158 Proc. Jpn. Acad., Ser. B 86 (2010) [Vol. 86, Review Superalgebra and fermion-boson symmetry By Hironari MIYAZAWAÃ1,y (Communicated by Toshimitsu YAMAZAKI, M.J.A.) Abstract: Fermions and bosons are quite dierent 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 dierent 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 reection, 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 dierent states. nary Lie group theory is insuf cient. A new mathe- Lambda, one of the strange particles, diers 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 Ã1 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 denite angular momentum. A multip- ments show that in reactions of nucleons and pions let is a set of degenerate states of a denite 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 dierent 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 reection is a separate transformation being isospin 1. In fact, this reaction is found to group. It is known that the space reection 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 dier 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 dierence, 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 dierent 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 à 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 dier- 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 articial 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 dierent spins or isospins are clas- glet and an octet of vector mesons have been found. sied 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.