Nambu mechanics of non-Abelian dyons

P H Y S I C A L R E V I E W D V O L U M E 1 7 , N U M B E R 2 1 5 J A N U A R Y 1 9 7 8

Nambu mechanics of non-Abelian dyons

Minoru Hirayama* and Hsiung Chia Tzet Stanford Linear Accelerator Center, Stanford Universi?~, Stanford, California 94305

(Received 19 August 1977)

Non-Abelian toplike realizations of Nambu's mechanics are found in nonrelativistic bound systems of a point singular SU(2) monopole and a test particle bearing G spin embedded in a general compact gauge group G . The exactly soluble cases of G = U(l), SU(2). and SU(3) dyons are studied in detail. The naturalness of the new formalism for these Keplerian systems is exhibited.

I. INTRODUCTION

In the current climate of superunified and quark- confining theories, there i s a renewed feeling that some generalization of our idea of space-time1 and/or extension of quantum mechanicsZ may be needed to reach a conceptually unified scheme for all interactions. If one chooses to proceed from Dirac's Hamiltonian method3 a s a f i r s t step to a quantum theory, then extensions of classical mechanics should be made. It i s in this spir i t that not long ago, with the Liouville theorem a s his guiding principle, Nambu4 wrote down a new analytic mechanics and discussed i t s quantization. This mechanics is remarkable in several respects . F i r s t , it t rea ts all conserved quantities of a mechanical system on the same footing. This i s clearly a most attractive feature from a quantum perspective. Second, i t allows fo r a phase space of odd a s well a s even dimensionality, a property shared by another new mechanics, pseudomechan- i c s , whose phase space i s partly spanned by anti- commuting Grassmann variables ."he lat ter permit a consistent inclusion of spin and other internal degrees of freedom into classical mechan- i c s . It has been shown that at the classical level this new mechanics i s connected to a singular Hamiltonian mechanic^.^ As regards quantization, Nambu showed that the usual Heisenberg equations a r e recovered in several examples. This lends further support to the idea of the unique- ness of quantum mechanics. Notably, he found that both of Jordan's nonassociative algebras, be they special or exceptional, can be readily in- corporated in his new mechanics. Thus it was later demonstrated that most of all the irreducible Fock representations of Green's trilinear algebra a r e consistent with Nambu's q ~ a n t i z a t i o n . ~ This invites the conjecture that quarks may well find their natural dynamical description within the new mechanics. Our work advances a small s tep in fulfilling such an expectation: We show that beyond the asymmetric top, Nambu mechanics finds i ts simplest physical realizations in non-Abelian

dyons. The lat ter have provided many models for strong-interaction dynamic^.^

Before stating our results , it i s helpful to re- call the essential elements of the new mechanics." In a general Nambu mechanics, the phase space is spanned by an n-tuple of dynamical variables x i , i = 1 , 2 , . . . , n. There a r e (n - 1) Hamiltonians HI, . . . , Hn-, which a r e constants of motion and which define the dynamics via the generalized Hamiltonian equations

= E i j k . . . , a jH l ' ' ' a ,Hn- , , (1 . I)

where E ij,. . . , i s the Levi-Civita symbol and ajna/ax,. The Fi =Fi(x, , . . . ,x,) i s an arbitrary differentiable function of the variables x i . The totally antisymmetric 12-linear bracket

{F ,,.. . , F , } ~ E ~ ~ . . . , ~ ~ F , ~ , F ; . . a , F n (1.2)

takes the place of the usual Poisson bracket. Like the lat ter , i t stands a s the fundamental algebraic object of the theory. Instead of the canonical Poisson brackets { q i , p j ] = bi,, one has the Nambu bracket

{xl ,xz , . . . , x n } = 1 , (1.3)

and in place of {qj ,qi) ={pi ,pj} = 0 , one has for instance

{x1,x~,x3 , a . . , x n ) =(I , (1.4)

etc. The corresponding Nambu-bracket-pre- serving canonical transformations a r e defined a s such that

{x: , . . . ,xA} ={xl, . . . ,xn} = 1 . (1.5)

From Eq. (1.1) it ensues that the velocity field 2 i s divergenceless o r equivalently the Liouville theorem i s fulfilled in the phase space. This fact opens the door to the corresponding new statistical mechanics yet to be investigated. As the simplest generalization of the usual canonical doublets (pn , qn), Nambu considers the interesting situation

17 N A M B U M E C H A N I C S O F N O N - A B E L I A N D Y O N S 547 -

of N canonical triplets (P,, Q,, R,), n = 1 , 2 , . . . , N Then one h a s f r o m Eq. (1.2) the t r i l inear bracket

a (A,B,C) CA,B,C1 =C

n ,1 a(Pn, Q,, R,)

and

where F = F ( P , Q, R), H I P , Q,R), and H2(P, Q, R) a r e two Hamiltonian functions. T o make a c a s e f o r the physical relevance of h i s new formal i sm, Nambu pointed out a specific realization f o r the n = 3 , one-triplet c a s e , namely the asymmetr ic top. H e r e the t r iplet 3 is naturally identified with the angular momentum 'i: in the body-fixed f r a m e . T h e r e a r e two guaranteed considered quantities: Hl = iC L,'/z, is the total kinetic energy (L, =Z,w,) and Hz =$c2 is the C a s i m i r invariant. The Nambu equations Eq. (1.7) a r e given by

which a r e just the Euler fo rce- f ree rigid-body equations. We observe that they correspond to a special c a s e of the L ie equationg in the "Lax form" when the algebra i s SO(3,R) of the rotation group.

This example is important on two s c o r e s . F r o m the s t ruc tura l viewpoint it i l lustrates one essent ial advantage of the new mechanics which revea l s Prima facie the key s t ruc tures of constrained sys - t e m s , i.e., their Lie algebra a t the level of the equations of motion. This situation cont ras t s with Dirac 's mechanics where the constraints appear a s subsidiary conditions. We reca l l that the essential s t r u c t u r e shared by c lass ica l and quantum mechanics l i es in their participating in the s a m e Lie algebra. While the usual bilinear bracket s t ruc ture is connected to some Lie o r Jordan algebra, the n- l inear bracket can in principle ac - commodate other a lgebras a s well. It is this alge- b ra ic naturalness and flexibility of Nambu mechan- i c s which make it mos t a t t ract ive in one's s e a r c h f o r generalized dynamics encompassing both clas- s ica l and quantum mechanics .

F r o m the physics viewpoint, toplike s y s t e m s a r e plentiful in nature. It suffices to mention the Bohr- Mottelson nuclear model and the dynamics of colored, e lectr ic s t r ings in the non-Abelian la t t ice gauge theory ." Since i t s inception, Nambu mechanics has been the object of a few investigations, a l l focused solely on i t s fo rmal aspects.6111112 What i s c lear ly m o r e des i rab le to the model- building physicist a r e fu r ther real izat ions of the new mechanics , possibly ones which b e a r direct ly on our cur ren t ideas about the basic constituents

of m a t t e r . In this work we depart minimally f r o m Nambu's s imple example of the top and yet we do exhibit real izat ions which obey the mentioned c r i t e r ion of selection.

Basically our work s t e m s f r o m F i e r z ' s old ob- servationL3 that a dyon, a sys tem of a charged particle interact ing with a magnetic monopole, has the algebraic s t r u c t u r e of a symmetr ic rigid top. Hence we infer that natural real izat ions of Nambu mechanics a r e to be found in these Keplerian sys- t e m s . We part icular ly have in mind their non- Abelian generalizations which have been the ob- jects of severa l recent investigations.

In a shor t paper , one of u s (M.H.) '~ studied in the context of Nambu mechanics the motion of a test par t ic le bearing isotopic spin ? in the field of a 't ~ooft-Polyakov15 monopole. Such a c lass ica l dyon is conceived a s a nonrelativistic, point- s ingular l imit of the field theory model of a n SU(2) monopole interacting with a Higgs field ca r ry ing a representat ion Ta of S U ( ~ ) . While entirely bosonic in composition such a complex yet admi t s integral o r half-integral total angular momentum depending on the tensor o r spinor nature of Ta'O In the la t ter c a s e we witness a rea l i s t i c example of the dynamical fermion-representat ion phenom- enon which has i t s counterpart in the duality between the sine-Gordon soliton and the mass ive fermion of the Thirr ing model.17 The spin and s ta t i s t i cs connection f o r such s y s t e m s h a s been elucidated by Goldhaber.18 We have a sys tem with superselect ion-rule s e c t o r s s e t by Dirac 's o r Schwinger's quantization condition. The super- selection r u l e mani fes t s itself in a s t r iking group- theoret ical manner which makes the dyons distinct- ly different f r o m the usual top. While in the l a t t e r only the Lie algebra is of importance, with the dyon, the self-adjointness of the Hamiltonian and rotational invariance a r e real ized only if the algebra of the sys tem can be irttegrated to yield the group. This globality constraint is nothing but the monopole quantization condition.lg

In this wor!~, we extend the work of Ref. 14 to the general c a s e s where the composite Su(2) monopole @ t e s t par t ic le , a n SU(2) dyon is embedded in a compact gauge group G .

We do not lose any nontrivial topological fea ture by res t r i c t ing ourselves to the limiting c a s e of dyons involving a point-singular monopole and a test par t ic le with internal quantum numbers . We will proceed r a t h e r systematical ly f r o m the s tandard problem of the ~ ( 1 ) dyon to that of the SU(2) dyon which rece ives h e r e a n improved t reatment over Ref. 1 4 . Then we go on to the natural extension of the SU(2) problem, namely when SU(2) is embedded in SU(3) in two possible ways, respectively the SO(3) and the U(2) embeddings. All

548 h l I N O R U H I R A Y A M A A N D H S I U N G C H I A T Z E - 17

these exactly soluble sys tems a r e c a s t into Nambu mechanics . The physical motivation f o r studying this SU(3) extension is enticing. In the c a s e of the S U ( ~ ) / U ( ~ ) dyon the resul t ing monopole quantization r e s t r i c t s the e lec t r ic charge of the t es t par t ic le to be quarkl ike. So the resulting Nambu sys tem descr ibes a very massive composite quark-monopole complex endowed with dynamical fermionic sp in . Such a s t ruc ture might be of eventual in te res t . In part icular , f o r such a n SU(3) dyon we obtain the corresponding SU(3) top equations of motion in the fo rm conjectured by Nambu. Our analysis readily general izes to the instance of a G dyon.

Our paper is organized a s follows: In Sec. 11, we introduce the U(l) dyon in i ts new role a s a Nambu mechanics . In Sec. III we f i r s t reca l l the essent ial resu l t s of the embedding SO(3) point- s ingular monopole in a general compact L ie group. Then we s e t up the dynamical equations in the usual Hamiltonian mechanics . We consider the dynamics of the Su(2) and Su(3) dyons a s Nambu mechanics of non-Abelian tops. Our resu l t s a r e in such a f o r m that generalization to a G dyon i s manifest . Finally in Sec. IV we d iscuss the pos- s ible advantages of the new formal i sm and i t s pending problems.

11. THE U ( 1 ) DYON

In many ways, magnetic monopoles may play a role in part ic le physics. On one hand, they a r e soliton solutions to gauge theories of weak and electromagnetic interact ions. They can a l so be identified with the end points of vortex l ines which then s e r v e a s field-theoretical building blocks f o r dual s t r ing theories .'O A s non-Abelian analogs of Cooper p a i r s in a color supervacuum, they may play a controlling ro le in the confining plasma phase of the quark-gluon dynamics." On the other hand, even if these physical pictures fai l to ma- ter ial ize, the possibility s t i l l exis ts that the relevance of their underlying algebraic s t ruc tures will survive under a different guise. So i t is hoped that when r e c a s t into a new mathematical f ramework , the explicit examples given h e r e may yet lead to possible generalizations of monopole s y s t e m s . These observations a r e justification enough f o r the reformulation of monopole dynamics into Nambu mechanics .

In this work we deal exclusively with dyons by which we mean composites of an electr ical ly charged particle moving in the field of a fixed monopole. As the kernel of a Nambu mechanics of dyons i s present in the Abelian c a s e , it will be our f i r s t concern to study the U( l ) dyon.

The dynamics of a nonrelativistic dyon h a s r e -

ceived ample t reatment in the l i t e ra ture ; we only gather the ingredients necessary f o r our

Since a t rue dyon does not have bound s ta tes , to s e c u r e binding we shal l assume an added nonelec- tromagnetic potential V(Y) taken spherically sym- met r ic and f r e e of s ingular i t ies f o r rad i i r > O .

The Hamiltonian of the sys tem i s

with

Notably ? x 2 differs f rom the field =g?/r3 of a fixed monopole by the Dirac s t r ing singularity along the unit direct ion E. F o r our problem the group kinematical s t ruc ture of the sys tem i s of central importance. F i r s t , we identify the con- s tants of motion, foremost among them, the angu- l a r momenta. Owing to the contribution - p7.' ( p = eg) of the gauge field, the conserved angular momentum is

? = ; x ( c - e Z ) - p? , (2.3)

a lready considered by P0incar6 . '~ The constraint ? .?=- /.I implies that the motion

of the test charge is confined to a cone. If we a r e to use the f a s a new dynamical var iable , we a r e nat_urally led to introduce a n object complementary to J , the radial dilatation genera tor

D=;.(c-eX) . (2.4) +

This is s o s ince J does not c a r r y any information about the radial component of :. Then one can check that the Poisson brackets of the system can be cas t into a closed Lie algebrazz:

{ x i , ~ , ) = o 7

{Ji,xj) = ' i j k X k , {Ji,Jj} = ( i j k J h (2.5)

{ D , x i } = - x i ,

{J~ ,D} =o . So the corresponding group i s the three-dimensional group made up of the Euclidean group E, = ~ ( 3 ) x T, of rotations and t ranslat ions plus the dilatations. The i rreducible representat ions a r e labeled by the "helicity" invariants

And since f .i. = - eg, they provide a n algebraic derivation of Dirac 's quantization condition. The la t t e r is required f o r rotational invariance and self-adjointness of the ~ a m i 1 t o n i a n . l ~ Ikstead of the momentum variables c , we can use J and D.

17 - N A M B U M E C H A N I C S O F N O N - A B E L I A N D Y O N S 549

Then

Since {F2, J) =0, we see that for fixed rZ, the system is isomorphic to a symmetric top, where- fore the natural motivation is to reformulate i t a s a Nambu mechanics.

F rom the above ingredients, we deduce the following equations of motion:

The geometry of the motion leads u s to define two angles 0 and I$ such that

and

Owing to the constraint cos8 =- p / J , 0 is not an independent variable. From (2.12) we have

So the equations of motion f o r F and a r e equivalent to the s e t s (2.8)-(2-10) and (2.13). We note that besides the tr iplet J , the variables r , D, and @ form another triplet. So these kinematics of the dyon system naturally invite a reformulation into a Nambu mechanics. Yet the variables r, D, and C$ a r e still unsuitable fo r that purpose. F rom Eqs. (1 . l ) and (1.2), any differentiable function F(x,) serves just a s well a s a new coordinat?. It follows that if an F(xi ) can be found such that F(x i ) = 0 , we then know i t s orbit to be restr icted to a submanifold of the phase space, one defined by F(xi ) =const. One such constant is clearly the Hamil- tonian Ht (2.71, which defines the energy shell. Besides J and H, the remaining fifth constant of motion of our problem can be chosen to be the angular variable

given a s an Abelian integral-where f ( r ) is given by

Qsually the five constants of motion a r e taken a s J , H, and J L J = m )FxF/ . However, @ is a better choice than le 1 since it i s conserved in the SU(N)

(N 32 ) dyon problem while IE 1 is not. This point will be made clear in Sec. III.

The equations of motion a r e now 2 . . J=H=Q,=O (2.16)

and the corresponding five constants of motion determine a one-dimensional submanifold, the orbit of the six-dimensional phase space. Fo r zur sixth Nambu coordinate completing the se t J , H, Q, , we can define

such that S = 1. Exploiting the two-triplet s tructure of the U(l)

dyon kinematics, contact with Nambu mechanics can be established in the following manner. We denote the two tr iplets a s (J,, J,, J,) = (P,, Q,, R,) and (S, Q,, H) = (P,, Q,, R,). Then the equation for an arb i t ra ry F =F(Q, P, R) i s given by

+ = [ F , H ~ , H z I , (2.18)

with H, =Q2=$ and H,=R,=H and

Then we get

[Pn ,Qn ,RnI = I , etc. (2.20)

The above equations (2.18)-(2.20) realize Nambu mechanics fo r the ~ ( 1 ) dyon system.

In subsequent sections, it will turn out that for systems with correlation between spatial and internal degrees of freedom, there exists a more convenient time variable 7 defined by

Then instead of the t derivative =dF/dt, we make use of the T derivative

It i s of course easy to rewrite (2.18)-(2.20) into Nambu mechanics with the proper time variable T

F T =[F,H,,H21 . (2.23)

The only modification necessary is that Pz should be identified not with S but with

which obeys

U , = l . (2.2 5)

Next we proceed with the instance of non-Abelian dyons .

M I N O R U H I R A Y A M A A N D H S I U N G C H I A T Z E

111. NON-ABELIAN DYONS

A. Generalities

In this section we formulate in the context of Nambu mechanics a general c lass of non-Abelian toplike sys tems . They a r e given by the nonrelativistic dynamics of a c lass ica l point par t ic le with m a s s MI and G spin T, in the field of a fixed SU(2) monopole embedded in a general compact semi- s imple gauge group G. T, a r e the genera tors of some representat ion of G. F o r this task we need to know a l l spherically symmetr ic three-dimensional point-monopole solutions f o r the simply connected universal covering of G. This purely group-theoretical problem has been analyzed by many people and has been in part icular solved completely in a very s imple and compact manner by Wilkinson and G ~ l d h a b e r . ~ ~ Here we only gather the resu l t s relevant to our work. Generally one seeks s tat ic solutions in the Coulomb gauge A, = O of the Higgs-Kibble field equations

where G,,=ac,A,,-ie[A,,A,] and D , 9 = a u @ - ie[A,, +] . A, and @ a r e mat r ix fields in some faithful representat ion of G. V(@) is the s tandard G-invariant quart ic polynomial. Specifically the point-monopole solutions a r e obtained in the Lon- don limitz5 when 6V/69 =a9 = O everywhere except a t the singularity positioned a t the or igin. The s imple recipez4 for constructing these solutions comes f r o m using a s ingular Abelian gauge

with the Dirac s t r ing singularity pointing f rom the pole along ii to infinity. Thus a solution to Eq. (3.1) is

+, and Q a r e constant mat r ices such that (aV/ a@),, = 0 and [ Q , 9, ] = 0. Then the generalized Dirac quantization condition is given by

Provided the eigenvalues of Q a r e a l l integers o r half-integers, the Dirac s t r ing is unobservable and with 5 being simply connected, i t can be removed bv way of a singular gauge transformation to get to a no-string gauge.

The spherical symmetry of the problem i s under-

scored by the conserved angular momenta

3i;x;;- i e r 2 g , (3.5) +

with = - eA and the associated gauge covariant propert ies

[J i ,J j l=ki jkJk ,

[ J i , ~ j l =iEijkrk, (3.6)

[ J i , ' + ] = o . The main resu l t of Ref. 24 is to provide the nec- e s s a r y and sufficient conditions fo r t ransferr ing the solution (3.3) to a regular gauge A; with mani- fes t spherical symmetry , i.e.,

when E =F X; and T i generate some Su(2) subgroup of G. We shal l u s e such a radial gauge in the following examples of the SU(2) and SU(3) dyons. The above summary constitutes the minimum knowledge we need. F o r fu r ther detai ls on actual determina- tions of point monopoles f o r a general group G, we r e f e r to the r e a d e r to the l i t e ra ture and shal l only quote the resu l t s needed below.

The equations of motion of a t es t par t ic le with m a s s m and isospin T i (i = 1 , 2 , 3 ) interacting with a n SU(2) magnetic monopole

with

The notation is self-explanatory. These equations can be derived f rom

~ ( ~ , P , T ) = ( F , H ) , (3.11)

with

where a generalized Poisson bracket is defined a s

(A,B) ={A,B} + ( A , B ) . H e r e

17 - N A M B U M E C H A N I C S O F N O N - A B E L I A N D Y O N S

is the standard bracket and where D i s given by

is characterist ic of classical t rue spin systems .26

The lat ter involve only the variables T, and no others. This is certainly not the case for a rigid body o r our Su(2) dyon since the T, do not form a complete se t of coordinates, neither in the kine- matic nor dynamic sense.

In the radial gauge and the London limit where the Higgs field i s rigid, i.e., everywhere constant, we have

1 Aq(x)=-- < 'k iak 7 (3.16)

for al l FE R3 - (0). This potential corresponds to the only spherically symmetric point monopole for SU(2).24 The conserved angular momentum i d 6

* J = F x T ~ + T , (3.17)

and since

J . r = T . r ,

the total angular momentum takes on integral o r half-integral values depending on the tensor o r spinor character of the representation of T.

We define the angles 9 and C$ just a s in the pre- ceding section. 9 and $ fix the direction of F rela- tive to 3. It is convenient to introduce three orthonormal vectors G l , G,, and G3 such that

and -. ,. m,=r . (3.2 1)

They satisfy - - m, .mj=tj i , (3.22)

and - . - * m , x m , = m , , etc . . (3.2 3)

Next we define the variables w,, w,, and w, by -. - * * T = wlml + w2m2 + w,m, . (3.24)

It should be noted that w, i s given by

w, = JcosS . (3.25)

After some manipulations we find that 6 can be expressed a s

+ .. D = r .p . (3.27)

Using (3.12), (3.16), (3.24), and (3 -26) we have

It i s easy to check that the se t of nine variables J1, J 2 , J , , Y, C$, D, wl, w2, and u), constitutes independent variables defining the SU(2) dyon system.

The nine equations of motion for F, c, and T a r e then equivalent to14

. a . J =J = J = o

1 2 3 9

In Eq. (3.29) we recognize the U ( l ) problem in the subset {f, Y, 4 , ~ ) . The remaining se t of {w ,,w ,, w,) relates specifically to the SU(2) problem. We note that while L'= (Fxfi)' i s conserved in the U(1) problem, i t i s no longer s o here. We .find L' = T' + 2w1J sine. T2, J, and sin0 a r e conserved but w, i s not, hence L~ i s also not conserved.

Jus t a s in Sec. 11, we a r e led to define new variables

and

where

We further define u and a by

I* = ( w ,2 + w ~ ' ) ~ ' ~ (3.33)

and

Then Eqs. (3.29) a r e equivalent to * . . . . . . J = H z @ = Q = u = g = O (3.35)

and

S = l .

552 M I N O R U H I R A Y A M A A N D H S I U N G C H I A T Z E - 17

This fo rm of the equations allows an immediate transcription into Nambu mechanics. Here we have a three- tr iplet s t ruc ture of dynamical var i- ables. The s imples t identification is (P,, Q,, R,) = (J, ,J , , . / , ) , (P, ,Q,, R , )=(S ,Q , ,H) , and (P , ,Q3 ,R3) = (fc, o , 0) , and H, = @ and H,=H. In general we can identify the eight var iables Q,, Q,, P, , P,, P,, R,, R,, and R 3 with any independent eight functions of the @, u, a , H , 8, 5 By setting Q , = S, H , = P,, and H , = R , any a rb i t ra ry differentiable function f ( P , Q , R ) will obey Eq. (1 .7) .

An alternative and m o r e elegant choice of var ia- ab les i s provided by the se t i2={J, I T , @ , H , G ) , where ~ = ( u ~ , , w , , t u , ) and

The motivation i s apparent. The f i r s t s ix variables constitute the U ( l ) dyon sys tem while w descr ibes the internal rotation of the isospin. The nine equations of motion in (3 .29) can be r e - expressed a s

tolr = w ~ I A ~ ~ , w2,= - w 3 w l , and w 3 , = 0 ,

where the subscript 7 denotes the T derivative in- troduced a t the end of Sec. 11.

It should be noted that % real ly sat isf ies the " S U ( 2 ) top" equation of motion:

It is now easy to c a s t (3.37) into the fo rm of Nambu mechanics. By makingthe identifications

(P,, Q,, R 2 ) = ( U , @ ,H),

( P ~ , Q ~ , R ~ ) = ( W ~ , W ~ , W S ) ,

H,= $ ( W ~ ~ + W , ~ + W ~ ~ ) + @ ,

and

H z = $ w ~ ' + H ,

we ge t to

In Eq. (3.38) we have observed that < behaves just like the angular momentum vector of a force-

f r e e rigid body in the body-fixed f rame. The coordinate system defined by G I , G,, and G, corresponds to the body-fixed f r a m e of a rigid body. The interconnection between internal and spatial motions of the t e s t par t ic le r e s t s in the space-dependent definition of and in the redefi- nition of the t ime variable.

C. The SU(3) dyon

We now proceed to the S U ( 3 ) dyon, which turns out to have some interesting new structure. By an S U ( 3 ) dyon we understand the composite of an S U ( 2 ) monopole embedded in the group SU(3 ) and a t es t par t ic le ca r ry ing an SU(3 ) unitary spin A,. F r o m the pre l iminar ies in Sec. IIIA, we know that all the SU(3 ) monopole solutions mus t be invariant under simultaneous spatial and isospatial rotations generated by J+T. T h e r e a r e two distinct embeddings of SU(2 ) into SU(3),'¶ the " U spin" U ( 2 ) - S U ( 2 ) B U ( 1 ) c a s e with ? identified with ( $ A , , ah,,_$ A,) and the "nuclear physicg" S O ( 3 ) c a s e with T = (&, - A , , A,) .

The corresponding nonrelativistic Harniltonian f o r the SU(3 ) dyon i s then given a s

The c number $ h , (a = 1 , 2 , . . . , 8 ) i s the unitary spin c a r r i e d by the t e s t par t ic le . V ( r ) is a spherically symmetr ic binding potential.

The equations of motion a r e derived f rom

where the generalized Poisson bracket i s given by

with fa,, being the totally ant isymmetr ic s t ruc ture constants of SU(3 ) . F r o m Eqs. (3.42) and (3.43) we obtain

which gives

where

17 - N A M B U M E C H A N I C S O F Y O N - A B E L I A N D Y O N S 553

Following the detailed analysis of Cor r igan et 1 2 1 . ~ ~ the two distinct Ansiitze for the asymptotic f o r m s of the gauge potentials a r e

(a) eA; % =i[u;, a,, y,L] . '3

T = A a = 1 , 2 , 3 2 '

and

F o r the point-singular l imit of interest h e r e , the above f o r m s a r e valid over the whole space. We have for both c a s e s (a) and (b)

The conserved angular momentum i s J = F X g+ T which obeys the O(3) algebra (J, , J,) = E,,,J, and ? identified with the X,'S according to the re- spective embeddings.

It follows from Eqs. (3.42) and (3.50) that

which i s identical to the SU(2) problem. The s truc- tu re of the Poisson bracke ts fo r 7, 6 , and 'i? i s a l so exactly that of the SU(2) dyon problem. Thus there i s no difference between the Su(3)/U(2) and S ~ ( 3 ) / S 0 ( 3 ) dyons a s f a r a s the quantities if, 6 , and ? a r e concerned. A s for the solutions for the remaining SU(3) genera tors , we select to analyze in detail the U(2) embedding case . Our motivation i s a s follows. The ~ ~ ( 3 ) / S 0 ( 3 ) monopole i s a s table pure monopole and the quantization condition ass igns to the t es t par t i c le multiplet integral charges. So this sys tem differs minimally f rom the SU(2) problem. On the other hand, the SU(3)/ U(2) dyon i s endowed with a m o r e interesting new structure. Indeed i t s monopole i s colored a s it c a r r i e s both electromagnetic and colored iso- magnetic charges with the consequence that the quantization condition i s no longer Dirac 's . The charges of the test par t i c le a r e then q ~ a r k l i k e . ~ ~ So this SU(3) dyon i s an object bearing both quark- like quantum numbers and dynamical half- integral spin.

Denoting To = A, /2, a = 4 , 5 , . . . , 8 , the equation (3.45) becomes

o r

Fo =ffacKbTc = (Ta ,K) , where

and

The second equality in Eq. (3.53) follows from the fact that

To make a n appropriate choice of isospin v_ari- ables , we go to the "body-fixed frame." A s J i s conzerved, we can, without loss of general i ty , fix J to be paral le l to the third axis. In that sys tem, 3 and ; a r e

and

s in8 cos+

Then G,, G2, and 6, a r e

COSO COS+ - s in@

G l = cos8s in+ m,= c o s q ,

and

( s i n . ) - ( 0 ) (3.59)

-. T and a r e related by

554 M l N O R C l H I R A Y A M A A N D H S I U N G C H I A T Z E

where XI,, UI, , and w, a r e given by (3.24). XI,, ze,, and w3 a r e the isospin variables in the body-fixed f rame.

We define the variables w,,u~,, . . . ,XI, which take the place of T,, T,. . . . , T , a s follows:

and

T,= I U , ; - - u i 7 s a r e Paul i matr ices . The definitions (3 .61) and (3 .62) should be regarded a s the isospinor and iso- sca la r var iants of ( 3 . 6 0 ) , respectively. The decomposition of T,,T,, . . . , T, into a t r iplet G, a complex doublet W, and a singlet w, corresponds to the SU(2 ) decomposition of an SU(3 ) octet:

8=3+2+2_+1_. - Now it tu rns out that LV and w, satisfy

and

war = O . (3 .64)

The tr iplet $ of course sat isf ies

wlr = wZw3 , ulBr = -W,U)~ , and w,, = 0 . (3.65)

It i s s t raightforward to recognize that (3 .63 ) , ( 3 . 6 4 ) and (3 .65 ) can be combined in a s imple form

where 8

and

G -' 2 2 - 2 w 3 . (3.68)

Equation (3.66) could indeed be called the " S U ( 3 ) top" equation.

By making the identifications

and

Hz =Gz+ H , (3 .72)

we a r e then led to thegeneral ized Nambu mech-

anics

where F i s an a r b i t r a r y differentiable dynamical quantity. In Eq. (3 .73) u,, u,, . . . , I ( , and I),, v5,. . . , t i , , . . . , I J , a r e regarded a s the frozen degrees of freedom and reflect the three dimensionality of physical space. We should a l so mention that an equation of the form of (3 .73) was suggested to us by Nambu a s the natural generalization of his original scheme.28 It 1s very pleasing to s e e such a realization among non-Abelian dyon sys tems a s we reca l l Dirac 's invitation to exploit the kinship between mathematical and physical s t ruc tures . a discourse which prefaced h i s celebrated introduction of monopoles into theoretical The form (3 .73) of the S U ( 3 ) dyon dynamics then invites the abs t rac t a lgebraic generalization in which none of the IA's , ? \ ' s , and w's a r e frozen out. What would be a physical realization of such a n extension?

1V. FINAL REMARKS

A s evidenced by s tudies of dynamical groups and supersymmetr ies , the underlying Lie algebra plays an essent ial ro le both in the formulation and the solutions of dynamical problems. It i s a striking feature of Nambu mechanics that the Lie algebraic s t ruc ture appears a t the very level of the equations of motion. Another notable feature i s the equal s ta tus of the space-t ime and internal-symmetry dynamical variables. This has a l l the hal lmarks of a unified theoret ical framework. ' Indeed while the generalized Poisson bracket s t ruc ture f o r the dyons looks ra ther unnatural both in the Dirac fo rmal i sm and in pseudomechanics, no such ob- jection can be ra i sed against the Nambu brackets . Of course to achieve this uniform symmetry with respec t to a l l the variables in the new machanics, a p r ice mus t be paid. In genera l , our classical intuition i s lost which may not be a handicap if one i s seeking to general ize quantum mechanics not via any correspondence principle but, say, through some algebraic s t ruc tures and there de- f o r m a t i o n ~ . ~ ~ In the examples considered in our work, the choice of dynamical var iables and con- se rved quantities i s relatively easy s ince we understand the physics behind our sys tems and can solve for the matheniatical s t ructure. This r e - moves some of the a r b i t r a r i n e s s of the choice of dynamical quantities inherent to Nambu mechanics . Though we only consider the instance of the U ( l ) , S U ( 2 ) , and SU(3 ) dyons, our t reatment makes obvious the s t ruc ture of Nanlbu mechanics of the SU(N) and the G dyons. Namely if G s S U ( 2 ) with g,,, ( a , b . c = 1. ... , n ) a r e the s t ruc ture con-

17 - N A M B U M E C H A N I C S O F N O N . A B E L I A N D Y O N S 555

s tan t s , then defining the w, in the body-fixed completely novel a lgebraic dynamical s t r u c t u r e s ? f r a m e , we have The generalized toplike s t r u c t u r e s we have ob-

n tained h e r e a r e certainly suggestive. Indeed G ~ = $ C w a 2 , G Z = $ ~ 1 3 Z , (4.1) Nambu mechanics s t i l l h a s to come of age; i t s

O=I relevance awaits fu r ther and m o r e rad ica l illu- and s t ra t ions than thoseprovided here. Moreover , the

quantization of dyons in the new formal i sm should H l = G , + @ , H , = G , + H ; (4'2) be studied. Already a t the c lass ica l level of the

then the Nambu equations of motion a r e dyons we s e e inviting connections of the new mechanics to dynamical to pseudomechanics,5

aF aHl aH, F ~ = gabc K c . (4.3) and i n a l a r g e r context t o the r i c h symplect ic s t ruc-

X'U, v , w t ~ r e ~ ~ of the usual canonical formalism. We

In the Introduction we give the main motivations fo r reformulation of the dynamics of dyons in the new mechanics. T h e r e the gauge field degrees of freedom a r e frozen down to a s tat ic potential. To go to a Nambu field theory i s no easy task. A s a f i r s t s t ep beyond the Keplerian sys tems studied h e r e , it would be natural to go over to the sys tem of a Nambu str ing with monopoles at their ends.31J0 Here the gauge field i s dynamical but s t i l l frozen into the line geometry of a string.

The hope in any reformulation of old problems in a new language l i es in new possibilities which might show up in the new context. In the dyon sys- t e m s studied here the correlat ion between internal symmetry and space-t ime degrees of f reedom shows a r ichness and beauty of a lgebraic s t ruc- tu res . Seen a s Nambu mechanics , do they suggest

a l so note the analogy between dyon sys tems with ~ a n g - ~ i l l s ~ ~ and gravitational pseudoparticles; 35 Would a Nambu stat is t ical mechanics of non- Abelian tops be helpful in our understanding of the physical vacuum^?^ Our investigations a r e con- tinuing in these directions.

ACKNOWLEDGMENTS

This work w a s supported by the U.S. Energy Research and Development Administration. We would like to thank Professor Y. Nambu for dis- cussions and a reading of the manuscript . We a r e part icular ly grateful to him f o r telling us about h i s generalization of Eule r ' s equations to any Lie group. We a l so wish to thank Sid Drel l for encouragements and for h i s w a r m hospitality at SLAC.

*Permanent address: Toyama University, Toyama 930, Japan.

?Present address: Department of Physics, Yale Uni- versity, New Haven, Connecticut 06520.

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Nambu mechanics of non-Abelian dyons