158 Proc. Jpn. Acad., Ser. B 86 (2010) [Vol. 86,
Review Superalgebra and fermion-boson symmetry
By Hironari MIYAZAWA 1,y
(Communicated by Toshimitsu YAMAZAKI, M.J.A.)
Abstract: Fermions and bosons are quite di erent kinds of particles, but it is possible to unify them in a supermultiplet, by introducing a new mathematical scheme called superalgebra. In this article we discuss the development of the concept of symmetry, starting from the rotational symmetry and nally arriving at this fermion-boson (FB) symmetry.
Keywords: supermultiplet, Lie algebra, hadron spectroscopy
with di erent spins are grouped as symmetry mem- Introduction bers of a supermultiplet. Feza Gursey€ and Luigi Symmetry is invariance under transformation. Radicati and Bunji Sakita extended the idea to in- Right-left symmetry, the invariance under the space clude the strange particles. Here all important mesons re ection, is one of the most elementary examples. are grouped in one supermultiplet, and baryons are Spherical symmetry is the invariance under the space in another. rotation. Our three dimensional space is supposed to The next question is: is it possible to incorporate be invariant under the space rotation. mesons and baryons in one symmetry multiplet? The above examples are transformations of Mesons obey Bose statistics and baryons obey Fermi space coordinates. There are other symmetries con- statistics, so they cannot be regarded as the same nected with transformations of the matter of our particles. Nevertheless, by extending the notion of world. Proton and neutron are almost identical ex- symmetry, even bosons and fermions can be grouped cept for the charge. They can be regarded as the together. For this fermion-boson symmetry, the ordi- same particle, the nucleon, in two di erent states. nary Lie group theory is insuf cient. A new mathe- Lambda, one of the strange particles, di ers in matical concept, called superalgebra, must be intro- mass from the nucleon by 12 percent, and can hardly duced. In the following sections we shall see how the be regarded as the same. However, we can separate idea of symmetry was extended to the FB symmetry. the hamiltonian into symmetric and asymmetric Space symmetry parts, rst consider the symmetric part only and then consider the asymmetry. In this way the Our three dimensional space is spherically sym- Lambda becomes a symmetry member of the nu- metric. The spherical symmetry means the invari- cleon. This symmetry is arti cial and is not an intrin- ance under the rotation, represented by the group sic one given by god. Anyway, this way of separating SO(3). In this article we shall mainly use the term the system into symmetric and asymmetric parts ‘‘invariance’’ to mean the invariance of the hamil- worked very well in strange particle physics. tonian of the system. The rotation group SO(3) is So far, the space symmetry and the internal mathematically equivalent to the special unitary symmetry are separate and independent. Eugene group SU(2), where the fundamental representation Wigner combined these two into one symmetry and consists of two complex elements. introduced the supermultiplet. In this scheme, nuclei q " : ½1 q 1 Professor emeritus, Faculty of Science, University of Tokyo, # Tokyo, Japan. Physically q " and q # are two states, the states y Correspondence should be addressed: H. Miyazawa, Faculty of Science, University of Tokyo, Hongo 7-3-1, Bunkyo-Ku, Tokyo of spin up and down of the fundamental spin 1/2 110-0003 (e-mail: [email protected]). particle. Combination of these elements and its com- doi: 10.2183/pjab.86.158 62010 The Japan Academy No. 3] Fermion-boson symmetry 159 plex conjugate, or bound state of this particles and Charged and neutral pions form an I ¼ 1mul- their antiparticles, produce group representations or tiplet which is almost degenerate in mass. Experi- multiplets of de nite angular momentum. A multip- ments show that in reactions of nucleons and pions let is a set of degenerate states of a de nite quantum the isospin is conserved. An example is: The deu- number, the angular momentum this case. All mem- teron and the alpha particle are known to have bers of a multiplet are the same particle, only the isospin zero. Then the reaction direction of its angular momentum being di erent in d þ d ! þ 0; ½4 space. The term fundamental means that any spin can never happen if the isospin is conserved, the left- state can be constructed from q " and q # and their antiparticles. hand side being isospin 0 and the right-hand side Space re ection is a separate transformation being isospin 1. In fact, this reaction is found to group. It is known that the space re ection symme- occur very rarely. try only holds approximately, violated by the weak Neglecting the electromagnetic interaction, which interaction. On the other hand, the rotational sym- is not large compared with the strong interaction, metry holds exactly, so far as there is no external the nucleon-pion physics is invariant under the rota- eld. In this article we shall mainly consider continu- tion in the isospace. This is an experimental fact. So, ous transformations. we might say that this symmetry is given by god. Symmetries in matter Strange particles Our space is spherically symmetric. Similar sym- Lambda is a companion of the nucleon. It also metry is found in the matter. The atomic nucleus has spin 1/2 and can turn into a nucleon. Further, consists of protons and neutrons. Proton and neu- lambda has a new quantum number, strangeness, tron are quite similar. Their masses di er only by which is conserved in the strong interaction. In addi- 0.1 percent. Except for the electric charge and the tion to the fundamental elements proton and neu- small mass di erence, proton and neutron are identi- tron (eq. [2]), we need a new element as the carrier of the strangeness. cal. They can be regarded as two di erent states of 0 1 one particle, which is called nucleon. Similarly to p B C eq. [1] we put @ A B ¼ n : ½5 p N ¼ : ½2 n This triplet cannot be fundamental. Other The dominant mass term in the hamiltonian can be strange particles, sigma and xi are quite similar to written in the form nucleon and lambda and must be grouped together X in one multiplet. Then the quark model is intro- y y H0 ¼ M ðap kap k þ anka nkÞ: ½3 duced. In this model the fundamental triplet consists k of up quark u, down quark d, and strange quark s, 0 1 y where a and a are annihilation and creation ope- u rators, respectively, and k is the momentum of the B C Q ¼ @ d A: ½6 particle. This H is invariant not only under the ex- 0 s change of p and n but also under the SU(2) transfor- mation of N (eq. [2]), It turned out, experimentally, Mesons are bound states of quark and anti- that the whole hamiltonian is invariant under the quark. Baryons, that is, nucleon, lambda and their SU(2). We introduce another three dimensional brothers, are bound states of three quarks. We con- space, which is called the isospace. Proton and neu- sider the symmetry among these particles. tron form an isodoublet, a state of isospin 1/2, pro- Now that we have the fundamental triplet of ton being the state of isospin up and neutron isospin elements, we want to consider the mixing among down. The nuclear system is invariant by the rota- the three, the SU(3) group. u and d form an isodoub- tion in the isospace. Here the small p-n mass di er- let and accept SU(2) transformation. However, s is ence and the coulomb interaction, which is small much heavier than u and d, and the triplet does not compared with the nuclear interaction, are neglected. form a degenerate multiplet. 160 H. MIYAZAWA [Vol. 86,
Still, we wish to stick to the SU(3). We split the two are independent. Wigner1) combined these two hamiltonian into SU(3) symmetric part and asym- symmetries. metric part, and consider only the symmetric part, The fundamental elements are as follows: disregarding the asymmetric part as perturbation. 0 1 p" This SU(3) symmetry is man-made, and is not given B C B p# C by god. We shall see how this arti cial symmetry : ½10 @ n A works in strange particle physics. " n Group theoretically the nine states of QQ split # into an eight-dimensional representation and a one- Since the masses of the four elements are equal, the dimensional one, mass term in the hamiltonian is invariant under the SU(4) transformation. Thus Wigner extended 3 3 ¼ 8 þ 1: ½7 the symmetry as follows: Each representation corresponds to a degenerate multiplet. In fact, an octet of pseudoscalar mesons SUð2ÞSp SUð2ÞIso ! SUð4ÞSI; ½11 and a singlet of pseudoscalar meson have been found The extended symmetry is applied to nuclear phy- experimentally. For spin 1 bound states, also a sin- sics. Nuclei with di erent spins or isospins are clas- glet and an octet of vector mesons have been found. si ed in a supermultiplet. Wigner showed that this Baryons are bound states of three quarks. Three symmetry gave a good t to light nuclear states quark products split into four representations. where the Coulomb interaction is unimportant. 3 3 3 ¼ 10 þ 8 þ 8 þ 1: ½8 This idea of combining the space and internal symmetries was extended to the strange particle The ten dimensional representation corresponds to physics by Gursey€ and Radicati2) and by Sakita.3) the decuplet of spin 3/2 baryons. The eight corre- The fundamental elements are sponds to the spin 1/2 baryon octet. The rest, 8 þ 1 0 1 have not yet been established. u" B C In this way all the known hadrons (baryons and B u C B # C mesons), can be classi ed as multiplets or representa- B C B d" C tions of SU(3), but each multiplet is not degenerate. q ¼ B C; ½12 B d C The mass di erence among a multiplet member can B # C @ s A be calculated by introducing the mass di erence " among the elementary triplet as perturbation. For s# the baryon octet, the mass formula is and they consider the symmetry SU(6)SI. 1 1 Mesons are bound state of q and q . The thirty- ½MðNÞþMð Þ ¼ ½3Mð ÞþMð Þ : ½9 2 4 six states decompose into two representations. It is remarkable that the mass formula [9] is satis ed 6 6 ¼ 35 þ 1: ½13 to within a percent of error. Similar formula can be The lowest-mass mesons, an octet of pseudoscalar derived for the octets of mesons, but they are not so mesons and a nonet of vector mesons, t precisely to well satis ed experimentally. the 35. Thus, we see that the man-made SU(3) works Baryons are bound states of three quarks. well for the classi cation of hadrons and for deriving 6 6 6 56 70 70 20: 14 relations of their properties. ¼ þ þ þ ½ A charming idea is that intrinsically the hamil- The octet of spin 1/2 baryons and the decuplet of tonian is symmetric under the SU(3), but the sym- spin 3/2 baryon resonances t precisely into the 56. metry is broken spontaneously. In this article, how- In this way, all of the known mesons belong to ever, this possibility will not be considered. one representation of the group SU(6), and the bary- ons belong to another representation. Members of Wigner’s supermultiplet each multiplet are not degenerate in mass. The sym- Let us disregard the strange particles for a mo- metry is man-made and cannot be exact. A mass ment. We have two symmetries, the space rotation formula can be written for each multiplet, and these
SU(2)Sp and the isospace rotation SU(2)Iso. These formulae are found to hold reasonably exactly. How- No. 3] Fermion-boson symmetry 161
ever, the most striking result is that the SU(6) theory ak;i;i¼ 1; 2; ;n. Here k denotes the kinetic mo- gives the ratio of the magnetic moments of the pro- mentum of the particle. The mass term of the hamil- ton and neutron to be tonian is 3 Xn X p ; 15 y ¼ ½ H0 ¼ M a ak;i: ½18 n 2 k;i i¼1 k assuming that each quark has a magnetic moment The operators proportional to its charge. The relation [15] holds X y almost exactly. Gij ¼ ak;iak;j; ½19 One additional comment is about the spin- k statistics relation. A quark is a spin 1/2 particle and commute with H0 and form a symmetry Lie algebra, must obey Fermi statistics. The baryon’s multiplet, Pwhich generates the unitary group U(n). The trace 56 in eq. [13] is totally symmetric under permuta- Gii is the total number of particles, and the re- tions of the quarks, violating the statistics. This is maining Gij form the special unitary algebra SU(n). evaded by introducing another internal variables called the color degrees of freedom. Each quark can Fermion-boson symmetry have three color states, red, blue and green. The ob- Starting from the rotational symmetry, SU(2)Sp, served baryons are totally antisymmetric in the color the concept of ‘‘symmetry’’ has been extended, variables so that they are totally symmetric in other partly arti cially but supported by experiments. variables. The resulting symmetry is SU(6). Here all important Now that the color degrees of freedom have been baryons are classi ed as a 56-dimensional multiplet, introduced, we are tempted to consider a large group and the low lying mesons are classi ed as a 35- SU(18)SIC, combining the color with other variables. dimensional representation. However, the color degrees of freedom are never ob- Can we go further? In other words, is it possible served in nature. to classify baryons and mesons in one multiplet? Since baryons are fermions and mesons are bosons, Symmetry and Lie algebra it is not possible to mix them by ordinary transfor- For continuous transformations, it is more con- mations. venient to consider in nitesimal transformations, or Let us assume that the fundamental particles in- the generators of the group. The generators often clude both fermions and bosons. Suppose that there have de nite physical meaning: For instance, for are n fundamental Fermi-like elements (denoted by SU(2) the generators are the three components of y Sp ai;ai ;i¼ 1; ;n) and m fundamental Bose-like the angular momentum. y elements (denoted by ai;ai ;i¼ n þ 1; ;nþ m). If the hamiltonian H is invariant under a contin- Their masses are not the same, but we write the fun- uous transformation group, its generators Gi com- damental hamiltonian as follows: mute with H. If two quantities Gi and Gj commute X nXþm y with H, the commutator of Gi and Gj also commutes H ¼ M ak;iak;i þ ¼H0 þ : ½20 with H. k i¼1
½Gi;H ¼0; ½Gj;H ¼0 ! It may appear that H0 is invariant under SU(n þ m) ½16 transformation, but this is not the case. Fermi-like G ¼½G ;G ; ½G ;H ¼0: k i j k element and bose-like element cannot be added; So, the set of Gi’s forms a Lie algebra with the com- they cannot be mixed. However, ‘‘generators’’ can mutation relations of the form. be de ned as eq. [19], and they commute with H0. X X ½Gi;Gj ¼ cij kGk: ½17 y y Gij ¼ ak;iak;j;Gij ¼ Gji; ½Gij;H0 ¼0: ½21 This set generates the continuous symmetry group. k
A set of operaters which commute with the hamilto- The operation of Gij makes H0 invariant and nian de nes a symmetry algebra. so the set of Gij de nes a symmetry of the system, Let us denote the creation and annihilation although the set does not generate a continuous y operators of the fundamental elements by ak;i and group. 162 H. MIYAZAWA [Vol. 86,
If one of the (i, j) is Fermi-like and the other is 6 F ¼ : ½25 Bose-like, that is, if 1 i n; n þ 1 j n þ m or 21 1 j n; n þ 1 i n þ m,theG is Fermi-like ij The adjoint representation of SU(6j21) or FF and commutators between them cannot be calcu- consists of, in terms of the SU(6) multiplets, lated but anti-commutators are, ð27; 27Þ¼ð6 þ 21; 6 þ 21Þ fGij;Gklg¼ jkGil ilGkj; Fermi Fermi case: 1 35 56 56 70 70 1 26 ½22 ¼ þ þ þ þ þ þ ½ Otherwise, the commutators can be written as þ 35 þ 405:
½Gij;Gkl ¼ jkGil ilGkj; other cases; ½23 The rst two terms, 1 and 35, are the well-estab- lished negative-parity mesons, that is, pseudoscalar or, in short, singlet, pseudoscalar octet, vector singlet and vector
½Gij;Gkl ¼ jkGil ilGkj: ½24 octet mesons. The next 56 is the baryon octet and decuplet. Relation [24] contains both commutators and The 56 is their antiparticles. 70 and 70 are multip- anticommutators so the set of operators Gij does not lets of baryons and antibaryons, but they are not form a Lie algebra. An algebra with anticommu- yet established. The remaining 1 þ 35 þ 405 are tation as multiplication is called a Jordan algebra. positive parity mesons. 1 þ 35 contains singlet and Our system is a mixture. Multiplication is de ned by octet of scalar mesons and singlet and octet of axial- the anticommutator between Fermi-like elements vector mesons. 405 contains tensor mesons which are and by the commutator for other combinations. not yet established. In this way, all of the well-estab- 4) Our ‘‘superalgebra’’ does not generates a con- lished hadrons are classi ed in a supermultiplet of tinuous group. Still, from the commutation-anticom- SU(6j21). mutation relations [24], we can construct represen- The Bose elements 21 added to the triplet of tations, determine multiplets and calculate matrix quarks to form SU(6j21) are actually two antiquark elements. The operators Gij and the relations [24] states, and cannot be regarded as fundamental. This are similar but, of course, not equal to the generators symmetry is not intrinsic but is arti cial. However, of the special unitary group of n þ m dimensions. We by using this FB symmetry, all known hadrons call this superalgebra SU(njm). (baryons and mesons) are classi ed as the same, and In this way fermion-boson symmetry can be relations between scattering amplitudes, for instance, achieved by introducing a superalgebra. Fermions can be derived. The symmetry is broken, of course. and bosons can be grouped in a multiplet. Each fer- Diquarks will be heavier than quarks. Then we have a mion and boson in a supermultiplet are the same relation among the masses of negative parity mesons, particle in di erent phases. positive parity mesons and baryons:
SU(6j21) Symmetry MðM ÞþMðMþÞ¼2MðBÞ: ½27 Having arrived at the idea of supersalgebra, we This relationship seems to be roughly satis ed ex- try to extend the S(6) symmetry to include baryons perimentally. and mesons in one supermultiplet. In addition to the One comment is that in this SU(6j21) scheme, elements of SU(6), eq. [12], a set of Bose particles there is no problem of statistics, and the color de- must be included in the fundamental elements. This grees of freedom are not necessary. All elements can set of bosons must be an SU(6) multiplet by them- have integral charges. selves. The simplest possibility is a singlet spin 0 par- ticle but this is insuf cient. The next possibility is 15 Is our space fermion-boson symmetric? or 21, and we see that the 21-dimensional multiplet Fermion-boson symmetry claims that a super- made from q q system is most convenient.5) The 21 multiplet contains both fermions and bosons. This is representation consists of an SU(3) triplet of scalar a convenient and useful way to classify and handle mesons and a sextet of axial-vector mesons. These, the existing hadrons. This is an arti cial symmetry together with the triplet of quarks (eq. [12]), form and cannot be regarded as intrinsic one given by god. the fundamental elements of SU(6j21). Our three dimensional space is spherically sym- No. 3] Fermion-boson symmetry 163 metric. Is it possible that the space is fermion-boson added elements are lorentz covariant quantities and symmetric, either intrinsically or arti cially? independent of internal symmetry. The total symme- The simplest superspace can be created from the try is the direct product of the internal symmetry fundamental elements, and the relativistic space symmetry. 0 1 q" Conclusion P ¼ @ q A; ½28 # Fermion-boson symmetry attempts to regard s 0 fermions and bosons as the same particles. Ordinary where the s0 stands for a spin 0 particle. The symme- symmetry is de ned as invariance under a transfor- try algebra SU(2j1), from eq. [28], is in parallel to the mation group. In the case of the fermion-boson sym- SU(3), eq. [6], which generates the internal symme- metry, there is no transformation group. The symme- try. If our space is FB symmetric as is given by try is de ned not by a group but by a superalgebra, SU(2j1), every particle must have its super partner. where the multiplication is de ned by anticommuta- For instance, electron must have boson partner of tor in some cases and by commutator in others. A similar mass. No such boson has been found. representation of the superalgebra (that is, a super- It is proposed that the space FB symmetry holds multiplet) contains both fermions and bosons. For at very short distance, that is, at very large energy example, all important hadrons can be classi ed in scale. All known particles are members of a super- one supermultiplet. multiplet of zero-mass states, although they are non- While the nonrelativistic fermion-boson symme- degenerate due to small symmetry breaking. Nothing try works well in hadron spectroscopy, the relati- is wrong with this scheme. However, relations similar vistic FB symmetry fails to do so. If the space is to eqs. [9], [15] or [27] are needed to support this pro- fermion-boson symmetric, all particles must have posal. their super partner of di erent statistics. No such Theoretically, the introduction of a spin 0 parti- partners are found. It is possible to expect that the cle as a fundamental element is not without doubt. space FB symmetry is seriously broken but would In the case of internal symmetry, the third element, hold at very short distances. We can split the system in eq. [5] or s in eq. [6], was necessary as a carrier into FB symmetric part and symmetry breaking of the conserving quantum number strangeness. part, and rst consider the symmetric part. To see if With regards to the symmetry of the space, there is this man-made symmetry works, further investiga- no such necessity. All spin states can be created tion is necessary. from spin 1/2 particles and its antiparticles and the spin 0 object is not necessary. The number of the References fundamental elements must be as small as possible. 1) Wigner, E. P. (1937) On the consequences of the We have been considering the space rotation symmetry of the nuclear hamiltonian on the spec- SU(2) . The Lorentz group SL(2,C) can be made troscopy of Nuclei. Phys. Rev. 51, 106{119 (1937). Sp 2) Gursey,€ F. and Radicati, L. A. (1964) Spin and uni- Fermion-boson symmetric by introducing boson ele- tary spin independence of strong interactions. ment in addition to the fundamental two-component Phys. Rev. Lett. 13, 173{175. spinor. Here the symmetry means the invariance of 3) Sakita, B. (1964) Electromagnetic properties of bary- the lagrangean of the system and not of the hamilto- ons in the supermultiplet scheme of elementary particles. Phys. Rev. Lett. 13, 643{646. nian. The symmetry can be de ned by a superalge- 4) Miyazawa, H. (1966) Baryon number changing cur- bra, that is, by commutation and anticommutation rents. Prog. Theor. Phys. 36, 1266{1276. relations. This scheme is called supersymmetry.6) 5) Miyazawa, H. (1968) Spinor currents and symme- It is to be noted that Wigner’s symmetry (eq. tries of baryons and mesons. Phys. Rev. 170, 1586{1590. [11]), and the subsequent extensions (SU(6) and 6) Wess, J, and Zumino, B. (1974) Supergauge transfor- SU(6j21)), cannot be made relativistic. The internal mation in four dimensions. Nucl. Phys. B70, variables transform as a representation of the special 39{50. unitary group while Lorentz spinor transforms as a representation of the special linear group, and they (Received Dec. 18, 2009; accepted Jan. 20, 2010) cannot be mixed in a larger group. The fermion- boson symmetry can be relativistic, in so far as the 164 H. MIYAZAWA [Vol. 86,
Pro le
Hironari Miyazawa was born in 1927, in Tokyo. He graduated from University of Tokyo in 1950 and obtained Dr. of Science Degree in 1953 under the supervision of Prof. Takahiko Yamanouchi. From 1953 to 1955 he was a research associate at Insti- tute for Nuclear Studies, University of Chicago, where he made research on theoreti- cal nuclear physics under Prof. Gregor Wentzel and Prof. Enrico Fermi. After return- ing to University of Tokyo he was promoted to Professor of Physics in 1968. In 1988 he moved to Kanagawa University and served there until 1998. During these periods he also served visiting professorship at University of Chicago and University of Min- nesota, and directorship at Meson Science Laboratory, University of Tokyo. Cur- rently he is a Professor Emeritus, University of Tokyo.