Quantum Mechanics as a Generalization of Nambu Dynamics to the Weyl-Wigner Formalism* **
Iwo Bialynicki-Birula3 and P.J. Morrisonb 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, 9-12 (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 brackets. Thus, a phase-space variational principle can be constructed. The purpose of this Letter is to provide a unified basis for various noncanonical Poisson brackets in- troduced in the past [1-8] and to give a new and II. Lie Algebraic Basis for Nambu Dynamics important example of such a bracket. This unification is obtained by extending the concept of the triple We begin by considering a semi-simple Lie algebra bracket introduced by Nambu [9] in his generalized with structure constants cand metric tensor gtj (see, version of Hamiltonian dynamics. Our extension of for example, [10]) the Nambu dynamics involves the replacement of the k l Lie algebra of the rotation group in three dimensions, 9ij = -c üc jk, (1) that features prominently in the original formulation, that is used to raise and lower indices. We have intro- by an arbitrary Lie algebra. When this algebra is cho- duced the minus sign here to make g^ positive for the sen to be the (infinite-dimensional) Lie algebra associ- rotation group. With Lie algebras we can associate ated with the Weyl-Wigner representation, we obtain dynamical systems whose states are described by the the phase-space formulation of quantum mechanics. elements w' L, of the Lie algebra. The w' are to be The noncanonical bracket for the Wigner function viewed as phase-space coordinates of the system, and that results from this formulation reduces in the clas- L, are the algebra generators. A natural Lie bracket sical limit to the well-known bracket for the classical can be constructed from the structure constants as distribution function. follows: Our extension of Nambu dynamics leads to a natu- . .. dA dB ral parametrization of the original noncanonical vari- (2) ables in terms of canonically conjugate pairs of new 5? 5?- variables. With this parametrization, the noncanoni- The Jacobi identity satisfied by the structure con- cal brackets are transformed into ordinary Poisson stants,
C?j Cmk + Cjk Cmi + Cki Cmj = 0, (3) * Presented at a Workshop in honor of E. C. G. Sudarshan's contributions to Theoretical Physics, held at the Univer- guarantees the Jacobi identity for the bracket {, }c, sity of Texas in Austin, September 15-17, 1991. ** Originally published in Physics Letters A 158,453 (1991); {A, {B, C}c}c + {B, {C, A}c}c + {C, {A, B}c}= 0. (4) reprinted with the kind permission of Elsevier Science Publishers B. V. Such structures were discovered by Sophus Lie (1890) Reprint requests to Prof. Dr. I. Bialynicki-Birula. 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 Nambu Dynamics ics of various continuous media [5]. Also, they have and H are distinct since S must be the Casimir in order been studied in the context of Poisson manifolds for {A, B}c to satisfy the Jacobi identity, and thus S [11, 12]. can be viewed as a kinematical object. On the other The fact that the structure constants have three hand, the Hamiltonian H is free to be chosen to gener- indices hints at the existence of a geometric bracket ate the dynamics of the system of interest. 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 III. Triple Bracket Formulation of Quantum Mechanics of the structure constants [10], Now, we generalize the triple bracket formulation LJjk „im jn e k » y mi (5) of the previous section, which is based on finite dimen- Thus we introduce the following triple bracket: sional Lie algebras, to acommodate a particular in- finite dimensional Lie algebra that we extract from the dA dB 0C [A, B, C] — c,J —.—.—r. (6) Weyl-Wigner representation [13, 14] of quantum me- OH'1 CWJ 0W chanics (For recent references see e.g. [15].) We begin by defining the following family of operators: A simple relationship exists between [A, B, C] and Hr p p T),h Hr p {A, B}c, which is made manifest by inserting the E(r', p') = Jdre ~ e '*~ ' (11) Casimir of the Lie algebra, where dF = d" r d" p/(2 n h)n, n is the number of dimen- 1 • • sions, and is used to indicate operators. The opera- S = 2 9ij w' wJ> (7) tors £(r, p) can be viewed as a basis spanning the space of all quantum mechanical operators, a basis from into one of the slots of the triple bracket, i.e. which the Weyl-Wigner representation is derived. The
[A, B,S] = {A,B}C. (8) Wigner function is obtained by projecting the density operator q onto the basis E, i.e. Due to this relationship time evolution can be repre- sented as follows: W(r, p)=Tr{Q £(r, p)}, (12)
d F e = jdrW(r, p)£(r, p). (13) — = [F, H, 5], (9) at In analogy with the finite-dimensional case, Wigner functions play the role of coordinates for the Lie alge- where F is an arbitrary dynamical variable. In this bra spanned by the operators E. In the simplest case formulation the dynamics is determined by two gener- of a pure state, we obtain Wigner's original formula, ating functions, the Hamiltonian H and the Casimir which we shall call the entropy. W(r, p)= 7>{| f>
Nambu chose 5 to be the rotational kinetic energy l (ih)~ [E(z1),E(z2)] = \dr3C(zl,z2,z3)E(z3), (15) and H to be the Casimir, the square of the total angu- lar momentum. In this way he obtained Euler's equa- where the "structure kernel" C(zx, z2, z3) is given by tions from his triple bracket (10). It is interesting to note that for this case the roles of the Hamiltonian and 2-4" 2 C(z ,z ,z )= - sin -{[z z ] + [z z ] + [z z ]), the entropy can be interchanged as exemplified by l 2 3 1 2 2 3 3 l Nambu's choice. However, this is a peculiarity of the (16) and rotation group for which any function S in (8) pro- duces a genuine Lie bracket. In general, the roles of S [z,. Zj] = ri • p;- —Pi • Tj. (17) I. Bialynicki-Birula and P. J. Morrison • Quantum Mechanics as a Generalization of Nambu Dynamics 11
We have chosen the name structure kernel since C{zl, Similarly, there is a natural choice for the entropy Sf, jk z2,z3), like c' , is antisymmetric under the inter- since the group of transformations generated by the change of its arguments, structure kernel C has a continuous version of the Casimir used in Section II. Indeed, owing to the anti- C(z ,z ,z ) = -C(z ,z ,z ) = C(z ,z ,z ), (18) 1 2 3 2 1 3 2 3 1 symmetry of C, and satisfies the Jacobi identity, y}c =0 (27) jdr[C(z ,z ,z)C(z,z ,z') + C(z ,z ,z)C(z,z ,z') 1 2 3 2 3 1 for all functionals stf if
+ C(z3,z1,z)C(z,z2,z')] = 0. (19) 6^=\\dr W2(z). (28) The structure kernel C(zl, z2, z3) defines a Lie alge- bra for phase-space functions as follows: Equations (23)-(25) and (28) generate the usual (cf. e.g.
{A, B}M(z) = \dri dr2 C(z, zl,z2)A(z1)B(z2). (20) [15, 16]) time evolution of the Wigner function, i.e. dW The subscript M is used since this bracket can be - = -{W, H}M (29) shown to be equivalent to In the classical limit, when h 0, the Moyal {A, B} (t, p)= \a{ r, p) sin * (S • g -6 • 0 ) B( r, p), M r p p r bracket reduces to the standard Poisson bracket and (21) the bracket (22) for functionals of Wigner functions which is the usual form of the Moyal bracket [16]. This reduces to that for classical distribution functions. algebra generates a bracket on functionals which is an Such brackets were previously introduced [3, 8] to cast the Vlasov equation into Hamiltonian form. 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 a fds/ d&l = drwl W- , —} . JJ law/' A w ( In this section we introduce a general method for dW(z2) dW(z3) ' [dW'dW\M expressing noncanonical brackets of the form (2) and Now, in complete analogy with Sec. II, we intro- (22) in terms of canonically conjugate variables. The duce the triple bracket, method is a generalization of that used to express the Vlasov equation in canonical Hamiltonian form [4], < {*/, ^}=Jdr1 dr2 dr3 c(Zl,z2,z3) The new canonical variables, which may be viewed as d^ a® generalized Clebsch potentials, are related to the orig- (23) inal phase-space coordinates by the following expres- ' aw(zt) aw(z2) dw(z3)' sions, for finite and infinite dimensional cases, respec- where s4, 3b, and ^ are arbitrary functionals of W. tively: Following Nambu we obtain the evolution equation w' = 4V Pj, (30) for a functional Jjr[ W] in terms of our triple bracket in the form W(z)=$drC(z,zl,z2)Q(zl)P(z2). (31)
dJ^ Using the chain rule je, (24) ~dt a f a f a f _ a f ik 7 4V- Pk (32) The natural choice for the Hamiltonian is Jtf, the dpj a w ' energy of the system, and the Jacobi identity, the following relationship be- J?[W)=\drW(r, p) H(r, p), (25) tween the two brackets is obtained: dA aB dA aB where A(w(q, p)), ß(w(g, p))}qp(q, p)= — l — - — — 3q dPi dPi 8q{ H(r, p)=JZ- + V(r). (26) ..dA dB 2m k r C^ä^"(/l(vv)'ß(vv)}f(w)- (33) 12 I. Bialynicki-Birula and P. J. Morrison • Quantum Mechanics as a Generalization of Nambu Dynamics
Similarly, V. Conclusion
{s/[W[Q, P]L®[W[Q, P]]}QP[Q,P] We have generalized Nambu dynamics to include / 5 s/ m 5 s/ hffl \ arbitrary Lie algebras, even of infinite dimension. An = J r \5Q(zj 5P(z) ~~ 5P(z) 5£(z)/ important example of this construction is quantum mechanics in the Weyl-Wigner representation. Hence,
= Jdr1dr2dr3 ^(z1)C(z1,z2,z3) already quantized theories fit into this framework and 6W/(Z ) 5VF(Z ) 2 3 the quantization attempted by Nambu appears to be (34) unnecessary. We have shown how noncanonical brack- ets arise in a natural way from our triple bracket that Since the canonical brackets generate the equations are based on structure constants of Lie algebras. Fur- of motion in the usual way, we can also derive these ther, we have given a general construction for intro- equations from the phase-space action principle. In ducing canonical variables with the corresponding particular, for the Wigner function the action is phase-space action principle. In this way we have ob-
S[Q,P]=$dt{SDRP(z,t)dTQ(z,t) (35) tained the action principle for the Wigner function. The authors would like to acknowledge the hos- -Jdr1dr2dr3H(z1)C(z1,z2,z3)ö(z2,r)P(z3,o}, pitality of the Aspen Center for Physics where this where Q and P are varied independently. If Q and P work was started and completed. PJM was sup- satisfy the resulting Hamiltonian equations, the W ported by the U.S. Department of Energy Contract constructed from (31) obeys (29). No. DE-FG05-80ET-53088.
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