\noindent We briefly discuss some algebraic and geometric aspects of the
generalized Poisson bracket and non--commutative phase space for generalized
quantum dynamics, which are analogous to properties of the classical Poisson
bracket and ordinary symplectic structure.Comment: 10pages,revtex, IASSNSHEP-93/5

E. Witten

R. Abraham

S. Okubo

S.L. Adler

Stephen L. Adler

V.I. Arnold

Yong-Shi Wu

English

arXiv

journal articleWe briefly discuss some algebraic and geometric aspects of the generalized Poisson bracket and noncommutative phase space for generalized quantum dynamics, which are analogous to properties of the classical Poisson bracket and ordinary symplectic structure

Wu, Yong-Shi

Adler, Stephen L.

The University of Utah: J. Willard Marriott Digital Library

PHYSICAL REVIEW D VOLUME 49, NUMBER 12 15 JUNE 1994 Algebraic and geometric aspects of generalized quantum dynamics Stephen 1. Adler Institute for Advanced Study, Princeton, New Jersey 08540 Yong-Shi Wu Department of Physics, University of Utah, Salt Lake City, Utah 84112 (Received 1 November 1993) We briefly discuss some algebraic and geometric aspects of the generalized Poisson bracket and noncommutative phase space for generalized quantum dynamics, which are analogous to properties of the classical Poisson bracket and ordinary symplectic structure. PACS number(s): 11.10.Ef, 03.65.Ca Recently, one of us (S.L.A.) proposed a generalization of Heisenberg picture quantum mechanics, termed gen-erolized quantum dynamics, which gives a Hamiltonian dynamics for general noncommutative degrees of free-dom [1,2]. The formalism permits the direct derivation of equations of motion for field operators, without first pro-ceeding through the intermediate step of "quantizing" a classical theory. In a complex Hilbert space, generalized quantum dynamics gives results compatible with stan-dard canonical quantization. It is also applicable to the construction of quantum field theories in quaternionic Hilbert spaces, where canonical methods fail, basically because the matrix elements of operators are themselves elements of the noncommutative quaternion algebra. It is hoped that the methods of generalized quantum dy-namics will facilitate answering the question of whether quantum field theories in quaternionic Hilbert space are relevant to the unification of the standard model forces with gravitation at energies above the grand unified the-ory (GUT) scale. As applied to quantum theory, generalized quantum dynamics is formulated by defining a Hilbert space V H (based either on complex number or quaternionic scalars) which is the direct sum of a bosonic space vjj and a fermionic space VIi. Next, following Witten [3]' one de-fines an operator (_l)F with eigenvalue +1 for states in vjj and -1 for states in VIi. Finally, one needs a trace operation Tr 0 for a general operator 0, defined by n It is easy to show that the trace Tr vanishes for operators o which anticommute with (_l)F, and so Tr 0 acts nontrivially only on the part of 0 which commutes with (_l)F. Let {qr (t)} be a finite set of time-dependent quan-tum variables, which act as operators on the underlying Hilbert space, with each individual qr of either bosonic or fermionic type, defined respectively as commuting or an-ticommuting with (_l)F. No other a priori assumptions about commutativity of the qr are made. The Lagrangian L [{ qr }, {qr }] is then defined as the trace of a polyno-mial function of {qr(t)} and its time derivative {qr(t)}, or as a suitable limit of such functions. The action S 0556-2821/94/49( 12)/6705(4)/$06.00 49 is defined as the time integral of L, and generalizations of the Euler-Lagrange equations follow from the require-ment that dS = 0 for arbitrary (same-type) variations of the operators. Derivatives of L with respect to qr and qr are defined by writing the variation of L, for infinitesimal variations in the {qr}, in the form (2) where cyclic permutations of operators inside Tr have been used to order dqr and dqr to the right. The momen-tum Pr conjugate to qr is defined by (3) and the Hamiltonian H is given by H = Tr L Prqr - L. (4) r In complete analogy with the Lagrangian derivatives defined in Eq. (2), for a general trace functional A, con-structed as the trace Tr of a (bosonic) polynomial func-tion of operator arguments, one can define unique deriva-tive dA/dqr with respect to the operator qr (and of the same bosonic or fermionic type as qr) by the relation (5) Again, cyclic invariance of the trace has been used to re-order all dqr factors to the right in the respective terms in which they occur. Using this derivative, one can then define generalized Poisson brackets, as follows. Let {qr}, {Pr} be the set of operator phase space variables in-troduced above, which for each r are either both bosonic or both fermionic, in the sense that they commute or anticommute with (-1) F. Again, no further a priori as-sumptions are made about their commutativity. If we now let A[{qr}, {Pr}] and B[{qr}, {Pr}] be two trace functionals of their arguments, then the generalized Pois-son bracket {A, B} is defined by {A, B} = Tr [L: Cr (dA 8B _ 8B dA)], (6) r 8qr dPr 8qr dPr 6705 ©1994 The American Physical Society 6706 STEPHEN L. ADLER AND YONG-SHI WU 49 with Er = +1(-1) according to whether qr and Pr are bosonic (fermionic). Using the generalized bracket, the time development of a general trace functional A[{qr}, {Pr}, t] takes the form [1,2] dA = aA {A H} dt at + , , (7) with H the total trace Hamiltonian. It was conjectured in Refs. [1] and [2] that the generalized bracket obeys the Jacobi identity, 0= {A, {B, C}} + {C, {A, B}} + {B, {C, A}}, (8) and this conjecture has recently been proved by Adler, Bhanot, and Weckel [4]. The key observation is that de-spite the absence of both commutativity and the product rule and the lack of a definition for the double derivative, pair~ise cancellations still occur in the right-hand side of Eq. (8) because of cyclic permutability inside the trace Tr. The proof of Eq. (8) is, in fact, independent of the Hilbert space arena on which the operators {qr} act. All that is used are the definition of derivative ofEq. (5), and the assumptions that operator multiplication is associa-tive, and that there exists a graded trace Tr permitting cyclic permutation of noncommuting operator variables, according to the formula (9) with the + (-) sign holding when 0(1) and 0(2) are both bosonic (fermionic). Evidently the generalized bracket of Eq. (6) can be viewed as an extension of the classical Poisson bracket, which permits the introduction of noncommuting phase space variables {qr}, {Pr}. Our aim in this note is to document a number of further algebraic and geometric properties of noncommutative phase space, which closely relate to the existence of the generalized Poisson bracket that satisfies the Jacobi identity of Eq. (8), but which do not enter into the proof given in Ref. [4]. The first of these involves the algebraic structure of the trace functionals, under the product operation used to construct the antisymmetric bracket of Eq. (6). Letting A and B be any two trace functionals defined on phase space, a product A 0 B that remains a trace functional can be defined by [ JA JB] A 0 B == Tr ~ Er Jqr JPr ' (10) in terms of which the generalized Poisson bracket takes the form of a commutator: {A,B}=AoB-BoA. (11) The algebra Ao of trace functionals under the product 0 can now be characterized in terms of the standard clas-sification [5] of nonassociative algebras. It is associative if the associator (A, B, C) defined by (A, B, C) == (A 0 B) 0 C - A 0 (B 0 C) (12) vanishes. It is flexible if the associator obeys (A,B,C) = -(C,B,A), (13) and it is Lie admissible if the associator obeys 0= (A,B,C) - (A,C,B) + (B,C,A) -(B,A,C) + (C,A,B) - (C,B,A). (14) Evidently, any associative algebra is Lie admissible, but the converse is of course not true. Now by substituting Eq. (12) into Eq. (14) and rearranging using Eq. (11), we find that Eq. (14) is equivalent to 0= {A,{B,C}} + {C,{A,B}} + {B,{C,A}}, (15) which is true by virtue of the Jacobi identity for the gen-eralized Poisson bracket. To see that Eq. (12) does not vanish and that Eq. (13) does not hold, it suffices to consider the special case in which the variables {qr} and {Pr} are commuting (bosonic) c numbers. This is just the classical case in which {A, B} is proportional to the stan-dard Poisson bracket, and a simple calculation of multiple derivatives (see, e.g., Ref. [5], Sec. 7.3) shows that both the vanishing of Eq. (12) and the identity of Eq. (13) are false for the product defined by Eq. (10). Hence the algebra Ao is neither associative nor flexible, and there-fore is only of secondary interest. But as in the case of its classical analog, Ao is Lie admissible by virtue of the Jacobi identity, and hence the resulting Lie structure de-fined by Eq. (11) is of primary importance. Thus, the trace functionals form a Lie algebra under the general-ized Poisson bracket of Eq. (11) and, in particular, the total trace conserved symmetry generators that commute with the total trace Hamiltonian form a Lie subalgebra [4]. The second aspect to be discussed relates to the tan-gent vector fields associated with the generalized dynam-ics. Let X A be the tangent vector field associated with a trace functional A, defined as a formal derivative opera-tor by (16) and defined operationally by its action on any trace func-tional B, (17) with (XAB) given by In terms of this operator, the time development of a gen-eral trace functional B [{ qr}, {Pr}], under the dynamics 49 ALGEBRAIC AND GEOMETRIC ASPECTS OF GENERALIZED ... 6707 governed by A as total trace Hamiltonian, can be rewrit-ten as [ef. Eq. (7)] dB dt = -(XAB). (19) Thus the tangent vector field XA can be viewed as (mi-nus) the directional derivative along the time evolution orbit (called the phase flow in Ref. [6]) of the point ( {qr }, {Pr }) in phase space, which is determined by the Hamiltonian equations of motion [1] dqr 8A dt = lOr 8Pr ' (20) with A acting as the total trace Hamiltonian. Following Ref. [6], we call a tangent vector field of the form of Eq. (16) a Hamiltonian vector field, the same name as for its classical counterpart. We note that with respect to the product defined by Eq. (10), the directional derivative XA does not obey the Leibniz product rule: (It is easy to verify that the same is true in the classical case.) However, it does obey the Leibniz product rule for the generalized Poisson bracket or the commutator defined by Eq. ( 11 ) , because, in view of Eq. (14), this equation is equivalent to the Jacobi identity of Eq. (8). What is the algebraic structure of the Hamiltonian vec-tor fields? Let us compute the action of the commutator of two tangent vector fields XA and XB on a third trace functional C: ([XA' XB]C) = (XA(XBC» - (XB(XAC» = {A, {B,C}} - {B,{A,C}} = {A, {B, C}} + {B, {C, A}}. (23) Using Eq. (14) with A replaced by {A,B} and B re-placed by C, we also get (X{A,B}C) = {{A,B},C}, (24) and subtracting Eq. (24) from Eq. (23) gives finally «[XA,XB]- X{A,B})C) = {A, {B, C}} + {B, {C,A}} + {C, {A, B}} = O. (25) Hence validity of the Jacobi identity for the general-ized Poisson bracket implies that the Hamiltonian vector fields X A defined by Eqs. (16)-(18) obey the commuta-tor algebra (26) and, therefore, form a Lie algebra that is isomorphic to the Lie algebra of trace functionals under the general-ized Poisson bracket, which is the generalized quantum dynamics analogue of a standard result [6] in classical mechanics. Finally, we address the geometric structure underly-ing generalized quantum dynamics. As is well known, there is a geometry which underlies classical Hamiltonian dynamics, namely the symplectic geometry of ordinary phase space. Can we generalize symplectic geometry to noncommutative phase space? If a generalized symplec-tic structure exists, is it preserved by phase space flows (or Hamiltonian time evolutions) as in classical mechan-ics [6]? In the following we present a discussion of these questions with affirmative answers, which is readable to physicists who are not familiar with differential forms [7]. Ordinary symplectic geometry is defined by a stan-dard (constant) antisymmetric metric in the tangent or cotangent spaces of a phase space. (By way of contrast, Riemannian geometry, which is perhaps more familiar to physicists, is defined by a symmetric metric in the tan-gent or cotangent spaces of a manifold.) To avoid differ-ential forms, let us consider the cotangent space, which is known to be spanned by covariant vectors whose com-ponents form the gradient (or differential) of a function on phase space. The standard (antisymmetric) symplec-tic metric, or the inner product, between two covariant vectors that are the gradients of two classical functions A(qr,Pr) and B(qr,Pr) on phase space, is provided by the classical Poisson brackets {A, B}. In a noncommu-tative phase space, the analogues of functions are trace functionals, and the analogues of the differentials of func-tions are the differentials of trace functionals, i.e., Eq. (5) adapted to phase space: With the generalized Poisson brackets of Eq. (6) avail-able, we can use it to define a generalized symplectic structure rl on the noncommutative phase space, through defining the inner product between two cotangent vectors 8A and 8B as follows: rl(8A,8B) = {A, B} [ (8A 8B 8B 8A)] == Tr L lOr - - - - - • (28) 8qr 8Pr 8qr 8Pr r To see that such a symplectic structure is preserved by any Hamiltonian phase flow of Eq. (20), we observe that the time derivative of the inner product along the phase-flow orbit is d d dt rl(8B,8C) = dt {B, C} = {{B, C}, A}, (29) while that of the differential 8B along the same flow is d . dt 8B == {jB (30) where the dot abbreviates the time derivative. Therefore, we have 6708 STEPHEN L. ADLER AND YONG-SHI WU 49 n(8S, 8C) + n(8B, 86) = {S, C} + {B, 6} = {{B, A}, C} + {B, {C, A}}. (31) Therefore the Jacobi identity of Eq. (8) implies d . . dt n(8B,8C) = n(8B, 8C) + n(8B, 8C); (32) that is, the symplectic structure is invariant under Hamil-tonian phase flow. This statement can be viewed as a (dual) form of the generalized quantum dynamics ana-logue of the Liouville theorem. Thus, generalized quantum dynamics, albeit with non-commuting operator phase space variables, has an under-lying generalized symplectic geometry which is preserved by the time evolution generated by any total trace Hamil-tonian. Basically this is due to the existence of a (graded) trace Tr that permits cyclic permutation of noncommut-ing operator variables, which implies the validity of the Jacobi identity for the generalized Poisson bracket. As in classical mechanics, we expect that the basic concepts and theorems of generalized quantum dynamics will be invariant under the group of symplectic transformations, [1] S.L. Adler, Nucl. Phys. B415, 195 (1994). [2] S.L. Adler, Quaternionic Quantum Mechanics and Quan-tum Fields (Oxford University Press, Oxford, in press). [3] E. Witten, J. Diff. Geom. 17, 661 (1982). [4] S.L. Adler, G.V. Bhanot, and J.D. Weckel, J. Math. Phys. 35, 531 (1994). [5] S. Okubo, Introduction of Octonion and other Non-Associative Algebras in Physics (Cambridge University i.e., under transformations which preserve the general-ized symplectic structure. To conclude, we have seen that in many algebraic and geometric aspects, the generalized quantum dynamics proposed in Refs. [1] and [2] is analogous to classical mechanics. It is really surprising that with the help of a cyclically permutable (graded) trace alone, so many features of classical mechanics can be generalized to a noncommutative phase space. (We remind readers once more that in Ref. [1] and in our present discussion, no phase space variable commutation relations such as com-mutativity, anticommutativity, or q commutators are as-sumed.) Further developments in generalized quantum dynamics, paralleling to some extent aspects of existing quantization schemes, are expected. The authors wish to thank F.J. Dyson for a discussion with S.L.A., and acknowledge the hospitality of the As-pen Center for Physics, where this work was done. S.L.A. was supported in part by the Department of Energy un-der Grant No. DE-FG02-90ER40542. Y.-S.W. was sup-ported in part by the National Science Foundation under Grant No. PHY-9309458. Press, Cambridge, England, in press). [61 V.l. Arnold, Mathematical Methods of Classical Mechan-ics (Springer-Verlag, New York, 1978), p. 211; R. Abra-ham and J.E. Marsden, Foundations of Mechanics, 2nd ed. (Benjamin/Cummings, Reading, MA, 1980), p. 194. [7] A more mathematical treatment, using the notions of gen-eralized differential forms in noncommutative phase space, may be possible. This is left for a future publication. 

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