L]~TT]~R]~ AL NUOVO CIMENTO VOL. 17, N. 5 2 0 t t o b r e 1976

Comments on the Nambu Mechanics.

G. J . RUGGERr

Department de l~isica, JFacultad de Ciencias, Universidad Central de Venezuela, Caracas, Venezuela (*)

( r icevuto il 2 Lugl io 1976)

NAMBU (~) p roposed some a l t e r n a t i v e fo rms of classical d y n a m i c s in wh ich t h e Liouvi l le t h e o r e m is t a k e n as t h e f u n d a m e n t a l gu id ing p r inc ip le and , consequen t ly , t h e poss ib i l i ty of c o n s t r u c t i n g a co r r e spond ing s t a t i s t i ca l m e c h a n i c s a long t h e usua l l ines is in p r inc ip le ensured . T he pu r pos e of t h i s n o t e is to m a k e some c o m m e n t s r e l a t ed to t h e s t r u c t u r e of one of t he poss ible r e su l t i ng fo rma l i sms a n d to i t s r e l a t i on to mor6 s t a n d a r d classical d y n a m i c a l theor i e s (~).

W e beg in b y a s s u m i n g a se t of evo lu t ion e q u a t i o n s of t h e fo rm ~ ~ X~(x), /~-~ 1 . . . . , n. F o r s impl ic i ty we r e s t r i c t ourse lves here , a n d in w h a t follows, to cases in

w h i c h t h e r e is no expl ic i t t i m e dependence . I t is a n e l e m e n t a r y r e su l t of t h e t h e o r y of f i r s t -order pa r t i a l d i f ferent ia l e q u a t i o n s (a) t h a t t h e Liouvi l le cond i t ion (**) O~X~(x)~ 0 is su]]icient as wel l as necessa ry for t he ex i s tence of a se t of i n d e p e n d e n t c o n s t a n t s of m o t i o n H1, H~, ..., H~_ 1 such t h a t A : X ~ z A can be p u t in t h e fo rm

(1) A = ~(A, H1 .. . . . H~_I) a ( x ~, x ~ . . . . . x~)

for a n y we l l - behaved d y n a m i c a l v a r i a b l e A. Here t h e H ' s p l a y t h e role of H a m i l t o n i a n s . E q u a t i o n (1) is one of t h e poss ib i l i t ies sugges ted b y NA~IBU who rea l ized t h a t t he Liouvi l le t h e o r e m necessar i ly follows f rom it . The fac t t h a t t h e converse is also t r ue ha s severa l i m p l i c a t i o n s : 1) I t shou ld be poss ib le to r educe some of t he o t h e r a l t e rna - t ives p roposed b y NAMBU to t h e fo rm of eq. (1) (***); 2) I t shou ld a lways b e possible to cas t t h e o r d i n a r y H a m i l t o n i a n mechan i c s in t h e N a m b u fo rm a l t h o u g h , as wil l b e c lear below, t h e fo rma l i sms h a v e assoc ia ted a lgebra ic s t r u c t u r e s wh ich are dif ferent . As shown

(*) Present address: Department of Mathematical Physics, University of Birmingham, Birmingham B15 2TT, England. (1) Y. N~..~rBu: Phys. Rev. D, 7, 2405 (1973). (~) See also h . J. K~I,.'~AY in Colloque Interrbationale de Geometric Syrnplectique et Physique Mathema- tique (Colloques Interaationaux C.N.R.S., Aix-en-Provence, France, 1974), and references cited therein. (3) See, for instance, Ch.-J. DE LA VALI~E POUSSIN: Cours d'Analyse In/initesimale (Paris, 1949). (**) The sum convention is everywhere used. (***) We arc referring to eq. (6) of ref. (1).

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170 G . J . RUGGERI

b y ~NAMBU the re is a t leas t one case, n a m e l y t he r ig id ro t a to r , in wh ich t he new scheme is a n a t u r a l one. T he r e d u c t i o n to t h e N a m b u form is also poss ible for all classical d y n a m i c s based on a Poisson- l ike b r a c k e t , r egu la r or not , because in those cases a se t of co-ord ina tes can be found in wh ich t he Liouvi l le t heo rem holds 0)- This set inc ludes D i rac ' s t h e o r y of c o n s t r a i n e d H a m i l t o n i a n sys tems.

W e now show t h a t t he b r a c k e t

~(A1, A2, ..., A , ) (2) {A1, A 2 . . . . . . .t~}_ -

~ ( x ' , x ~ . . . . . x ~)

has all t h e p roper t i e s needed for a cons i s t en t a lgebra ic i n t e r p r e t a t i o n of t h e fo rmal i sm. Consequen t ly , as a d v o c a t e d b y ~NAMBU, i t can be r i gh t l y cons idered as t h e r e l e v a n t a lgebra ic en t i t y . To th i s a im, we s t u d y t he fo rm of t he t r a n s f o r m a t i o n s x - + x ' w h i c h p rese rve t h e Liouvi l le condi t ion . E q u a t i o n (1) is t h e n va l id in t h e t w o - c o - o r d i n a t e s y s t e m w i t h two, no t necessar i ly equal , sets of H a m i l t o n i a n s . W e h a v e t h e n

~(.~'~, x '~ . . . . . x ' , , ) ~(H;, H~ . . . . . H~') (3)

~ ( x ~, x" . . . . . x '~) ~ (H~ , H2 . . . . . H~)

Clearly th i s set of t r a n s f o r m a t i o n s cons t i t u t e s a group�9 Sonic of t h e m , cal led gauge t r a n s f o r m a t i o n s b y NAMBI=, leave t he co-ord ina tes u n c h a n g e d and t r a n s f o r m the Hami l - t o n i a n s w i t h i n t he c o n s t r a i n t t ~ . H i / c H j ] ~ 1. O the r t r a n s f o r m a t i o n s can be classified accord ing to t h e b e h a v i o u r of t he H a m i l t o n i a n s . A m o n g s t t he s imples t are those for wh ich t h e y t r a n s f o r m as scalars�9 F r o m eq. (3) t h e y are t he (volume preserv ing) t r a n s - f o r m a t i o n s of j a cob i an u n i t y 0)

( 4 ) {x% x '~ . . . . . x ' " } = = 1 .

W e not ice t h a t , even if we r e s t r i c t ourse lves to those t r a u s f o r m a t i o n s w h i c h are con- t i n u o u s l y connec t ed w i t h the iden t i ty , th i s set is b igger t h a n t he co r r e spond ing subg roup in t he s t a n d a r d H a m i l t o n i a n mechan ics . Thus , as we m e n t i o n e d before, a l t h o u g h t h e H a m i l t o n i a n mechan ics can be p u t in t he form (1), t he r e su l t i ng scheme has a r i che r s t r u c t u r e (5).

The in f in i t es imal N a m b u t r a n s f o r m a t i o n s x ' s = x~ + 8x~ sa t i s fy ~g(Sxs) = 0. B y t he same a r g u m e n t used in con juc t ion w i t h eq. (1), t h e r e exis t gene ra to r s G 1, G 2 . . . . . G,_~, def ined up to a gauge t r a n s f o r m a t i o n , wh ich are such t h a t

(5) ~xt' = {xl,, G1, G2 . . . . . G,~-I) �9

E q u a t i o n (1) shows t he evo lu t ion e q u a t i o n s as t he unfo ld ing of a canon ica l t r a n s f o r m a - t ion g e n e r a t e d b y t h e H a m i l t o n i a n s . L e t us c o m m e n t now on t he a lgebra ic a spec t s of t he fo rmal i sm. To beg in w i t h we not ice t h a t t h e N a m b u b r a c k e t (2) is a l t e r n a n t , n m l t i l i n e a r and m u l t i d e r i v a t i v e {~). Then , any l inear and /o r (pointwise) mu l t i p l i c a t i ve r e l a t i on be tween d y n a m i c a l va r i ab le s will be p r e se rved d u r i n g t he evo lu t ion and , more genera l ly , u n d e r t r a n s f o r m a t i o n s (5). More i n t e r e s t i ng is t he fac t t h a t N a m b u b r a c k e t r e la t ions are also canon ica l ly s table . T h a t occurs genera l ly if and on ly if

(6) 3{A~, A 2 . . . . . A,~)__ - - { ~ A , , A 2 . . . . . An}_ + ... / {A~, A 2 . . . . . ~A,}_

(4) See for i n s t a n c e , E�9 C. G. SUDARSItAN al2d N. ~[L'KUND.4.: Classical Dynamics : A 3lode~'~ Perspective (New Y o r k , N. Y . , 1974). (~) I . COEE.~" a n d A. ft. I~ALNAY: Intern. J . Theor. P h y s . , 12, 61 (1975).

COMMENTS ON THE NAMBU MECHANICS 171

which for the canonical t ransformations is equivalent to the (~ Jacobi ident i ty

(7) {(A1 . . . . . A . } _ , B~ . . . . , B . _ ~ } _ =

= { ( A . B1 . . . . . , B._I~_, ~ . . . . . A.}_ + ... + {A1, ~ . . . . . (A . , B1 . . . . . B._~}_}_.

Due to eqs. (4) and (5) and to the derivation proper ty this ident i ty is satisfied by the Nambu bracket . The si tuation is then exact ly analogous to tha t of the s tandard Hamil tonian mechanics where the usual Jacobi ident i ty guarantees the dynamical s tabi l i ty of the Poisson bracket relations thus opening the way to define through them a consistent algebraic structure between the dynamical variables. This is crucial when a t tempt ing to quantizc the theory.

I t could happen in some cases tha t a certain set, say n - - k , of constants of motion should also be t rea ted as canonical invariants (*). The number of independent genera- tors in eq. (5) would then be reduced to k - - 1 and the bracket is effectively k-linear. The simplest possibil i ty is tha t for which this role is played by a subset of the Hamil- tonians, say H~, ..., H~_~. The k-linear bracket is then defined by

(8) {A , G1 . . . . . G,~_I} *_ =~ {A , H , . . . . . ~ . _ ~ , G~ . . . . . G~_I}_ .

Notice t h a t : 1) the null elements contr ibute to the de]inition of the bracket , and 2) cq. (7) reduces r the proper ident i ty for {, }*. If k = 2 the s tandard Jaeobi iden- t i t y is obtained so tha t {, )*_ is a Lie bracket (%

To conclude, we remark tha t the quantizat ion program for the Nambu mechanics should explicit ly incorporate ident i ty (7) within the relevant algebra(e). I t is easy to see, for example, tha t the tri l inear commuta tor proposed by NAMBV (1) as a counter- pa r t of (2) within an associative algebra does not satisfy (7). In this sense i t is not useful for defining, simultaneously, dynamical equations and algebraic relations. How- ever, this problem is avoided if the dynamics is constructed in the s tandard way, i.e. through a commutator .

The author wishes to thank Prof. A. J. K.&LNAY, in Caracas, for many st imulat ing discussions.

(*) T h a t would be t he ease, for example , i f t h e y express some cons t r a in t s be tween t he e l e m e n t a r y va r i ab l e s . (~) G. J . RUO(~Em: Intern. Journ. Theor. Phys. , I2, 287 (1975).