ouantum Theory and the Structures of Time and Space, Kil.s L. Caste)1, M. Drieschner and C.F. von Weizsäcker;Carl Hanser, Munich 1975 Such an interplay of internal and space-time symmetries is not un familiar in iwrticle physics. The charge conjugation operator С is related to space-time reflections /by the CPT theorem/ and it also acts as an internal-transformation group automorphism. Twistor theory predicts [19] another interconnections since here the squared rest-mass and spin magnitude are simultaneous Casimirians of the internal and l'oincaré groups;. R. Penrose, pr iva te newspaper t am indebted to Professor Taub for reminding me of this direct method .J. Rosen, Nuovo Ctm. 45Л, 234 /1966/ .I..I.tie Swart, Rev. Mod. Phys., 3J>, 916 /1963/ C.J. Goebel, Phys. Rev. Letters 16, 1130 /1966/ C. DulleiPond, Ann. Phys. 33, 214 /1965/ U.V.. Возе, Phys. Kev. ISO, 1231 /1966/ W. Mukunda, J. Math. Phys. 11., 1759 /197o/ E.G. Beltrametti and Л. Blasi, Phys. betters 20, 62 /1966/ S. Okubo, Progr.Theor. Phys. 21, 949 /1962/ The three-twistor structure of the spin operator has also been studied by G. Sparling /personal communication/ W. Pauli: Continuous Groups in Quantum Mechanics /lecture notes reprinted in Ergebnisse der Modernen Naturwissenschaften, Vol. 37; Springer, 1965. Cf . also Ref. [27"j G. Racah: Group Theory and Spectroscopy /lectures notes reprinted in the volume of Ref.[31] R. Mirman, J. Math. Phys. 10, 331 /196?/ G.W. Mackey: Induced Representations of Groups and Quantum Mechanic»; Uenjamin, 1968 M. Huszár, personal communication b.c. Biedenharn, Phye. Letters 3, 254 /1963/ and J. Math. Phys. 4, 436 /1963/ - 24 - p37] G.E. Baird and L.C. Biedenharn, J. Math. Phys. 4, 1449 /1963/ [38] A.Chakrabarti, J. Math. Phys. 9, 2087 /1968/ gives some of the matrix elements of the translation operators for inhomogeneous unitary groups [39} K.T. Hecht, Nucl. Phys. 62, 1 /1965/. I»> his Table 4 /p.31/, column (A',p') = (A,u),p=l, all signs have to be reversed when и = О. [40j R.M. Asherova and Vu.F. Smirnov, Nucl. Phys. B4, 399 /1968/ [41] D. Speiser: Theory of Compact Lie Groups and Some Applications to Elementary Particle Physics, in Group Theoretical Concepts and Methods in Elementary Particle Physics, Ed. P. GUrsey. Gordon and Breach, 1964 [42] Z. Perjés and G.A.C. Sparling, to be published [43З G.A.J. Sparling, personal communication. The globax connectivity properties of the bitwistor group had received much of discussion but. they have not been correctly sorted out until recently. [443 Z. Perjés, to be published [45З A.P. Yutsis, l.B. Levinson and V.V. Vanagas: Mathematical Apparatus of the Theory of Angular Momentum. Monson, 19o2 [4G3 H.A. Jahn, Proc. Roy. Soc. A205, 192 /1951/ [473 M. Resnikoff, J. Math. Phys. 8, 63 and 79 /1967/ [483 Representations of inhomogeneous unitary algebras have been studied by R. Mirman, J. Math. Phys. 8, 57 /1967/; 9, 39 and 47 /1968/. r.t. also Ref. [333 [49З E.P. Wigner, Ref. [1бЗ , Chap. 20 [50З M.E. Pearl: High Energy Hadron Physics. Wiley, 1974 [Si] J.D. Bjorken, lecture at the Dubna Conference on Interactions of Vector Mesons, 1969 [52j Cf. Particlee and Fields - 1974, Chap. 6. Ed. C.E. Carlson. A.I.P. 1975 [53З H. Lipkin, Physics Reports 8C, 173 /1973/ [54З G.A.J. Sparling, personal communication [553 М.У. Han and V. Nambu, Phys. Rev. 139B, 1006 /1965/ [56j R. Penrose, in Group Theory in Non-Linear Problems. Ed. A.O. Barut; Reidel, 1974 [57З L. Hughston, private newspaper U-J ___J leo« ^'^ Xiadja a Központi Fizikai Kutató Intézet FelelSs kiadót SzegS Károly tudományos igazgató Szakmai lektort Tóth Kálmán Nyelvi lektor t Simányi József Példányszámi 100 /second print/ Törzsszámi 76-901 Készült a KFKI sokszorosító Üzemében Budapest, 1977. február hó