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Quantum Mechanics as a Generalization of Nambu Dynamics to the Weyl-Wigner Formalism* ** Iwo Bialynicki-Birula3 and P.J. M o r r i s o n b

a Institute for Theoretical Physics, Polish Academy of Sciences, Lotnikow 32/46, 02-668, Warsaw, Poland

b Department of Physics and Institute for Fusion Studies, University of Texas at Austin, Texas, 78712, USA

Z. Naturforsch. 52a, 9 - 1 2 (1997)

It is shown that Nambu dynamics can be generalized to any number of dimensions by replacing the 0(3) algebra, a prominent feature of Nambu's formulation, by an arbitrary Lie algebra. For the infinite dimensional algebra of rotations in phase space one obtains quantum mechanics in the Weyl-Wigner representation from the generalized Nambu dynamics. Also, this formulation can be cast into a canonical Hamiltonian form by a natural choice of canonically conjugate variables.

I. Introduction

The purpose of this Letter is to provide a unified basis for various noncanonical Poisson brackets in-t roduced in the past [ 1 - 8 ] and to give a new and impor tan t example of such a bracket . This unification is obtained by extending the concept of the triple bracket in t roduced by N a m b u [9] in his generalized version of Hami l ton ian dynamics. O u r extension of the N a m b u dynamics involves the replacement of the Lie algebra of the ro ta t ion g roup in three dimensions, that features prominent ly in the original formula t ion , by an arbi t rary Lie algebra. When this algebra is cho-sen to be the (infinite-dimensional) Lie algebra associ-ated with the Weyl-Wigner representat ion, we obta in the phase-space formula t ion of q u a n t u m mechanics. The noncanonical bracket for the Wigner funct ion that results f rom this formula t ion reduces in the clas-sical limit to the well-known bracket for the classical distr ibution function.

O u r extension of N a m b u dynamics leads to a na tu-ral parametr iza t ion of the original noncanonica l vari-ables in terms of canonically con juga te pairs of new variables. With this parametr iza t ion , the noncanoni -cal brackets are t ransformed into ord inary Poisson

* Presented at a Workshop in honor of E. C. G. Sudarshan's contributions to Theoretical Physics, held at the Univer-sity of Texas in Austin, September 15-17, 1991.

** Originally published in Physics Letters A 158,453 (1991); reprinted with the kind permission of Elsevier Science Publishers B. V.

Reprint requests to Prof. Dr. I. Bialynicki-Birula.

brackets. Thus, a phase-space variat ional principle can be constructed.

II. Lie Algebraic Basis for Nambu Dynamics

We begin by considering a semi-simple Lie algebra with structure constants cand metric tensor gtj (see, for example, [10])

9ij = -ckücl

jk, (1)

that is used to raise and lower indices. We have intro-duced the minus sign here to make g^ positive for the rota t ion group. With Lie algebras we can associate dynamical systems whose states are described by the elements w' L, of the Lie algebra. The w' are to be viewed as phase-space coordinates of the system, and L, are the algebra generators. A na tura l Lie bracket can be constructed f rom the s tructure constants as follows:

. .. dA dB

5? 5?- (2)

The Jacobi identity satisfied by the s t ructure con-stants,

C?j Cmk + Cjk Cmi + Cki Cmj = 0 , (3)

guarantees the Jacobi identity for the bracket {, }c ,

{A, {B, C}c}c + {B, {C, A}c}c + {C, {A, B}c}= 0 . (4)

Such structures were discovered by Sophus Lie (1890) and are known to generate classical Eulerian dynam-

0932-0784 / 97 / 0100-0009 S 06.00 © - Verlag der Zeitschrift für Naturforschung, D-72072 Tübingen

10 I. Bialynicki-Birula and P. J. Morrison • Quantum Mechanics as a Generalization of N a m b u Dynamics

ics of various continuous media [5]. Also, they have been studied in the context of Poisson manifolds [11, 12].

The fact that the structure constants have three indices hints at the existence of a geometric bracket operation on three functions. It would be appealing if all three functions appeared on equal footing, which can be achieved by using the fully antisymmetric form of the structure constants [10],

Jjk „im jn k L » y e mi

Thus we introduce the following triple bracket:

dA dB 0C [A, B, C] — c,J —.—.—r.

OH'1 C W J 0 W

(5)

(6)

A simple relationship exists between [A, B, C] and {A, B}c, which is made manifest by inserting the Casimir of the Lie algebra,

1 • • S = 2 9ij w ' wJ> (7)

into one of the slots of the triple bracket, i.e.

[A, B,S] = {A,B}C. (8)

Due to this relationship time evolution can be repre-sented as follows:

d F — = [F, H, 5] , at

(9)

where F is an arbitrary dynamical variable. In this formulation the dynamics is determined by two gener-ating functions, the Hamiltonian H and the Casimir which we shall call the entropy.

In the special case where the structure constants are those of the rotation group in three dimensions, cijk = 8ijk, our construction reduces to that given by Nambu [9],

[A, ß, C] = VX • (VBx VC). (10)

In order to describe the dynamics of a rigid rotator Nambu chose 5 to be the rotational kinetic energy and H to be the Casimir, the square of the total angu-lar momentum. In this way he obtained Euler's equa-tions from his triple bracket (10). It is interesting to note that for this case the roles of the Hamiltonian and the entropy can be interchanged as exemplified by Nambu 's choice. However, this is a peculiarity of the rotation group for which any function S in (8) pro-duces a genuine Lie bracket. In general, the roles of S

and H are distinct since S must be the Casimir in order for {A , B}c to satisfy the Jacobi identity, and thus S can be viewed as a kinematical object. On the other hand, the Hamiltonian H is free to be chosen to gener-ate the dynamics of the system of interest.

III. Triple Bracket Formulation of Quantum Mechanics

Now, we generalize the triple bracket formulation of the previous section, which is based on finite dimen-sional Lie algebras, to acommodate a particular in-finite dimensional Lie algebra that we extract from the Weyl-Wigner representation [13, 14] of quantum me-chanics (For recent references see e.g. [15].) We begin by defining the following family of operators:

E(r', p') = J d r e H r p~p T),h eHr'*~p' (11)

where d F = d" r d" p/(2 n h)n, n is the number of dimen-sions, and is used to indicate operators. The opera-tors £(r, p) can be viewed as a basis spanning the space of all quantum mechanical operators, a basis from which the Weyl-Wigner representation is derived. The Wigner function is obtained by projecting the density operator q onto the basis E, i.e.

W(r, p)=Tr{Q £(r, p)},

e = j d r W ( r , p)£(r, p).

(12)

(13)

In analogy with the finite-dimensional case, Wigner functions play the role of coordinates for the Lie alge-bra spanned by the operators E. In the simplest case of a pure state, we obtain Wigner's original formula,

W(r, p)= 7>{ | f > < f | £ ( r , p)} = <«P|£(r, p ) | f >

= Jd" s e^"! 's 'p,h xjj (r + s/2) ip* (r - s/2). (14)

Here we have chosen to work with a dimensionless Wigner function, which requires a normalization that differs from that of other authors (cf. [15]).

The basis operators E(z) (z = (r, p)) obey the follow-ing commutator algebra:

(ih)~l [E(z1),E(z2)] = \dr3C(zl,z2,z3)E(z3), (15)

where the "structure kernel" C(z x , z2, z3) is given by

2 - 4 " 2 C(zl,z2,z3)= - sin - { [ z 1 z 2 ] + [z2 z3] + [z3zl]),

(16) and

[z,. Zj] = ri • p;- —Pi • Tj. (17)

I. Bialynicki-Birula and P. J. Morrison • Quan tum Mechanics as a Generalization of Nambu Dynamics 11

Similarly, there is a natural choice for the entropy Sf, since the group of t ransformat ions generated by the structure kernel C has a cont inuous version of the Casimir used in Section II. Indeed, owing to the anti-symmetry of C,

y } c =0 for all functionals stf if

6^=\\dr W2(z).

(27)

(28)

We have chosen the name structure kernel since C{z l , z2,z3), like c'jk, is ant isymmetric under the inter-change of its arguments ,

C(z1,z2,z3) = -C(z2,z1,z3) = C(z2,z3,z1), (18)

and satisfies the Jacobi identity,

jdr[C(z1,z2,z)C(z,z3,z') + C(z2,z3,z)C(z,z1,z')

+ C(z3,z1,z)C(z,z2,z')] = 0 . (19)

The structure kernel C(zl, z2, z3) defines a Lie alge-bra for phase-space functions as follows:

{A, B}M(z) = \dri d r 2 C(z, zl,z2)A(z1)B(z2). (20)

The subscript M is used since this bracket can be shown to be equivalent to

{A, B}M(t, p ) = \ a { r, p) s in * (Sr • g p - 6 p • 0 r ) B( r, p),

(21)

which is the usual form of the Moyal bracket [16]. This algebra generates a bracket on functionals which is an infinite dimensional analogue of the bracket {A, B}C

defined by (2),

{sf,%}c = \dridr2dr3 W(zl)C(z1,z2,z3) (22) IV. Transformation to a Canonical Hamiltonian Form

Equat ions (23)-(25) and (28) generate the usual (cf. e.g. [15, 16]) time evolution of the Wigner function, i.e.

dW - = - { W , H}M (29)

In the classical limit, when h 0, the Moyal bracket reduces to the s tandard Poisson bracket and the bracket (22) for functionals of Wigner functions reduces to that for classical distribution functions. Such brackets were previously introduced [3, 8] to cast the Vlasov equation into Hamil tonian form.

a fds/ d&l = drwl , — } . J l a w / ' A w ( W-

dW(z2) dW(z3) J ' [dW'dW\M

Now, in complete analogy with Sec. II, we intro-duce the triple bracket,

{*/, < ^ } = J d r 1 d r 2 d r 3 c(Zl,z2,z3)

d^ a ® ' aw(zt) aw(z2) dw(z3)'

(23)

where s4, 3b, and ^ are arbi t rary functionals of W. Following N a m b u we obtain the evolution equat ion for a functional J j r [ W] in terms of our triple bracket in the form

In this section we introduce a general method for expressing noncanonical brackets of the form (2) and (22) in terms of canonically conjugate variables. The method is a generalization of that used to express the Vlasov equation in canonical Hamil tonian form [4], The new canonical variables, which may be viewed as generalized Clebsch potentials, are related to the orig-inal phase-space coordinates by the following expres-sions, for finite and infinite dimensional cases, respec-tively:

w' = 4 V P j , (30)

W(z)=$drC(z,zl,z2)Q(zl)P(z2). (31)

dJ^

~dt je, (24)

Using the chain rule

a f a f

The natural choice for the Hamil tonian is Jtf, the energy of the system,

dpj a w 7 4 V -

a f _ a f ik Pk'

(32)

J?[W)=\drW(r, p) H(r, p),

where

H(r, p)=JZ- + V(r). 2m

(25)

(26)

and the Jacobi identity, the following relationship be-tween the two brackets is obtained:

dA aB dA aB A(w(q, p)), ß(w(g, p))}qp(q, p)= — — - — —

3ql dPi dPi 8q{

k ..dA dB r C ^ ä ^ " ( / l ( v v ) ' ß ( v v ) } f ( w ) - ( 3 3 )

12 I. Bialynicki-Birula and P. J. Morrison • Quantum Mechanics as a Generalization of Nambu Dynamics

Similarly,

{s/[W[Q, P]L®[W[Q, P]]}QP[Q,P]

/ 5 s/ m 5 s/ h f f l \ = J r \5Q(zj 5P(z) ~~ 5P(z) 5£(z)/

= J d r 1 d r 2 d r 3 ^ ( z 1 ) C ( z 1 , z 2 , z 3 ) 6 W / ( Z 2 ) 5VF(Z 3 )

(34)

Since the canonical brackets generate the equat ions of mot ion in the usual way, we can also derive these equat ions f rom the phase-space action principle. In part icular, for the Wigner funct ion the action is

S[Q,P]=$dt{SDRP(z,t)dTQ(z,t) (35)

- J d r 1 d r 2 d r 3 H ( z 1 ) C ( z 1 , z 2 , z 3 ) ö ( z 2 , r ) P ( z 3 , o } ,

where Q and P are varied independently. If Q and P satisfy the resulting Hami l ton ian equations, the W constructed f rom (31) obeys (29).

V. Conclusion

We have generalized N a m b u dynamics to include arbi trary Lie algebras, even of infinite dimension. An impor tan t example of this const ruct ion is q u a n t u m mechanics in the Weyl-Wigner representat ion. Hence, already quant ized theories fit into this f ramework and the quant iza t ion a t tempted by N a m b u appears to be unnecessary. We have shown how noncanonical brack-ets arise in a na tu ra l way f rom our triple bracket that are based on s t ructure constants of Lie algebras. Fur -ther, we have given a general const ruct ion for intro-ducing canonical variables with the corresponding phase-space action principle. In this way we have ob-tained the act ion principle for the Wigner funct ion.

The au thors would like to acknowledge the hos-pitality of the Aspen Center for Physics where this work was started and completed. P J M was sup-ported by the U.S. Depa r tmen t of Energy Cont rac t No. DE-FG05-80ET-53088.

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