Symmetry and supersymmetry in nuclear physics

RIVISTA DEL NUOV0 CIMENT0 VOL. 19, N. 7 1996

Symmetry and supersymmetry in nuclear physics

F. IACHELLO Center for Theoretical Physics, Sloane Physics Laboratory Yale University - New Haven, CT 06520-8120, USA

(ricevuto il 25 Novembre 1996)

15 17 18 19 19 20 20 21 24 25

1. Symmetry 2. Spectrum-generating algebras and dynamic symmetries 3. Dynamic symmetries in nuclear physics--Early work 4. Spectrum-generating algebras in nuclear physics--Early work 5. Spectrum-generating algebras and dynamic symmetry in particle physics 6. Spectrum-generating algebras and dynamic symmetries in nuclear physics: U(6)

and its implications 7. Internal degrees of freedom 8. Generalizations of U(6)--The search for higher symmetries 9. Other schemes

10. Supersymmetry 11. Spectrum-generating superalgebras and dynamic supersymmetries 12. Supersymmetry in particle physics 13. Supersymmetry in nuclear physics 14. U(6/~) and its implications 15. Implications of nuclear supersymmetry to other fields of physics 16. Role of dynamic symmetries

1. - S y m m e t r y

Symmetry is a wide-reaching concept used today in a variety of ways. Weyl in his book ,Symmetry, described its introduction in the ancient world, especially by the Greeks, and its reintroduction during the Italian Renaissance [1]. Symmetry began playing an important role in physics (and chemistry) in the second part of the 19th century, when it was used to describe certain geometric properties (geometric symmetries). Its use expanded considerably after the introduction of quantum mechanics, and it is used today in many ways. In addition to geometric symmetries which play an important role in the description of molecules, crystals, etc., there are a) kinematic (space-time) symmetries; b) dynamic symmetries; c) gauge symmetries; d) permutational symmetries, . . . . Among all these different types, one that has played an important role in the development of physics in the last 30 years is dynamic symmetry. Although implicit in early work by Pauli, Fock, Bargmann, Heisenberg, and Wigner, the concept of dynamic symmetry emerged explicitly in the early 1960's and it was fully exploited only in the 1970's. A mathematical definitions of dynamical symmetry is given

2 F. IACHELLO

in the following section. Here it suffices to say that these are situations in which the Hamiltonian H splits but does not admix the representations of a group G.

Since 1974, the most useful and extensive applications of the concept of dynamic symmetry have been in nuclear physics (interacting boson model). In this paper, some of these applications will be reviewed. These applications will also illustrate the similarities between particles and nuclear physics, especially from the point of view of symmetry, a point of view advocated by Dyson in his monograph ,,Symmetry Groups in Nuclear and Particle Physics), [2].

2. - Spectrum-generating algebras and dynamic symmetries

In order to introduce the concept of dynamic symmetry, it is convenient to start from the more familiar concept of (kinematic) symmetry. Consider the case of a non-relativistic particle moving in a central potential, with Hamiltonian

p2 (1) H . . . . . + V(r).

2 m

The Hamiltonian operator H commutes with the three components of the angular momentum operator, L,

(2) [H, L~] = 0, i = 1, 2, 3.

The three components L~ form a Lie algebra ~ = SO(3), the real, orthogonal algebra in three dimensions. (In this paper, following common notation, capital letters will be used for both algebras and groups.) Since the Hamiltonian H commutes with all the elements of ~ , the associated rotation group G is a symmetry group of H. The knowledge that G = S 0 ( 3 ) is the invariance group of H helps somewhat, but it does not provide the solution to the spectrum of H (which obviously depends on V(r)). For this reason rotational invariance is called here a kinematic symmetry.

In contrast with this, consider the case in which the Hamiltonian operator describes a particle moving in a Coulomb potential

p2 Ze 2 (3) H -

2 m r

This problem has a larger symmetry than rotational invariance [3]. This larger symmetry is of dynamical origin, since it arises from the special nature of the interaction, V(r). The Hamiltonian H commutes both with the angular momentum, L, and with the three components of the Runge-Lenz vector, A,

1 Zr (4) A = : ( p x L - L x p ) - - - .

2 r

L and A form a Lie algebra, G =S0(4) (the real, orthogonal, algebra in four dimensions). The Hamiltonian, H, can also be written in terms of an operator, called the invariant Casimir operator, C2, of SO(4) as

A (5) H -

C 2 + 1

SYMMETRY AND SUPERSYMMETRY IN NUCLEAR PHYSICS 3

Its eigenvalues can be simply found, since the invariant operator is diagonal in the basis provided b y e . Some simple manipulations lead to the eigenvalue expression

A (6) E ( n , l , m l ) = - - -

n 2

This simple example illustrates the importance of dynamic symmetries, since, once the symmetry has been identified, it leads to explicit expressions for the energy spectrum (and all other observables). The role played by eq. (6) (the Bohr formula) in the development of quantum mechanics cannot be underestimated.

It is possible to give a more precise definition of dynamical symmetry. It is that situation in which:

i) The Hamiltonian operator H is constructed in terms of elements of an algebra, ~ , called a spectrum-generating algebra (SGA).

Let G~ e ~ be the elements of $'. The spectrum-generating algebra is the algebra upon which H is expanded

(7) 14 = f ( Ga) .

Quite often, this expansion is into polynomials in the Ga's

(8) 14 = Eo + E ~ . G a + Eua~G.zGB + . . . . a aft

(A notable exception to this statement is the case of Coulomb-like problems discussed above, where 1 / H and not H is expanded onto powers of the Ga's.)

ii) The Hamiltonian operator H contains only certain combinations of the elements, called invariant (or Casimir) operators, of ~ and its subalgebra chains ~ ' ~$'"~, . . . ,

(9) H = Eo + a C ( ~ ) + a ' C ( ~ ' ) + a " C ( ~ " ) + . . . .

In this case the eigenvalue problem for H can be solved in explicit analytic form by introducing a basis for the representations of W~W' ~W"~ . . . . In this basis the Casimir operators are diagonal and the eigenvalues of H are

(10) E = E0 + a(C(~)} + a ' (C(~ ')> + a"(C(~")} + . . . ,

where (C($')} denotes the expectation value of C(~) in the appropriate representation o f~ . The situation described by eq. (9) is called a dynamic symmetry (DS). It is a special case of i).

In addition to the Hamiltonian operator H whose eigenvalues give the energies, another important set of operators is the transition operators T whose matrix elements give the intensities of transitions from one state to the other

(11) <~f ITl~oi> = T~.

4 F. IACHELLO

The transition operators are also expanded into elements of g',

(12) T = to + ~ t , , G, + . . . .

When a dynamic symmetry occurs, the matrix elements of T can also be evaluated explicitly in terms of the quantum numbers that characterize the states (representations of if).

3. - Dynamic symmetries in nuclear physics--Early work

Dynamic symmetries were introduced implicitly at the very beginning of Nuclear Physics, throughout the work of Heisenberg (1932). Heisenberg considered protons and neutrons as different states of the same particle, and introduced a variable r, later called the third component of the isotopic spin. The assumption that the forces between all pairs of particles are equal is equivalent to a dynamic symmetry in which the energy levels depend only on

(13) E ( T , T~) = E o + A T ( T + 1).

The importance of Heisenberg's work is not only related to the assumption of the equality of nuclear forces, but to the introduction of a new effective degree of freedom, which does not live in ordinary three-dimensional space but in an abstract ,internal,, space. It is in these abstract spaces that most of the dynamic symmetries of physics are found. An example of isotopic spin symmetry in nuclear physics is shown in fig. 1. There are two important observations that one can make. First, fig. 1 has been prepared by subtracting the Coulomb energies. Coulomb energies break isotopic spin symmettT, i.e. eq. (13), but in an obvious way. Second, there remains a small (few percent) breaking of isotopic spin symmetry. This is a general feature of all dynamic symmetries. These symmetries are not exact (in contrast to kinematical symmetries). Nonetheless, isotopic spin symmetries have been of great importance in understanding nuclear properties.

Isospin symmetry was enlarged a few years later (1936) by Wigner [5] who combined isotopic spin with the other intrinsic variable of nuclei, spin, into a more

4.05 0+;1 3.56 0+;1

3.09 0+;1

0 1+;0

6He 6Li 6Be

I I I

-1 0 +1

Fig. 1. - Isotopic spin symmetry in nuclei.


general scheme, known as Wigner supermultiplet theory. Since the degrees of freedom are four, p 1', p $, n T, n $, Wigner supermultiplet theory is based on the Lie algebra SU(4). The physical basis for Wigner theory is the independence of nuclear forces from both spin and isospin. This assumption is valid only approximately for light nuclei (and not for heavy nuclei). Wigner supermultiplet theory is equivalent to a dynamic symmetry in which the energy levels depend only on

(14) E(P ,P ' ,P" ;S , T;Sz, Tz)=Eo+A(P2+4P+P'2+2P'+P"2),

where P, P ' , P" label the representations of SU(4). This formula was tested by Franzini and Radicati [6], and its breaking for ground state energies analyzed. The importance of Wigner theory is not so much the suggestion that SU(4) is a good symmetry of nuclei (it is in fact badly broken) but rather the attempt to use dynamic symmetries to classify a large number of experimental data, thus providing a simple understanding of those data. It is also the first attempt to introduce larger ,,internal, symmetries in physics (larger than SU(2) ~ SO(3)). However, by introducing a larger symmetry, one introduces a new technical problem, namely that of developing a new mathematical tool, the explicit construction of Lie algebras and Lie groups, a subject introduced by Lie in the 1880's and developed by Cartan and others in the early 1900's but only from the abstract point of view. The problems open by Wigner were:

i) The explicit construction of the representations of $" and its subalgebras $ " , ~" , . . . .

ii) The solution of the branching problem, i.e. what representations of ~ 'c ff are contained in a given representation of ~ .

iii) The explicit evaluation of the matrix elements of the elements (generators) of in a given representation of s

iv) The explicit construction of the Casimir operators C($') and their eigenvalues.

For these larger symmetries it was no longer possible to use the statement, attributed to Bohr, and reported by Condon and Shortley, that ,,all results in quantum mechanics can be obtained by elementary methods and do not require the use of group theory,.

Another important use of implicit dynamic symmetry came in nuclear physics with the development of the Elliott model (1958) [7]. This model deals with space degrees of freedom, x,p, rather than with ,,intrinsic, degrees of freedom. It originates from earlier work by Racah, Jahn and Flowers. It exploits the dynamic symmetry of the three-dimensional isotropic harmonic oscillator with Hamiltonian

p2 (15) H = + kr 2 .

2m

As in the case of the Coulomb problem, the harmonic oscillator has a larger symmetry than just rotational invariance, characterized by the algebra of U(3), the unitary algebra in three dimensions. (In general, the degeneracy algebra (dynamic symmetry) of the oscillator in n dimensions is U(n), while the degeneracy algebra of the Coulomb problem in n dimensions is O(n + 1).) The nine elements of the algebra of U(3) in the

6 F. IACHELLO

Elliott model can be written as

H = r 2 + b e p 2 ,

(16) L , = (r • p ) ....

" " ( 4~'T ) 1/2 r, 114 2y(2)(~ -7 Q , = - g - { r e Y ( e ' ( O 9 , . ) + v p . , , . v , , , 9 , , ) } / b - .

Here L, are the three components of the angular-momentum operator (p = 0, _+ 1), while Q,, are the five components of the quadrupole operator (/~ = 0, +- 1, +_ 2).

The Elliott model occupies a very important role in the development of dynamic symmetry for several reasons: i) it deals with the space par t of the wave function of nuclei; ii) it illustrates a fur ther use of dynamic symmetries, in which dynamic symmetries can be used as a basis where the breaking of the symmetry can be analyzed. These aspects can be seen by considering the simple Hamiltonian

(17) H = Eo - g • Q , , ( i ) . Q , , ( i ' ) , tt

that is the case of particles interacting with a quadrupole-quadrupole interaction. This (semirealistic) nuclear Hamiltonian has a dynamic symmetry, since it can be rewrit ten in terms of the Casimir operators of the a l g e b r a s ~ > ~ ' , U(3)>O(3) , as

(18) H = Eo - g C 2 ( U ( 3 ) ) + g C e ( O ( 3 ) ) �9

E (MeV)

10

24 12Mg12

. , 8 +

6 +

~ 4 + 3 +

4 + ~ 2 +

,. 2 +

0 +

Exp

4 +

m 8 +

m 6 +

Th

4 + .4 + 4 +

3 +

2 + ,.2 +

0 +

Fig. 2. - Comparison between the experimental spectrum of 24Mg and the Elliott model ((8,4) representation of SU(3) ) .

SYMMETRY AND SUPERSYMMETRY IN NUCLEAR FHYSICS 7

It is thus precisely of the type (9). The eigenvalues of (18) in the basis SU(3)~0(3) , characterized by the quantum numbers ()[, it), L are given by

(19) E()~,/~; L; ML) = E0 - g(;~2 +/~2 + )~/~ + 3)~ + 3/~) + gL(L + 1 ).

An example of Elliott symmetry is given in fig. 2. The comparison between theory and experiment is not very good. However, Elliott showed that SU(3) can be used as a basis in which more realistic Hamiltonians can be diagonalized.

4. - Spectrum-generating algebras in nuclear physics. Early work

All examples presented in the previous sections, describe dynamic symmetries, sometimes called degeneracy algebras. In a more ambitious scheme, one may wish to describe not only a degenerate multiplet of states but the entire spectrum of H. In order to do so, one needs to introduce a spectrum-generating algebra of which the dynamic symmetry is a special case. This concept was also introduced implicitly in early work in nuclear physics by Goshen and Lipkin [8], who studied the spectrum- generating algebra of the harmonic oscillator. The Elliott model describes an oscillator shell, for example the (s, d) shell. If one wants to describe simultaneously all oscillator shells, one needs to enlarge the algebra of U(3). Goshen and Lipkin suggested to use Sp(6, R) as SGA of the oscillator. (This algebra describes the oscillator within two representations. An algebra which describes the oscillator within only one representation is U(3, 1).)

5 . - Spectrum-generating algebras and dynamic symmetry in particle physics

The concept of dynamic symmetries and spectrum generating algebras, implicit in the early work of Heisenberg, Wigner and Elliott were made very explicit in the work of Gell-Mann [9], Ne'eman [10] and others in the early 1960's. Gell-Mann and Ne'eman introduced the group SU(3) to describe the low-lying hadron spectrum. This group SU(3) acts on an (,internal- space and it is a generalization of Heisenberg's isospin. Gell-Mann-Ne'eman dynamic symmetry corresponds to the chain

(20) SU(3) ~SUr(2) | Uy(1) ~SOT(2) | Uy(1).

By writing the mass operator as

(21) M = Mo + aC1 (Uy(1)) + b

one obtains the Gell-Mann-Okubo mass

1 C~(Uy(1))], C2 (SUT( 2 ) ) - -4

formula [11]

(22) M(T, T z ; Y ) = M o + a Y + b [ T ( T + I ) - I Y 2 ] .

This mass formula has played a crucial role in the development of particle physics in the 1960's and 1970's. Figure 3 shows a comparison between the masses of the baryon decuplet and those obtained by using (22). Gell-Mann-Ne'eman SU(3) was enlarged to SU(6) by Gfirsey and Radicati [12] in a way similar to that used by Wigner to enlarge Heisenberg SU(2) to SU(4).

8 F. IACHELLO

E 1.8

(GeV) I. 6

1.4

1.2

1.0

Exp

Z*

A

Th

Z*

A

Fig. 3. - Comparison between the experimental masses of the baryons decuplet and the Gell-Mann-Okubo mass formula.

Gell-Mann-Ne'eman SU(3) and Gtirsey-Radicati SU(6) act on an ,dnternal, space. In subsequent years, Dothan, Gell-Mann and Ne'eman [13] and Barut and Biihm [14] attempted to extend the use of spectrum-generating algebras and dynamic symmetries in particle physics to cover space degrees of freedom and in so doing spelled out those concepts. In the first of those two papers it is said that in ,,an approximate symmetry theory ... it is sufficient that a set of operators be found that obey the equal-time commutation relations of some algebra .. . . . and that the stationary and quasi-stationary quantum-mechanical states have a tendency to fall approximately into irreducible representations of the algebra as a result of dynamics-. The algebras suggested were of the Elliott-Lipkin type.

6. - Spectrum-generating algebras and dynamic symmetries in nuclear physics: U(6) and its implications

Beginning in 1974, the most extensive use of dynamic symmetries and spectrum- generating algebras has been done in Nuclear Physics [15]. This use brought in a Renaissance in Nuclear Structure Physics. The main new aspect is that dynamic symmetries are no longer an observation on a previously known model or theory, but rather the starting point of new models on theories. The role of symmetry becomes thus a dominant role by reversing the logic of the approach, as expressed by Casimir in a letter to the author of the present article.

This new role of dynamic symmetry (and spectrum-generating algebras) became evident with the introduction in 1974 [16] of the Interacting-Boson Model (IBM) of Nuclear Structure Physics [17] which was, from the very beginning, formulated in the language of SGA and DS. The IBM describes collective low-lying states of nuclei in terms of correlated pairs of nucleons. In its original formulation no distinction was made between proton and neutron pairs and only pairs with angular momentum J = 0 and J = 2 were retained. The pairs were subsequently treated as bosons. The J = 0 bosons are called s, while the J = 2 bosons are called d. This leads to a description of low-lying collective states of even-even nuclei in terms of a system of interacting bosons. (The bosons here are Cooper like pairs as in the electron gas.) The IBM can be viewed as well as an approximation (truncation) of the nuclear-shell model. A large

SYMMETRY AND S U P E R S Y M M E T R Y I N N U C L E A R P H Y S I C S 9

literature exists on this subject, but it will not be reported here. Only the aspect related to dynamic symmetry will be briefly reviewed.

The interacting boson model can be directly formulated in terms of a spectrum generating algebra, by introducing boson creation and annihilation operators, s*, d~(/~=0, _+1, +2); s, d , ( /~=0, _+1, _2), generically denoted by b*,(a= 1, ..., 6), b a ( a = l , ..., 6). The Hamiltonian, H, and transition operators, T, of the model have the form

(23) aftra

T = to + ~ ta/~ b2 b~ + . . . .

But the bilinear products

(24) Ga~ = b~bz, a, fl = 1, ..., 6,

are the elements (generators) of a Lie algebra: the Lie algebra of U(6), the unitary algebra in six dimensions. They satisfy the commutation relations,

(25) [G@, Gra ] = Gaa 6 Zr - Grz 5 aa �9

The Hamiltonian and transition operators can be rewritten as

(26) H = E o + E e ~,~ Ga/3 + • uazra Ga~ Gfa + . . . .

a/~ a,~ r a

T = to + ~ ta~ Gap + . . . . af

They are thus of the form (8) and (12): U(6) is the spectrum generating algebra of the interacting boson model. Moreover, since there is in the nucleus a fixed number of particles and hence of pairs, treated as bosons, one must consider the totally symmetric representations of U(6) characterized by a fLxed number of bosons, N (taken usually as the number of active valence pairs), with Young tableau

N-times A

(27) [N] - O[3... [~.

The structure of the Hamiltonian of the interacting boson model is amenable to a simple study of its dynamic symmetries. The generic problem that one wishes to solve is then the following: Given a model (or theory) with SGA ~ , what are its possible dynamic symmetries? This problem, which was first solved in the context of the IBM, is now of interest in many other branches of physics. In most applications, one also imposes additional constraints. In the case of the IBM, the additional constraint is that, in view of the rotational invariance of H in three-dimensional space, any chain of subalgebras that starts from ~ must contain the physical angular momentum algebra, SO(3). Here one must find all possible paths that starting from U(6) end with SO(3).

10 F. IACHELLO

There are three (and only three) possible paths (subalgebra chains)[18]

7 U(5) ~ SO(5) ~ S0(3) ~ SO(2) (I),

(28) U(6) ~ SU(3) ~ S0(3) ~ SO(2) (II),

x~ S0(6) ~ SO(5) ~ S0(3) ~ S0(2) (III).

Corresponding to each of the three algebra chains, one can write energy formulas, similar to those discussed in the previous sections (Bohr-like formulas). This is done by v~Titing the Hamiltonian in terms of Casimir operators of a chain. For example, for chain I

(29) H = Eo + ~'C~ (U(5)) + aC._, (U(5)) + fiCz (0(5)) + yC~ (O(3)),

with eigenvalues

(30) E ( N , ~t, v, ~j , L , ML) = Eo + ~'~ + a~cl(~t + 4) + fiv(v + 3) + 7L(L + 1).

Here, C1 (U(5)) denotes the linear Casimir operator of U(5), C2(U(5)) the quadratic Casimir operator of U(5), etc. In view of rotational invariance, the S 0 ( 3 ) symmetry is not broken (i.e. there is no term containing the Casimir operator of 0(2)). If the Hamiltonian is quadratic in the elements of the algebra, eq. (29) is the most general expression containing Casimir operators of chain (I) that can be u~'itten. This is of the form (9). The eigenvalues of H are ~Titten explicitly in eq. (30) in terms of the number of pairs, N, the angular momentum, L, and its component, ML, and of additional quantum numbers which label the representations of ~ G ' ~..., including some missing labels not associated to Casimir operators and thus not appearing in the energy expression.

In a similar way, one can derive energy formulas for the other chains. The complete result is

E ( N , ~j, c, ~j , L , M L) =

= E o + ~ . n ~ l + a ~ ( t ( n ~ t + 4 ) + f i v ( v + 3 ) + T L ( L + l ) (I), (31)

E ( N , ~, , , K, L, M~) = Eo + K(2"- + t~2 + 2u + 32 + 3u) + K' L (L + l ) (II),

E ( N , a , v , , j , L , M L ) - E o + A o ( o + 4 ) + B v ( r + 3 ) + C L ( L + I ) (III) .

The three formulas (31), similar to the Gell-Mann-Okubo mass formulas of particle physics, have played a very important role in the development of Nuclear Physics in the last 20 years. Its main use has been in the analysis of experimental data. In view of the fact that nuclear spectra depend on how many particles (proton and neutrons) are present, it has been possible to investigate whether or not nuclei display dynamic symmetries of the type described in (31). Since their introduction in the middle 1970's, several examples of each of these s~Tnmetries have been found [19]. Figures 4, 5 and 6 show three examples, one for each chain.

The relevance of the dynamic symmetries of the Interacting Boson Model to Nuclear Physics and Physics in general is the following: i) The symmetries describe complex systems. Nonetheless, simplicity is found in these systems despite their complication; ii) The dynamic symmetries of eq. (31) are realistic, in the sense that they describe the experimental data within few percent. This is due to the new use of symmetry in which the Hamiltonian H does not commute with all the generators of the


E (MeV)

3-

2-

I--

O-

I10^ , 48tJa62

(rid,O)

4+ 3

(rid,l)

0+.__

Exp.

(ndi2'O)

2+._

4+--- 2 + 0+.~

2+__

O+__

Th.

(ndJ.O) (rid,l) (rid-2,0).

4 +-- 2+__ 0+---

2+.._

o~.... U(5)

Fig. 4. - An example of U(5) dynamic symmetry in nuclei 11~

group G

(32) [H, G,~] 4 0 ,

but ra ther is writ ten as in eq. (9) leading to a hierarchy of splittings and a pat tern in which the energy levels are not degenerate. In other words, all the states belonging to

E (MeV)

5-

2-

0

156 A . 6 4 ~ 0 9 2

(24,0) (20,2)

I0 +--- 4+.__2+_ _ / ~ 8+-.-- 0 - ' -

6+..... 4t_.._ 2+.._ 0-r

Exp.

(16,4) (18,0)

Th. ( 2 4 , 0 )

i0+.__

8 ~ 2 ~ 3 ~ 0 r 2 6+.__

4+____ 2+____ 0 ~---

(20,2) (16,4) (18,0)

4 +-E- 4~--- 4 +-- +._. 3~ 02.--2

4+___ ~

su(3)

Fig. 5. - An example of SU(3) dynamic symmetry in nuclei 15~Gd.

12 F. IACHELLO

E (MeV)

2-

K-

O-

196 78Pt118

Exp.

(6,0) (6,t) (4,0) (2,0)

6 2+__ O+--- e+._

3+._ 4+._.

2~._

2+_._

0+._.

Th.

(6,0) (6,1)

0+._ 4+_..

2+._ 2+_._ o+_...

(4,0)

2*--- 0+_..

0(6 )

(2,0)

I 0+__

Fig. 6. - An example of S0(6) dynamic symmetry in nuclei l'~;Pt.

the representation N (a given nucleus) are not degenerate. They are split (dynamically) by the interactions, but this splitting is diagonal.

The diagonal nature of the splitting and thus the occurrence of dynamic symmetries can be tested not only by a comparison of energy levels, as in figs. 4, 5, 6, but also, and most importantly, by a comparison of transition intensities. A consequence of the occurrence of dynamic symmetries is that i) selection rules exist for certain transitions and ii) those transitions that are allowed depend only on certain group theoretic quantities (the Clebsch-Gordan coefficients or their generalization) and are parameter free. This parameter-free nature of the transition is what makes the dynamic symmetry particularly appealing in analyzing experimental data.

As an example of these symmetry rules, and in order to illustrate another important point when making use of symmetry considerations, consider the case of electric quadrupole transitions (E2 transitions). For problems with rotational invariance, it is convenient to construct operators that have definite transformation properties under rotations (and inversions). This is done by introducing tensor operators with respect to rotations, in the sense of Racah. The Hamiltonian operator is a scalar under rotations, while the electromagnetic transition operators transform as scalars, vectors, tensors, etc., or, in general, as tensors of rank k. The construction of tensor operators is greatly aided by the use of spectrum-generating algebras. Instead of introducing the generators of the Lie algebra in the uncoupled form (24), it is possible to introduce from the very beginning tensor operators in the Racah form. Denoting by b/t, ., a tensor creation operator with angular momentum 1 and component /~, and bl, z the corresponding annihilation operator, /J~., = ( - ~/- t' ~ ~, ~,, the operators

(33) G~(~)(/, l ' ) = [bl * • bl]~ ~) ,

generate the same Lie algebra (here U(6)) in the Racah form. The introduction of tensor operators has the following advantages: i) It allows one to identify immediately


the operators in the algebra with physical operators; ii) it reduces the number of possible terms in H and T. For example, the electric quadrupole operator (E2), when written in terms of tensor operators, has the generic form

(34) T(~ E2) = a2(dr • ~ + s r • cl)(~2) + f i2 (d t • d)z~ (2) .

One can see immediately from this expression that all electromagnetic E2 transitions depend only on two quantities a2 and fi2, no matter whether or not there is a symmetry, as long as U(6) is the SGA of the problem.

If there is a dynamic symmetry all matrix elements can be evaluated explicitly, and the analysis of experimental data simplifies considerably. As an example, consider the case of symmetry I, U(5). The intensities of transitions, usually quoted as

1 (35) B(E2; J i -* Jf) - - - I(Ji II T(E2) iijf>e,

2 J i + l

take the following form:

(36)

B(E2;

B ( E 2 ;

B (E2;

B ( E 2 ;

B (E2;

2r162 = a ~ N ,

4r - * 2 r ) = a ~ 2 ( N - 1),

22 -*2r = a ~ 2 ( N - 1),

02 ---~ 2r ) = a ~ 2 ( N - 1),

22 --~ 0r = 0 ,

and similar expressions for other states. One can see that in the presence of the symmetry, the transition 2e + --~0r is forbidden, while for allowed transitions one has

B(E2;4~-- )2r = 2 ( N - l ) (37)

B(E2; 2r - * Of ) - - ~

(a parameter-free expression). Dynamic symmetries thus provide a wealth of relations that have been repeatedly

tested in the last 20 years and have confirmed the ,,goodness, of dynamic symmetries in nuclei within a few percent.

Finally, another important aspect of the dynamic symmetries of the interacting boson model, or, in general, of models (or theories) where the interactions depend on some external variable (here the number of pairs, N) is that they provide a framework for the classification of all situations that can be encountered in practice. Consider, again, a Hamiltonian of the type (26) where the coefficients e,~, u~7~ depend on some external parameter N

(38) H = E0 + ~ e,~ (N) G,~ + ~ u,z~ (N) G,7 Gf~ + . . . .

For some values of N, the coefficients are such that a dynamic symmetry arises. In general, as N varies, one may cover a region in the parameter space which spans

14 F. IACHELLO

SYMMETRY AND SUPERSYMMETRY IN NUCLEAR PHYSICS

Fig. 7. - A classification of nuclear species in the region 50 <~ N ~< 82, 82 <~ N ~< 126. The color code is indicated in the triangle on the lower right-hand corner. From ref. [18].

situations which are intermediate between two symmetries. In the case of the interacting boson model, with three possible dynamic symmetries, it has become customary to describe the situation by drawing a triangle called Casten's triangle, where the symmetries are located at the vertices and the intermediate situations inside the triangle or along the sides. To make things clear, dimensionless parameters are introduced and a combination of parameters is used. A parameter set then determines a point in the triangle. This method has been used to provide a global classification of nuclear spectra. A portion of this global classification is shown in fig. 7 [20]. In this figure each nuclear species is color coded according to the dynamic symmetry to which it belongs or its departure. A study of the available spectra has shown that approximately 20% of all nuclear species are well described by a dynamic symmetry, while the remaining 80 % are in intermediate situations (transitional nuclei). For these situations, the use of the full spectrum generating algebra is necessary rather than the dynamic symmetries. Nonetheless, the dynamic symmetries are still useful in these cases, since they provide a basis within which the Hamiltonian can be diagonalized.


7. - Internal degrees of freedom

In the description of the previous section, no distinction is made between proton and neutron pairs. An improved description[21] can be obtained by explicitly distinguishing proton (pp) and neutron (nn) pairs and thus introducing proton and neutron bosons, s~, d~ and s,, d~. The proton-neutron degree of freedom is here an ,internal, degree of freedom. This degree of freedom can be formally described by considering proton and neutron bosons as members of a doublet

j, 1> j l 1) (39) ' I z } = ~-, + ~ , Iv}= 2 ' 2 '

called F-spin (to distinguish it from usual isospin). The algebraic structure of the corresponding problem (called interacting boson model-2) is the direct sum of the algebraic structure of protons and neutrons, ~ | ~ , , with Hamiltonian

(4o) H=H~+H,+V~,,

where

(41)

f ~T r , _ x-~ (Jr),w(n)_ X~ (n) ~(3)~(:r) 1-1:r = 12~o: r t 2..~Eafl ( . ra f t t L U a f i 7 6 i r a Y Irfi6 ,

@ a#7~ rr ~ , ~ ( v ) , ~ ( v ) _ x~ (v) ,~(v),~(v)

l ' l v = ~ o v t ~ 8 aft I-iaft + .~' Uafly~ I-~a7 0-fl~ , aft afiy6

~-' ~ (3) ~ (v) V~v : 2.J W a f i r ~ l r a # t r y 6 �9

ctp~o

The study of the dynamic symmetries of the interacting-boson model-2 is more complex than that of the interacting-boson model-1. It can be done either by treating explicitly proton and neutron bosons, as in (41), with algebraic structure U~(6)O U~(6), or by introducing F-spin and rewriting the Hamiltonian in terms of it with algebraic structure U(6) |

The interacting boson model-2 has been the subject of many investigations. From the point of view of symmetry that is discussed here, there are two new aspects. The first is the extent to which F-spin symmetry is a good symmetry. This aspect has been only partially investigated. The ground-state configuration of many nuclei is that with maximum F-spin. Admixtures of states with F < Fmax in the ground state are typically of the order 10% or less. These admixtures increase with excitation energy.

The second aspect that has received most of the attention is the occurrence of additional states. This aspect can be simply seen by considering the case of one proton, N~ = 1, and one neutron, N~ = 1. The tensor product

(42) [] | [] = [][] �9 [][] []

yields not only totally symmetric states (these are the same as in sect. 5) but also states with mixed symmetry. The search for these states has been an important component of nuclear structure physics. Unambiguous results were finally obtained in 1984 [22]. A simple illustration of these states is given in fig. 8 where the situation for one of the

16 F. IACHELLO

>

(6,0)

1 -

6 + ~

4 + ~

2 + ~ 0 - 0 + ~

[3,o1

(2,2) (0,0)

4 + ~ 3 +

2 + _ 2 + 0 + ~

[2,1]

(4,1) (2,2) (1,1)

I / \ ' 2 + 4 + 1 + ~

5+...- 3 + ~ 2 + 2 + ~ o+

2+....- 1 I

SUrt+v(3)

Fig. 8. - A typical spectrum with U, + ;. (6) ~ SU~ + ,, (3) symmetry and N: = 2, N, = 1.

possible dynamic symmetries of IBM-2 is illustrated, that corresponding to the chain

(43) U~(6) �9 U~(6) ~ U~+ ,,(6) ~ SU(3)~ + ,. ~ S0(3)~ § ,, ~S0(2)~ + ~.

In the combination of the two U(6)'s, there appear new states, here in particular :~ + v states with angular momentum and parity 1+, which correspond to motion of protons relative to neutrons. This aspect, namely the study of dynamic symmetries of coupled systems, is that of major interest from a general point of view. It shows that even where one goes up in complexity from single systems (however complex that may be, here U(6) ) to coupled systems (here U(6) | U(6)) simplicity remains. This simplicity may however not be invariance under the ~dntrinsic, transformations (here F-spin) but ra ther a dynamic symmetry in the , intrinsic, variables. In other words, intrinsic degrees of freedom must be t reated in the same footing as space degrees of freedom.

A fur ther expansion of the internal degrees of freedom of pairs was introduced in 1981 by considering proton-neutron pairs with T = 1, s(~, d~[23]. This expansion is needed if one wants to discuss nuclei where protons and neutrons occupy the same


valence shells (light nuclei). The pairs (bosons) form now a triplet under isospin

(44) { I x ) = I1, +1) ,

16} = I1, 0},

Iv) = I1, - 1 ) .

In this case, an explicit construction of the Hamiltonian and other operators in terms of ~, 5, v bosons is not very useful and it is convenient to formulate the problem directly in terms of U(6)| This model Hamiltonian (Interacting Boson Model-3) in this formulation can be written as [24]

(45) H= Sd~d-- Y, fc(t) Q(t).Q(t) + aT(T+ 1), t - 0 , 1 , 2

where the scalar product is with respect to both angular momentum and isospin. The Interacting Boson Model-3 is an unescapable consequence of isospin invariance of the nuclear forces. Although the dynamic symmetries of IBM-3 have been theoretically studied, their experimental study has been hindered by the lack of data in heavy nuclei with N ~ Z where IBM-3 applies. In recent years some of these data have become available. At the present_time, the study of the dynamic symmetries of IBM-3 is one of the most active areas of research for symmetries in nuclei. IBM-3 incorporates the dynamic symmetries of the Interacting Boson Model with Heisenberg's isospin symmetry.

Finally, another enlargement of the internal degrees of freedom of pairs was introduced in 1982125] by considering not only nucleon pairs with S = 0 , T = I (z, ~, v) but also nucleon pairs with S = 1, T = 0 (pn T = 0 pairs 0). The introduction of (pn) T = 0 pairs is not dictated by isospin invariance but is a dynamic effect, which may or may not be present in heavy nuclei. The algebraic structure of the model Hamiltonian is U(6) | UST(6) since there are now six degrees of freedom both in space and in spin-isospin space. The dynamic symmetry of this model (interacting boson model-4) have been investigated theoretically, but, as in the case of IBM-3, not experimentally. The new data, which hopefully will become available in the future, will shed some light on the question whether or not Usv(6) is a good symmetry of heavy nuclei. IBM-4 is the ultimate theory in the attempt to classify low-lying states of nuclei with an even number of particles, since it encloses both the dynamic symmetries of the interacting boson model and Wigner supermultiplet theory which is contained as a subalgebra of UST (6).

8. - Generalizations of U(6)--The search for higher symmetries

U(6) and its dynamic symmetries are a description of low-lying state of nuclei in terms of monopole and quadrupole degrees of freedom. At higher excitation energies, other degrees of freedom became active. A search for higher dynamic symmetries has occupied a fraction of the nuclear physics community in recent years. Among the generalizations of U(6) there are two which are worth mentioning:

i) A generalization in which hexadecapole J -- 4 degrees of freedom are included, g-bosons. This leads to a model with U(15) structure (sdg IBM)[26].

18 F. IACHELLO

ii) A generalization in which octupole, J = 3, and dipole, J = 1, degrees of freedom are included, f and p bosons. This leads to a model with U(16) structure (sdpf IBM) [27].

The dynamic symmetries of these models have been theoretically investigated although experimentally they appear to be somewhat broken. The study of these models presents a theoretical challenge since it is technically quite difficult (unitary algebras of large dimension).

9. - O t h e r s c h e m e s

In addition to interacting boson models, other schemes have been extensively investigated in the last few years. In particular, two fermionic schemes have been studied and developed. The first scheme is a continuation of the Elliott model. SU(3) is the degeneracy algebra of the three-dimensional harmonic oscillator. Introducing boson operators that create or destroy a quantum of vibration, ai t, at( = 1, 2, 3), one can ~Tite the elements of the Lie algebra of U(3) as

(46) Gij = ai; ai, i , j = 1, 2, 3.

In order to construct the spectrum-generating algebra of the harmonic oscillator, one adds to the operators in (46) the bilinear products of creation and of annihilation operators

(47) { Tij = ai * aj * + aj ai ~ ,

Mij = ai aj + aj a t , i>~j.

The 21 operators of (46) and (47) form the Lie algebra of Sp(6, R). This Lie algebra can be used to calculate properties of nuclear spectra [28]. The algebra Sp(6, R) has an important advantage in that it includes not only the valence shell but all excitations with 2 h(o, 4h~o, .... An important result is that no renormalization of effective charges is needed (contrary to the case of models based on valence shells where a renormalization is always needed). The study of symplectic models presents a technical challenge since the algebra Sp(6, R) is non-compact and its unitary representations are infinite-dimensional. In recent years major progress has been made in the representation theory of these algebras and the problem can be said to be to a large extent solved.

From the point of view of dynamic symmetry, the chain of particular interest here is

(48) Sp(6, R) ~SU(3) ~SO(3) ~SO(2),

which contains the Elliott model. As already mentioned in sect. 3, this symmetry is not realistic for heavy nuclei. Sp(6, R) can be viewed instead as a spectrum-generating algebra to be used for calculation of nuclear spectra. Attempts to merge the U(6) scheme with Sp(6, R) have been made and are still being studied.

Another possible scheme starts from the most general spectrum generating algebra of fermions in a shell with degeneracy ~ = E(2ji + 1), U(tg). The algebra here is the


bilinear products of fermion creation and annihilation operators

(49) G~j = a~ aj, i, j = 1, . . . , t ) .

Within this algebra one now searches for all possible dynamic symmetries (fermion dynamic symmetries) [29], U(~9) ~ ~ ~ . . . . There are many such possibilities, since t9 is usually large. Among these, there are some that resemble the symmetries of the bosonic space, for example S0(6). Although the dynamic symmetries of U(tg) are broken, nonetheless, the fermion dynamic symmetry approach is useful in providing bases in which a realistic fermionic Hamiltonian can be diagonalized. These bases can be optimized by an appropriate choice of the chain U(t?) ~ ~ ~ .... Work in this direction is still in progress.

10. - Supersymmetry

The concept of symmetry, which, as discussed in the previous section, was enlarged from simple rotational invariance of non-relativistic quantum mechanics to ,,internal,, symmetry, such as Heisenberg's isospin, SUT(2), Wigner spin-isospin, SUsT(4), Gell-Mann-Ne'eman flavor symmetry, SUF(3), and to rather complex ,,space,, symmetries, such as the symmetries of the interacting boson model, U(6), and complex combinations of ,,space- and ,internal, symmetries, such as U(6) | was further enlarged in the early 70's to encompass more complex situations. While symmetry in the ordinary sense applies separately to a system either of bosons or of fermions (such are the symmetries discussed in the previous chapters), this new type of symmetry, called supersymmetry, applies to situations in which one deals simultaneously with a system of bosons and fermions. For this type of symmetry, one can again think of a) kinematic (space-time) supersymmetries; b) dynamic supersymmetries, . . . . Again here, dynamic supersymmetries have played an important role in the development of physics in the last 20 years. In the following sections, a brief account of some of these developments will be given.

11. - Spectrum-generat ing superalgebras and dynamic supersymmetries

The definition of dynamical supersymmetry traces the steps of sect. 2 which led to the definition of a dynamic symmetry in the ordinary sense. A dynamical supersymmetry is that situation in which:

i) The Hamiltonian operator H is constructed in terms of elements of a superalgebra, ~* . The algebra ~ * is called a spectrum-generating superalgebra (SGSA). Let G,, Fie~J'* be the bosonic, G,, and fermionic, F~, elements of &o.. The spectrum-generating superalgebra is the algebra upon which H is expanded

(50) H = f(Ga, Fi) .

ii) The Hamiltonian operator H contains only certain combinations of the elements, called invariant (or Casimir) operators of ~ * and its subalgebra chains ~ * ~G'* ~ " * ~ . . . .

(51) H = E o + aC(b ~ + a ' C ( ~ ' * ) + a" C(~ r '* ) + . . . .

20 F. IACHELLO

In this case the eigenvalue problem for H can be solved in explicit form, as in sect. 2. This situation is called dynamical supersymmetry (DSS).

12. - Supersymmetry in particle physics

Supersymmetry was implicitly introduced in particle physics by Miyazawa (1966) [30] who tried to put hadrons with spin 0, 1/2, 1, 3 /2 all in the same multiplet. Later (1971), within the context of dual resonance models Ramond [31], Neveu and Schwarz [32] introduced it more explicitly, and finally a class of supersymmetric relativisitic field theory were introduced by Wess and Zumino (1974)[33]. The latter supersymmetry is of the (kinematic) space-time type, similar to rotational invariance in non-relativistic quantum mechanics or Poincar~ invariance of relativistic quantum field theory. The Miyazawa or Ramond supersymmetries are instead of the dynamical type. Supersymmetry, especially of the Wess-Zumino type, is one of the most active areas of research in particle physics at the present time.

13. - Supersymmetry in nuclear physics

Beginning in 1981 [34], the most extensive use of dynamical supersymmetries and spectrum-generating superalgebras has been done in nuclear physics. Supersym- metries in nuclear physics are based on the interacting boson-fermion model (IBFM) [35]. This is a model in which the low-lying states of nuclei are treated in terms of correlated pairs of nucleons with angular momentum J = 0 and J = 2 treated as bosons (s, d bosons) and additional fermions (the unpaired particles) with angular momentum j (or in general with several possible values of the angular momentum j). Introducing boson and fermion operators, generically denoted by b~i, b , (a = 1, ..., 6), ai ~, a~(i = 1, ..., Y2), the Hamiltonian and transition operators of the model have the form

(52) f f i

H (B)= E o + Y.e,,/~b,~ b/~ + Y. u,/~.j,~b, b/~ b~,b,~,

H (F) Co+E~l i ka i ' ak+ ~',ui~..~tai aka~at , ik ikst

V(~F) = ~,, w~,/3ik b[ b/~ ai k, ~zflik

H = H (B) + H (F) + V (BF) ,

and

(53) I T R)= to + ~t , / jb~ b/3 ,

t T (~) = td + ~t ika i a~., ik

T = T (B)+T (F).


In addition, there are here transition operators of the type

(54) P = ~,pai(b~ ai + air b,) , ai

called, for reasons to be explained in the following section, transfer operators. But the bilinear products

(55) t t:_(B) = b~ be ' U afl

G~ F) = ai t ak ,

Ftai = b: ai ,

Fi, = ai ~ b a,

generate the Lie superalgebra (graded Lie algebra) U(6/D). The operators of eq. (55) are quite often put into a matrix form

(56) b2 be I b2 ),

b. I a: ak where b~ b e, aitak are the Bose sector and bat ai, a t b, are the Fermi sector.

The Hamiltonian and transition operators of (52) and (53) can then be rewritten in terms of the elements of U(6/~9). Hence this algebra is the spectrum-generating superalgebra of the problem.

14 . - U(6/~2) and its implications

The classification of all possible dynamical symmetries of U(6/~2) is a rather complex problem which, however, has been solved for several values of ~2 of interest in the spectroscopy of nuclei. The first study dealt with j = 3 /2 , ~2 -- 4 [36]. For each value of f2 there are three families of symmetries, one for each of the three chains of (31). In early work, the chain

(57) U ( 6 / 4 ) ~ u B ( 6 ) o u F ( 4 ) ~ S O B ( 6 ) o s u F ( 4 ) ~

SpinBF (6) ~ SpinBF (5) ~ SpinBF (3) ~ SpinBF (2),

was considered. The combination of bosonic and fermionic algebras leads to spinor algebras, Spin(n). By writing the Hamiltonian in terms of Casimir operators of (57),

(58) H = E o + e l C I ( U B ( 6 ) ) + e 2 C 2 ( U B ( 6 ) ) + e ~ C i ( U F ( 4 ) ) + e 4 C 2 ( U F ( 4 ) ) +

+e~CI(UB(6))C1 (uF(4)) + ~]C�89 (OB(6)) + ~' C~ (SpinB~(6)) +

+tiC2 (SpinBF(5)) + yC2 (SpinBF(3))

and finding its eigenvalues it is possible to write down energy formulas describing this

22 F. IACHELLO

Fig. 9. - A supermultiplet (a set of states in nuclei belonging to a representation of U(6/4)).

situation. For example, in the case in which there are N bosons and 1 fermion, one obtains

(59) E(NB = N , N~ = 1, E,(O-1, 02, u 3 ) , ( r l , r2 ) , v j , J , ~?V/j) :

=E01 + 21l~:(2: + 4) + 2/] '[o1(ol + 4) + o2(02 + 2) + o~] +

+2fi[rl(r~ + 3) + r2(r2 + 1)] + 27J(J+ 1).

The relevance of U(6/f2) (or, in general, of supersymmetry) is that it provides a classification scheme for mixed systems of bosons and fermions, in particular here it provides a classification of odd-even nuclei. It elevates the concept of symmetry one step further in the complexity of the systems that can be treated.

An implication of supersymmetry is that now it is possible to classify simultaneously several nuclei, composing a supermultiplet. These nuclei belong to the same representation of U(6/f2) characterized by the total number of bosons and fermions, ~ = ~/~B + A/'~. Figure 9 shows a supersymmetric U(6/4) multiplet. The energy formula (59) gives a simultaneous description of several nuclei. An example is shown in fig. 10. In view of the fact that it relates states in different nuclei, supersymmetry is a very ambitious scheme.

The supersymmetric scheme for the analysis of nuclear spectra has been extended to cover many situations [37] with r ranging from 2 to 20. Several examples have been found of supersymmetry in nuclei, some of which are indicated in fig. 7. The scheme has also been extended to situations where the <<internal,> degrees of freedom (F-spin or isospin) are explicitly introduced, in particular to a version of the interacting boson-fermion model where the proton-neutron degrees of freedom are explicitly introduced (IBFM-2).

A question that has been extensively investigated is to what extm~t supersymmetry is a good description of nuclear spectra. In order to answer this question it is necessary to study not only the spectra but also the intensities of transitions. In supersymmetry, there are two different types of possible transitions: those induced by the Bose sector of the algebra, such as electromagnetic transitions, and those induced by the Fermi


E (MeV)

0

(2,0~

(I ,01]

+ 179~0S I14 / 7r9~ I'114 Th

F

- - - I T '~'~. ----'----_ (~0) - - - - " - - - _ I I T 1312+ - " - - - - -

- e + - - - - _ I - - - - I - t -H ' /2 § ----

~ L ~1 " ' ' - - ,/2 + " - - - _ _ , ~ ~

E q

(MeV) 190 76 0s 114

&~P---..

- - 6 §

-. T 4§ T o §

(,.o'~ "[~/L . . . -_ -Ir 2+ ~-..... ~,o)•

191 Exp 771r114

1 5 1 2 ~ " ~ ~ ~

T I I /2 ~ - . .

; ~ - - T'~'~'I 2-"~" --" 7/2;" "-- . / "~-.. /~ . 712 : y

' -="

~ 2 2 ~ 2 § ..

Fig. 10. - An example of supersymmetry in nuclei: the pair of nuclei 19~

sector. The latter transitions change a boson into a fermion or viceversa. They are experimentally accessible through the study of transfer reactions, such as (d, p) or (p, d). To begin with electromagnetic transitions, consider the electric quadrupole transitions. The corresponding operator in the Racah form and for U(6 /4) can be

24 F. IACHELLO

~Titten as

~,1(2) ~ ~)(2) r 7, ~(2) (60) T ( E 2 ) = a 2 ( d ; x s + s ~ • . . . . +f i2 (d x +y,z(a3/ ,~x~/ ,2#,

There are thus in this case three quantities that determine the intensities of transitions, a2, fi2 and y 2. However, the amount of experimental data available for testing the supersymmetric scheme is now much larger since in the same multiplet there are both even and odd nuclei. Tests of supersymmetry indicate that intensity relations are satisfied within 20 % or better. This breaking is larger than in the case of ordinary symmetry but nonetheless still leaves supersymmetry as a useful tool to analyze the situation.

Another set of operators is, as mentioned above, the set of transfer operators. Consider for example the transfer of a particle with j = 3/2. The corresponding operator (addition operator) can be written as

' * ~)u.~/2) (61) p(:~/2) = po(a:~'/,. • ~)(3/.~) + p2(a:~/2 • ,

and similar expression for the removal operator. Tests of superselection rules and intensity rules for these operators have been performed indicating again that supersymmetry is good within ~ 20%.

Supersymmetry poses a technical challenge since, although in principle straightforward, the evaluation of matrix elements of the elements of the superalgebra E* between states which are representations ~ * is a formidable problem, especially for superalgebras of large dimensions. In recent years, new impetus in the study of supersymmetry in nuclei has been given by the development of computer programs capable of algebraic manipulations. The use of these programs has allowed one to extend even further the applications of supersymmetry and has confirmed the usefulness of this scheme.

15. - Impl icat ions of nuclear supersymmetry to other fields of physics

Since supersymmetry in nuclei is to date the only explicit experimental example of supersymmetry in physics, it is of interest to discuss possible implications to other fields of physics. In nuclear supersymmetry, the supersymmetric partners are composite bosons (Cooper-like pairs) and individual fermions. Several authors have commented on the possibility that these supersymmetries are the only supersymmetries that can be realized in physics. This idea has been used by Nambu [38] to construct effective supersymmetries in type-II superconductors where the supersymmetric partners are electrons and Cooper pairs. Catto, Giirsey [39] and others have applied the same idea to hadronic spectroscopy where the supersymmetries partners are diquarks (D) and quarks (q). Gfirsey's scheme is a version of Gfirsey-Radicati, SU(6). Since the diquark transforms as the representation 21 of SU(6), the appropl~iate superalgebra here is U(6/21). This scheme is thus similar (if not identical) to the U(6/ t2) scheme of nuclear physics, except that the role of U(6) is interchanged since in nuclear physics U(6) refers to the Bose sector and U(t2) to the Fermi sector, while in Catto-Giirsey supersymmetry U(6) refers to the quark (Fermi part) and U(21) to the diquark (Bose part). Gfirsey's scheme implies the occurrence of diquark-antidiquark states, for which there is some evidence, although not compelling.


The question on whether or not supersymmetries in which the bosons are fundamental objects, not related to fermions, occur in nature awaits answer from the search at CERN and other laboratories for supersymmetric particles. In any event, it appears at this stage, that, if fundamental supersymmetries exist, they are badly broken.

The composite nature of the bosons makes it also clear that the dynamic symmetries and supersymmetries which are useful in nuclear physics are effective symmetries (i.e. symmetries of the effective degrees of freedom which play the dominant role in the low-lying states of nuclei). As the excitation energy increases, the order in spectra decreases and so does the usefulness of symmetry considerations. The breaking of dynamic symmetries can be associated with the onset of chaos. This relationship (not discussed here) has been the subject of many investigations in recent year.

16. - Role of dynamic symmetr ies

As is evident from the discussion in the previous sections, the main role of dynamic symmetries, as used since 1974, is that of providing a classification of complex systems. The atomic nucleus is one of these systems but similar concepts have also been used in other fields, for example in the study of complex molecules. This use of symmetry goes well beyond the early use of symmetry as an invariance of the Hamiltonian, but it is that which has proven most useful in the last 20 years.

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26 F. IACHELLO

[19] See, for example, CIZEWSKI J. A., CASTEN R. F., SMITH G. J., STELTS M. L., KANE W. R., B6RNER H. G. and DAVIDSON W. F., Phys. Rev. Lett. 40 (1978) 167.

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(World Scientific) 1994, p. 431. [25] ELLIOTT J. P. and EVANS J. A., Phys. Lett. B, 101 (1981) 216. [26] For a review of the sdg interacting boson model, see DEvI Y. D. and KOTA V. K. B., Pramana,

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Symmetry and supersymmetry in nuclear physics