Features of Moyal trajectoriesNuno Costa Dias and João Nuno Prata Citation: Journal of Mathematical Physics 48, 012109 (2007); doi: 10.1063/1.2409495 View online: http://dx.doi.org/10.1063/1.2409495 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/48/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Quantum dynamics in phase space: Moyal trajectories 2 J. Math. Phys. 54, 012105 (2013); 10.1063/1.4773229 Characteristic particle trajectories for an eigenfunction J. Math. Phys. 53, 122107 (2012); 10.1063/1.4770045 Response to “Comment on ‘Bohmian mechanics with complex action: A new trajectory-based formulation ofquantum mechanics’” [J. Chem. Phys.127, 197101 (2007)] J. Chem. Phys. 127, 197102 (2007); 10.1063/1.2798762 Multidimensional quantum trajectories: Applications of the derivative propagation method J. Chem. Phys. 122, 164104 (2005); 10.1063/1.1884606 Moyal–Nahm equations J. Math. Phys. 40, 2539 (1999); 10.1063/1.532713

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Features of Moyal trajectoriesNuno Costa Diasa� and João Nuno Pratab�

Departamento de Matemática, Universidade Lusófona de Humanidades e Tecnologias,Avenida Campo Grande, 376, 1749-024 Lisboa, Portugal and Grupo de Física Matemática,Universidade de Lisboa, Avenida Professor Gama Pinto 2, 1649-003 Lisboa, Portugal

�Received 3 July 2006; accepted 16 November 2006; published online 31 January 2007�

We study the Moyal evolution of the canonical position and momentum variables.We compare it with the classical evolution and show that, contrary to what iscommonly found in the literature, the two dynamics do not coincide. We prove thatthis divergence is quite general by studying Hamiltonians of the form p2 /2m+V�q�. Several alternative formulations of Moyal dynamics are then suggested. Weintroduce the concept of star function and use it to reformulate the Moyal equationsin terms of a system of ordinary differential equations on the noncommutativeMoyal plane. We then use this formulation to study the semiclassical expansion ofMoyal trajectories, which is cast in terms of a �order by order in �� recursivehierarchy of �i� first order partial differential equations as well as �ii� systems offirst order ordinary differential equations. The latter formulation is derived inde-pendently for analytic Hamiltonians as well as for the more general case of locallyintegrable ones. We present various examples illustrating these results. © 2007American Institute of Physics. �DOI: 10.1063/1.2409495�

I. INTRODUCTION

In recent years there has been a lot of work devoted to noncommutative structures in math-ematics and physics.1,2 In most cases one deals with the Moyal product, which first appeared in thecontext of deformation quantization.3–6 If we consider the position and momentum operatorsgenerators of the Heisenberg algebra A, then we can define the associated universal envelopingalgebra in quantum phase space U��A�, where c functions are multiplied by resorting to a twistedconvolution which admits a formal � expansion �We shall only consider one-dimensional systemsin this work. The generalization to higher dimensions is straightforward.�,

f�q,p��g�q,p� =�exp� i�

2� �

�q

�

�p�−

�

�p

�

�q�� f�q,p�g�q�,p���

�q�,p��=�q,p�, �1�

or an alternative kernel representation valid in the set of phase space square integrable functionsL2�R2 ,d��, d�=dqdp,7

f�q,p��g�q,p� =1

����2 dq�dp�dq�dp�f�q�,p��g�q�,p��

�exp�−2i

��p�q� − q�� + p��q� − q� + p��q − q���� . �2�

This is the so-called Moyal �or Groenewold� product. One can also construct a Lie algebraicbracket—the Moyal bracket—according to

a�Electronic mail: ncdias@mail.telepac.ptb�Electronic mail: joao.prata@ulusofona.pt

JOURNAL OF MATHEMATICAL PHYSICS 48, 012109 �2007�

48, 012109-10022-2488/2007/48�1�/012109/23/$23.00 © 2007 American Institute of Physics

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�f�q,p�,g�q,p�� =2

��sin��

2� �

�q

�

�p�−

�

�p

�

�q�� f�q,p�g�q�,p���

�q�,p��=�q,p�. �3�

These provide the basic structure for the formulation of quantum mechanics in the phase space.7–18

One of the remarkable properties of these expressions is the fact that they constitute formaldeformations of the usual commutative product and of the Poisson bracket, respectively,

f�q,p��g�q,p� = f�q,p� · g�q,p� + O��� ,

�f�q,p�,g�q,p�� = f�q,p�,g�q,p�� + O��2� . �4�

The quantum phase space endowed with the Moyal bracket admits a Heisenberg evolution for�real� c functions according to

A�q,p,t� = �A�q,p,t�,H�q,p���q,p�, A�q,p,t = t0� = A0�q,p� , �5�

where H�q , p� is the Hamiltonian symbol �taken to be real and � independent�. In particular, forthe fundamental variables �q , p�, we have

Q�q,p,t� = �Q�q,p,t�,H�q,p���q,p�, P�q,p,t� = �P�q,p,t�,H�q,p���q,p�, �6�

subject to the initial conditions,

Q�q,p,t = t0� = q, P�q,p,t = t0� = p . �7�

The subscript �q , p� in Eq. �6� stresses the fact that the Moyal bracket is to be evaluated withrespect to the canonical set �q , p�. From Eqs. �6� and �7� we may then write at time t= t0,

Q�q,p,t = t0� =�

�pH�q,p�, P�q,p,t = t0� = −

�

�qH�q,p� . �8�

Equations �8� are reminiscent of the classical Hamilton equations,

Q�q,p,t� =�

�PH�Q,P�, P�q,p,t� = −

�

�QH�Q,P� . �9�

It is commonly claimed that, because of Eq. �8�, the fundamental �q , p� evolve classically. Thedifference between Moyal and classical evolutions would then only become apparent once func-tions of q’s and p’s are considered. One of the aims of this paper is to prove that this is not true.

The subject of quantum trajectories has a long history. The topic can be traced back to Dirac’scelebrated book19 where the unitary transformations in quantum mechanics were considered to bethe analogs of canonical transformations in classical mechanics. Since then the subject developedin several directions. One of the most well-known approaches to quantum trajectories stems fromthe de Broglie–Bohm formulation of quantum mechanics.20–22 In this approach the time evolutionof the fundamental variables �q , p� is determined by a modified Hamiltonian that includes a state�i.e., wave function� dependent quantum potential. One of the most interesting features of the deBroglie–Bohm formulation is that it casts quantum mechanics in a causal form.22 On the otherhand, the main approach to semiclassical dynamics is based on average values of the form

qQ�t� = ���q�t���� and pQ�t� = ���p�t���� ,

where q�t� and p�t� are the Heisenberg time evolution of the fundamental variables. The study ofthe behavior of qQ and pQ has been an active field of research �see, for instance, Refs. 23 and 24�with emphasis in the case where � is a coherent state25 �or a squeezed state�.26,27 One can provethat for a general quadratic Hamiltonian, a coherent state wave packet follows the classical tra-jectory, while in the case of interactions, the quantum flows �qQ�t� , pQ�t�� are well approximated

012109-2 N. C. Dias and J. N. Prata J. Math. Phys. 48, 012109 �2007�

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by the classical trajectories during the Ehrenfest time.26–30 Recently, the relation between theaverage value of the fundamental variables in a coherent state and the semiclassical trajectoriesstemming from effective actions in quantum field theory was investigated.31 Closely related to thetopic of semiclassical dynamics and by itself an interesting field of research is the investigation onformal and mathematical aspects of unitary transformations in quantum mechanics �see Ref. 32and references therein�.

A third alternative approach to quantum dynamics is that of Moyal trajectories �Eqs. �6� and�7��. These are just the phase space representations of the Heisenberg evolution of the fundamentalvariables and, contrary to the two previous �better known� approaches, they are, by construction,state independent. In spite of this the Moyal trajectories are closely related both to the deBroglie–Bohm33 as well as to the average value approach to semiclassical dynamics �see Ref. 31and below�. Moreover, Moyal dynamics provides an alternative �classical-like� characterization ofthe action of unitary transformations in quantum mechanics. The subject of Moyal dynamics hasbeen scarcely investigated. Furthermore, the topic is subtle and there is some controversy in theexisting literature. A very common mistake is the claim that Moyal and classical trajectoriescoincide for the fundamental variables, a statement that can be found in several recently publishedpapers and which is wrong except for a small class of Hamiltonians.

One of the oldest papers in this field is Ref. 9 where the difference between classical andMoyal dynamics for general observables is pointed out, although no concrete statement is madeconcerning the canonical variables. In Ref. 34 the statement that, in the Moyal framework, Hamil-ton’s equations for the fundamental variables �Eq. �8�� should be solved in the noncommutativealgebra U��A� �in which case they will determine a time evolution that differs from the onestemming from Eq. �9�� is made explicit. The formulation of Weyl-Wigner quantum mechanics inthe Heisenberg picture �a key ingredient for the discussion of Moyal trajectories� has been studiedin many papers �see, for instance, Refs. 14 and 35�. One of the most important papers in the fieldof Moyal dynamics is Ref. 35, where the Moyal correction to the classical Hamiltonian equationsis derived order by order in � by using a cluster-graph representation of the star exponential aswell as a recursive hierarchy of classical transport equations �see also Refs. 36 and 37�. We shallreview this method in Sec. III A.

There are two major motivations to study Moyal dynamics: First, it provides an interestingand accurate semiclassical correction to classical evolution.34 Second, Moyal trajectories areclosely related to other approaches to semiclassical dynamics for which they may provide impor-

tant insights. In Ref. 31 we proved that the expectation value of an observable A �typically theposition operator� in a state with density matrix � is given by

A�t� � Tr�A�t��� = �F�− i�

�z�AM�z,t��

z=0. �10�

In the previous equation z= �q , p�, � /�z= �� /�q ,� /�p�, AM�z , t� is the Moyal evolution of A�z� �inother words, the Weyl symbol of the Heisenberg operator A�t�� which is the solution of Eq. �5� and

F�a� is the symplectic Fourier transform �or chord function� of the Wigner function F�z�,38–40

F�a� = dzF�z�ei��a,z�, ��a,z� = aqp − apq , �11�

where � is the symplectic form. If the state of the system is described by a coherent state withwave function

���x� = �m

���1/4

exp�−m

2��x − q�2 +

ip

��x −

q

2�, � =�m

2�q +

ip�2m�

, �12�

then Eq. �10� reduces to

012109-3 Features of Moyal trajectories J. Math. Phys. 48, 012109 �2007�

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A�t� = exp� �

4m

�2

�q2 +�m

4

�2

�p2AM�q,p,t� . �13�

This expression is particularly well suited for the semiclassical expansions in powers of Planck’sconstant. It is well known that expectation values such as Eq. �13� �or more generally Eq. �10��play an important role in quantum optics8 and in determining effective actions in quantum me-chanics and quantum field theories.41,42 Moreover the method of trajectories finds important ap-plications in quantum hydrodynamics23 and transport models in quantum chemistry and heavy-ioncollisions43 and it is also expected to be crucial if one aims at fully grasping the nonlocal natureof noncommutative quantum field theories.1

The subject of trajectories generated by a bracket on a symplectic �or more generally aPoisson manifold� with a star product is a notoriously difficult one. The present work is devotedto the simpler case of the flat phase space of a one dimensional system T*M �R2 with a globalDarboux chart and symplectic form =dq∧dp. Our results are developed using noncommutativephase-space methods while never resorting to operator methods. In this paper we will demonstratethe following.

�i� Prove that the claim about the coincidence of Moyal and classical trajectories is false formost Hamiltonians. We will also identify the specific features of Moyal trajectories whichmake them distinct from their classical �Hamiltonian� counterparts, namely, the facts that,in general, Moyal evolution does not act as a coordinate transformation �i.e., phase-spacefunctions do not transform as scalars under Moyal evolution� nor as a symplectomorphismin phase space.

�ii� Introduce the concept of star function and use it to formulate Moyal dynamics in terms ofa system of two first order ordinary differential equations in the noncommutative phasespace. These equations are reminiscent of the classical Hamiltonian equations and preservetheir functional form through time evolution.

�iii� Formulate Moyal dynamics in terms of a � hierarchy of first order partial equations.�iv� Formulate a � hierarchy of systems of first order ordinary differential equations. This result

will be obtained for the general case of locally integrable Hamiltonians.�v� Study in detail the Hamiltonians of the form H= p2 /2m+V�q� �with V�q� a smooth func-

tion� and derive explicit expressions for the Moyal corrections to the classical evolution.

Point �i� is included to clarify some aspects of Moyal trajectories which should be known buthave been often misinterpreted in the literature. Point �iii� is a review of some previous work35

which is included for completion. The other points are new results which are of conceptual �point�ii�� and more practical �points �iv� and �v�� relevance.

This paper is organized as follows. We start by recapitulating some well known features ofclassical dynamics, namely, that classical evolution is a canonical and a coordinate transformation,and that the Hamiltonian is a constant of motion when not explicitly time dependent44,45 �Sec. II�.We then show how these facts translate to Moyal dynamics �Sec. II�. In Sec. III we propose newformulations of Moyal equations and new strategies for solving them if not completely then atleast formally or semiclassically �in a � power series�. All the results will be illustrated withexamples �Sec. IV�. The Hamiltonian of the form H= p2 /2m+V�q� will be studied in detail�Sec. V�.

II. CLASSICAL AND MOYAL DYNAMICS

Some of the results of this section are well known. We present them here for completeness andto highlight the differences between Moyal and classical evolutions. We will give simple proofs ofthe results whenever we feel that there is some subtlety that should be emphasized. We shall usethe subscripts C and M for classical and Moyal evolutions, respectively. Before we begin, let usestablish our conventions concerning the various transformations which will appear in this paper.This is an important point as one can find at times contradictory terminology in the literature. Letthen T : �q , p�→ �Q=Q�q , p� , P= P�q , p�� be some diffeomorphism in phase space. We shall say

012109-4 N. C. Dias and J. N. Prata J. Math. Phys. 48, 012109 �2007�

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that T acts as a “coordinate transformation” in the algebra of complex valued functions in phasespace, if every element A�q , p� of that algebra transforms as �we adopt the active point of view�

A�q,p� → A��q,p� = A � T�q,p� = A�Q�q,p�,P�q,p�� .

The transformation T will be denoted a “symplectomorphism” if it is a coordinate transformationwhich leaves the Poisson bracket unchanged,

q,p��q,p� = Q�q,p�,P�q,p���q,p� = 1.

In Moyal mechanics we shall say that T is a “canonical transformation,” if the Moyal bracket ispreserved,32

�q,p��q,p� = �Q�q,p�,P�q,p���q,p� = 1.

There is a subclass of canonical transformations which plays a significant role in Moyal dynamics,namely, the “unitary transformations,”

Q�q,p� = U�q,p��q�U*�q,p�, P�q,p� = U�q,p��p�U*�q,p� ,

where U�q , p� is such that

U�q,p��U*�q,p� = U*�q,p��U�q,p� = 1.

We shall have more to say about these different transformations in due course.One possible formulation of classical dynamics is in terms of a classical transport equation.

Let then AC�q , p , t� be the solution of the first order partial differential equation,

AC�q,p,t� = AC�q,p,t�,H�q,p���q,p�, �14�

subject to the initial condition

AC�q,p,t = t0� = A0�q,p� . �15�

Then it is easy to prove the following.Theorem 2.1: The solution is

AC�q,p,t� = A0�QC�q,p,t�,PC�q,p,t�� , �16�

where �QC�q , p , t� , PC�q , p , t�� are the solutions of Hamilton’s equations �Eq. �9�� with initialconditions �Eq. �7��. By resorting to Jacobi’s identity and Theorem 2.1, we can also show thefollowing.

Theorem 2.2: The transformation

�q,p� → �Q = QC�q,p,t�,P = PC�q,p,t�� �17�

is a symplectomorphism, i.e.,

A�QC�q,p,t�,PC�q,p,t��,B�QC�q,p,t�,PC�q,p,t����q,p�

= A�QC�q,p,t�,PC�q,p,t��,B�QC�q,p,t�,PC�q,p,t����QC,PC�, �18�

for general phase-space functions A ,B.Lemma 2.3: The Hamiltonian is a constant of motion.Proof: For an arbitrary observable we have

AC�q,p,t� = AC�q,p,t�,H0�q,p���q,p� � F�q,p,t� , �19�

where we introduced the notation H0�q , p�=H�q , p , t= t0�. Differentiating the previous equationwith respect to time, we obtain

012109-5 Features of Moyal trajectories J. Math. Phys. 48, 012109 �2007�

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F�q,p,t� = AC,H0� + AC,H0� = F�q,p,t�,H0�q,p���q,p�. �20�

From Theorem 2.1 it then follows

F�q,p,t� = F�QC�q,p,t�,PC�q,p,t�,t = t0� . �21�

In particular, let us choose A=H. Then, since F�q , p , t= t0�=0, we conclude from the previousanalysis that F�q , p , t�=0 and from Eq. �19�,

H�q,p,t� = H�q,p,t�,H0�q,p���q,p� = 0. �22�

An obvious corollary of the previous result is the following. �

Corollary 2.4: The functional form of the Hamiltonian remains unchanged in the course ofclassical evolution,

H�QC�q,p,t�,PC�q,p,t�� = H�q,p� . �23�

The next theorem is the crux of the matter when we compare classical and Moyal evolutions. Itstates that the equations of motion are the same at any time t.

Theorem 2.5: The partial differential equations �Eq. �14�� for the position and momentum areequivalent to the set of ordinary differential equations,

Q = Q,H��Q,P� =�H

�P, P = P,H��Q,P� = −

�H

�Q. �24�

Proof: We prove the statement for Q. From Eq. �14�, Theorem 2.2, Lemma 2.3, and itscorollary we get

Q�q,p,t� = Q�q,p,t�,H�q,p���q,p� = Q�q,p,t�,H�Q�q,p,t�,P�q,p,t����q,p�

= Q�q,p,t�,H�Q�q,p,t�,P�q,p,t����Q,P� ⇔ Q = Q,H�Q,P���Q,P�, �25�

where we omitted the subscript C for simplicity and used Eq. �18�. �

Remark 2.6: Notice that this presentation of classical dynamics is somewhat unusual. Weposited the evolution to be given by the classical transport equation �Eq. �14�� even if AC standsfor the position or the momentum. For instance,

QC�q,p,t� = QC�q,p,t�,H�q,p�� =�QC

�q

�H

�p−

�QC

�p

�H

�q, QC�q,p,t = t0� = q , �26�

and a similar equation for PC�q , p , t�. This is a partial differential equation. It is only throughTheorem 2.5 that we show the coincidence of the solutions of the partial differential equation �Eq.�26�� and of the system �Eq. �24�� of ordinary differential equations. We have deliberately chosenthis presentation to render the passage to Moyal dynamics more transparent. As a word of cautionwe stress the double role played by the position and momentum in this approach. They aresimultaneously the variables with respect to which one performs the partial derivatives in, say, Eq.�14� as well as the initial conditions �Eq. �7��.

Let us now turn to Moyal dynamics ��6� and �7��. The following is well known.Theorem 2.7: The formal solution of the Moyal equation is given by

AM�q,p,t� = U�t��A0�q,p��U*�t�, U�t� � e�i�t−t0�H�q,p�/�. �27�

Here �� ,q , p��e��B�q,p� is the noncommutative exponential solution of

�

��= B� = �B, �� = 0,q,p� = 1, ∀ �q,p� � T*M � R2. �28�

012109-6 N. C. Dias and J. N. Prata J. Math. Phys. 48, 012109 �2007�

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Remark 2.8: Contrary to what happens in classical mechanics �cf. Eq. �16��, Moyal evolutiondoes not act as a coordinate �local� transformation. In general,

AM�q,p,t� � A0�QM�q,p,t�,PM�q,p,t�� . �29�

The exception are flows generated by Hamiltonians which are at most polynomials of degree 2 in�q , p�. We shall construct an explicit example which corroborates Eq. �29� in due course �seeRemark 4.4 below�.

An immediate consequence of Theorem 2.7 is the analog of Lemma 2.3.Lemma 2.9: The Hamiltonian is conserved under Moyal evolution.The following, well known result, is the Moyal counterpart of Theorem 2.2.Theorem 2.10: Moyal evolution,

�q,p� → �Q = QM�q,p,t�,P = PM�q,p,t�� , �30�

being a unitary transformation, preserves the Moyal bracket,

�QM�q,p,t�,PM�q,p,t���q,p� = 1. �31�

Theorem 2.10 states that it is the Moyal bracket rather than the Poisson bracket that is preservedunder time evolution. Let us clarify this point. There are, obviously, transformations which areboth canonical and symplectomorphisms.

Example 2.11: A transformation which is both canonical and a symplectomorphism. Let usconsider the symplectic transformation,

Q�q,p� = aq + bp, P�q,p� = cq + dp , �32�

where

det�a b

c d� = 1. �33�

This transformation is both canonical as well as a symplectomorphism,

�Q,P��q,p� = Q,P��q,p� = 1. �34�

However, there are transformations which are canonical but not symplectomorphisms and viceversa.

Example 2.12: A transformation which is canonical but not a symplectomorphism. Let usdefine the transformation

Q�q,p� = �eq/�, P�q,p� = e−q/��p + � sinh�2��p

�� , �35�

where � and � are positive real constants with dimensions of length and momentum, respectively.This transformation is invertible by the inverse function theorem. The Jacobian reads

��Q,P���q,p�

= Q,P��q,p� = 1 +2���

�cosh�2��p

�� 1. �36�

We conclude that this is not a symplectomorphism. On the other hand we have

Q�q,p�� ��

�q

��

�p−

��

�p

��

�q�2n+1

P�q,p� = ���2�

��2n+1

cosh�2��p

��, n � 1, �37�

where the derivatives �� and �� act on Q and P, respectively. Consequently,

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�Q,P��q,p� = Q,P��q,p� + �n=1

��− 1�n

�2n + 1�!��

2�2n

Q�q,p�� ��

�q

��

�p−

��

�p

��

�q�2n+1

P�q,p�

= 1 +2��

�cosh�2��p

��sin � = 1, �38�

and so this is a canonical transformation.As the latter example shows, canonical transformations are not symplectomorphisms in gen-

eral. Likewise, symplectomorphisms are not canonical transformations in general �see Example 1below�. Therefore, one should expect Moyal and classical trajectories to diverge. Indeed, this isstated in the following theorem.

Theorem 2.13: At an arbitrary instant t� t0 we have, in general, for Q=QM and P= PM,

Q�t� � �Q�t�,H�Q,P���Q,P� =�

�PH�Q,P�, P�t� � �P�t�,H�Q,P���Q,P� = −

�

�QH�Q,P� .

�39�

Proof: For A0�q , p�=q the solution of Eq. �5� takes the form �cf. Eq. �27�� Q�q , p , t�=U�t��q�U*�t�, and so

Q�q,p,t� = U�t���q,H�q,p���q,p��U*�t� = U�t��Fq�q,p��U*�t� , �40�

where Fq�q , p�=�H /�p. Unless Fq�q , p� is linear, we have �cf. Remark 2.8�,

U�t��Fq�q,p��U*�t� � Fq�Q�q,p,t�,P�q,p,t�� . �41�

Thus, in general,

Q�t� ��

�PH�Q,P� . �42�

A similar argument holds for PM�q , p , t�. �

Before ending this section let us try to clarify the following apparent paradox. Infinitesimaltransformations in phase space fully determine finite transformations. Moreover, Moyal and clas-sical �Hamiltonian� infinitesimal transformations coincide. Hence Moyal and classical trajectoriesshould be identical. However, it follows from the previous theorem that this statement is false. Infact, infinitesimal classical transformations �determined by the Hamiltonian equations �Eq. �9���coincide with infinitesimal Moyal transformations �determined by Eq. �8�� at the initial time only�with the obvious exceptions of quadratic Hamiltonians�. For later times Eq. �39� shows explicitlythat the two infinitesimal transformations do not coincide.

III. OTHER APPROACHES TO MOYAL DYNAMICS

The main conclusion of the previous section is that the partial differential equations �Eq. �6��governing Moyal dynamics are not equivalent to the system of ordinary differential equations �Eq.�9��. Since Eq. �6� of arbitrary order �possibly infinite� the problem of solving them exactly isnotoriously difficult. In this section we investigate various alternative formulations of Moyalmechanics. One obvious approach is that of evaluating semiclassical corrections �order by order in�� to the classical solutions. In Sec. III A Moyal dynamics is written in terms of a � hierarchy ofrecursive first order linear partial differential equations. This hierarchy has been previously pre-sented in Ref. 35. Here we rederive it in other �possibly simpler� terms. In Sec. III B Moyalequations are rewritten as a system of ordinary differential equations in the space of noncommu-tative phase-space functions. This is an interesting formulation allowing one, in some cases, tosolve Moyal equations exactly �see Sec. IV�. Furthermore, as a by-product, Moyal dynamics willbe written in terms of a � hierarchy of systems of ordinary first order linear and inhomogeneous

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differential equations in Secs. III C and III D. In a first step this result is derived for analyticHamiltonians �Sec. III C� and then generalized to the case of locally integrable ones �Sec. III D�.In this section we shall write QM =Q and PM = P for simplicity.

A. A hierarchy of first order partial differential equations

If we regard Moyal evolution as a partial differential equation, then we can easily derive anequivalent infinite hierarchy of linear inhomogeneous partial differential equations. Indeed letQ�q , p , t� and P�q , p , t� be the solutions of Eqs. �6� and �7�. A key feature of these solutions is thatthey display a regular behavior on � and admit a formal asymptotic series,

Q�q,p,t� = �n=0

+�

�nQ�n��q,p,t�, P�q,p,t� = �n=0

+�

�nP�n��q,p,t� , �43�

where, obviously, Q�0��q , p , t� and P�0��q , p , t� are the classical solutions. A simple analysis revealsthat only the even order terms are nonvanishing, provided the Hamiltonian is independent of �. Ifwe substitute Eq. �43� in the Moyal equations, we get the classical Hamilton equations for n=0and

Q�2n� = �k=0

n

�Q�2k�,H�2�n−k� = Q�2n�,H� + �k=0

n−1

�Q�2k�,H�2�n−k�, n � 1,

P�2n� = �k=0

n

�P�2k�,H�2�n−k� = P�2n�,H� + �k=0

n−1

�P�2k�,H�2�n−k�, n � 1, �44�

where we used the notation

�A�q,p�,B�q,p��2n = � �− 1�n

�2n + 1�!4n� �

�q

�

�p�−

�

�p

�

�q��2n+1

A�q,p�B�q�,p����q�,p��=�q,p�

. �45�

Here Q�2n��t� and P�2n��t� satisfy the initial conditions,

Q�2n��q,p,t = t0� = P�2n��q,p,t = t0� = 0, ∀ n � 1, ∀ �q,p� � R2. �46�

The important thing to remark in Eq. �44� is the fact that they constitute an infinite hierarchy oflinear inhomogeneous partial differential equations. For each order 2n they are in fact a system oftwo independent Hamiltonian transport equations with inhomogeneous terms,

Fq�2n��q,p,t� = �

k=0

n−1

�Q�2k�,H�2�n−k� and Fp�2n��q,p,t� = �

k=0

n−1

�P�2k�,H�2�n−k�, �47�

exclusively dependent on the solutions of order 0 ,2 , . . . ,2n−2. The solution of Eq. �44� is then35

Q�2n��q,p,t� = t0

t

d�Fq�2n��Q�0��q,p,��,P�0��q,p,��,�� ,

P�2n��q,p,t� = t0

t

d�Fp�2n��Q�0��q,p,��,P�0��q,p,��,�� . �48�

B. Ordinary differential equations in the noncommutative phase space

We start by introducing the concept of � function. Intuitively, these are functions whoseexpressions are written in terms of � products, i.e., the fundamental operations used to express the

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� functions are summations, multiplication by scalars and � products. The most interesting featureof these functions is that their functional form is preserved through Moyal time evolution. Hencethey provide the right objects to cast Eqs. �8� in an invariant form.

The main question is then what kind of functions f :T*M→C can be written as � functions. Tobegin with the simplest case, let us consider an analytic function N�q , p� in the phase-spacevariables. It is trivial to check that34 �let O1=q and O2= p�

N�q,p� = �n=0

�1

n! �1�i1,. . .,in�2

� �nN

�Oi1, . . . ,�Oin

��q,p�=�0,0�

�Oi1� ¯ �Oin

�S � �N�q,p��S, �49�

where S stands for total symmetrization. We used the fact that

and introduced the notation �N�q , p��S to denote the � function associated with N�q , p� which iswritten in a completely symmetric form. Notice that the objects �N�q , p��S are naturally charac-terized as elements of the universal enveloping algebra U��A� of the Heisenberg-Weyl Lie algebra,i.e., as series in the noncommutative coordinates.

Let us now extend the concept of � function beyond the case of phase-space analytic func-tions. We start by noticing that an arbitrary locally integrable function F�q , p� admits a �general-ized� Fourier representation,

F�q,p� = d� d����,��ei�q+i�p, �50�

in terms of possibly distributional coefficients ��� ,��. The set of functions of the form �50�comprises the set of all phase-space analytic functions as well as the set of square integrablefunctions �in fact formula �50� provides a representation for generalized functions46�.

All functions of the form �50� can now be written as � functions,

�F�q,p��S = d� d����,��e�i�q+i�p. �51�

For example, if F�q , p�=q2p, then we have ��� ,��=−i����������. It then follows that �F�q , p��S

= �q�q�p+q�p�q+ p�q�q� /3. Notice that the identity F�q , p�= �F�q , p��S follows immediatelyfrom

e�i�q+i�p = e�

i�q�e�i�pe�i�/2��� = ei�q+i�p, �52�

where we used the Baker-Campbell-Hausdorff �BCH� formula.To proceed we notice that � functions can be cast in many different functional forms which is

a direct consequence of ordering ambiguities �e.g., q�p= p�q+ i��. An important alternative to thecomplete symmetrization is the standard-antistandard symmetrization that we shall use in Sec. IVto study a specific example. In this case we write a combination of a term, where all the q’s standto the left and its conjugate. For F�q , p�=q2p, we have �F�q , p��SAS= �q�q�p+ p�q�q� /2. This canbe expressed as

�F�q,p��SAS =1

2 d� d����,��sec��

2

�2

�q�p��e�

i�q�e�i�p + e�

i�p�e�i�q� . �53�

Lemma 3.1: For a phase-space function satisfying Eq. �50�, the following holds:

�F�q,p��SAS = F�q,p� . �54�

Proof: Again using the BCH formula, we have

012109-10 N. C. Dias and J. N. Prata J. Math. Phys. 48, 012109 �2007�

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1

2 d� d����,���e�

i�q�e�i�p + e�

i�p�e�i�q� =

1

2 d� d����,��e�

i�q+i�p�e−i���/2 + ei���/2�

= d� d����,��ei�q+i�p cos����

2�

= cos��

2

�2

�q�p�F�q,p� , �55�

from where the identity �53� follows. �

Now let us go back to the equations of motion. At time t= t0 Eqs. �8� hold. However, they areonly valid at t= t0. Let U�t� be the symbol of the evolution operator and let �Fq�q , p��O be the �function associated with Fq�q , p�=�H /�p and written in the order O=S or O=SAS. We then have

Q�t� = �Q�t�,H��q,p� = U�t���Q�0�,H��q,p��U*�t� = U�t���Fq�q,p��O�U*�t� = �Fq�Q�t�,P�t���O,

�56�

and a similar equation for the momentum. The important thing to remark is the fact that theequations of motion Eq. �56� are now valid for all times in some interval �t0−� , t0+�� �� 0�. Wemay now regard these equations as a system of ordinary differential equations for the position andmomentum. We just have to bear in mind that we have to look for solutions in the set of �functions. This is a subtle point that deserves some further discussion. Consider the differentialequation �Eq. �28��

�

�tE�t� = H�E�t� ,

where H is the Hamiltonian. Strictly speaking this is a partial differential equation because the �product hides partial derivatives with respect to q and p. However, it may be useful to regard it asan ordinary differential equation in the algebra of � functions. That is, as a differential equationthat displays derivatives with respect to the singular variable t except for the ones which areincluded in the product of the algebra. By looking for solutions in the noncommutative algebra wefind, in analogy with the commutative case, that

E�t� = e�tH,

which is the best known � function. The expression of the � exponential is very useful, bothconceptually as well as in a more practical sense. There are a number of powerful methods14,35 todetermine the exact form of � exponentials �i.e., its expression as a function in the commutativealgebra� as well as its semiclassical expansions which �depending on the method� may not implythe resolution of the partial differential equation above.

Let us go back to our Eq. �56�. In analogy with the defining equation for the � exponential, thefact that Eq. �56� may be seen as an ordinary differential equation in the noncommutative algebrais conceptually appealing. However, in some cases it is also useful from a more practical point ofview. The example of Sec. IV will illustrate this point. We will see that it is possible to solve Eq.�56� without having to deal with the partial differential equation but instead �i� by solving it as anordinary differential equation in the noncommutative phase space �by analogy with the resolutionof the corresponding commutative equation�, finding its � solution, and �ii� translating this �function to the space of commutative functions.

C. A hierarchy of ordinary differential equations for analytic Hamiltonians

As a direct application of Eq. �56� we now study the �-semiclassical expansion of the Moyalevolution of position and momentum �Eq. �43��. Let us consider the case where the HamiltonianH�q , p� is a real analytic function, independent of �. In this case �Fq�q , p��S can be written as

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�Fq�q,p��S = �n,m=0

�

Cn,m�qnpm�k� , �57�

where k=n+m−1 for �n ,m�� �0,0� and k=0 for n=m=0, Cn,m are the Taylor coefficients, and

�58�

Using this notation Eq. �56� is written as

Q = �n,m=0

�

Cn,m�QnPm�k� . �59�

Let us then expand this equation in powers of �. We consider the expansions �43� and introducethe additional notation,

� = �n=0

+�

�n�n, A�q,p��nB�q,p� =1

n!� i

2�n� �

�q

�

�p�−

�

�p

�

�q��n

�A�q,p�B�q�,p����q�,p��=�q,p�.

�60�

It then follows that

�QnPm�k� = �r=0

�

�r�QnPm�k�r, �61�

where

�QnPm�k�r = ��

�=1

ni�+ �

�=1

mj�+ �

�=1

kl�=r

i�,j�,l�� 0,. . .,r�

�Q�i1��l1Q�i2��l2

, . . . ,�ln−1Q�in��ln

P�j1��ll+1, . . . ,�lk

P�jm��S, �62�

and so the leading order equation �Eqs. �59� and �62�� is the classical Hamiltonian equation, andto order O��r�, r�1 it reads �cf. Eq. �43��

Q�r� = �n,m=0

�

Cn,m�QnPm�k�r = �n,m=0

�

Cn,m�nQ�r��Q�0��n−1�P�0��m + m�Q�0��nP�r��P�0��m−1�

+ fr�Q�0�, . . . ,Q�r−1�,P�0�, . . . ,P�r−1�� , �63�

together with the initial conditions,

Q�r��t = t0� = P�r��t = t0� = 0, r � 1, �64�

and the inhomogeneous term is

fr�Q�0�, . . . ,Q�r−1�,P�0�, . . . ,P�r−1��

= �n,m=0

�

Cn,m ��

�=1

ni�+ �

�=1

mj�+ �

�=1

kl�=r

i�,j�� 0,. . .,r−1�;l�� 0,. . .,r�

�Q�i1��l1Q�i2��l2

, . . . ,�ln−1Q�in��ln

P�j1��ln+1, . . . ,�lk

P�jm��S.

�65�

A similar set of equations is valid for P and together with Eq. �63� they constitute a � hierarchyof systems of two first order ordinary differential equations. Finally, it is easy to verify that the

012109-12 N. C. Dias and J. N. Prata J. Math. Phys. 48, 012109 �2007�

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initial conditions will impose the odd contributions to the semiclassical expansions of Q , P to beidentically zero.

D. The hierarchy of ordinary differential equations in the general case

We now generalize the previous approach to locally integrable Hamiltonians �still real andtime, � independent�.

Theorem 3.2: Let the Hamiltonian be locally integrable. Then its partial derivatives admit a�possible generalized� Fourier representation as in Eq. �50�. Then in the space of formal series����� the Moyal evolution of position and momentum is dictated by a hierarchy of systems of twoinhomogeneous first order linear differential equations.

Proof: Let us write z= �q , p� and Z= �Q , P�. The Moyal equations at time t= t0 read

Zi�t = t0� = Fi, Fi = Jij� jH�z� , �66�

where sum over repeated indices is understood, H is the Hamiltonian, and we used the compactnotation �i= �� /�q ,� /�p�, Jqq=Jpp=0, and Jqp=−Jpq=1. Under the assumptions of the theorem His locally integrable and so it makes sense as a �regular� distribution.46 Its partial derivatives arealso well-defined distributions and so admit a �generalized� Fourier representation. Hence, thevector function F may be written in the form

F�z� = d�f���ei�·z, �67�

where �= ��q ,�p�, � ·z=q�q+ p�p, and, in general, f��� is also a distribution. Following the proce-dure discussed in Sec. III B, we may write at time t� t0,

Z�t� = d�f���e�i�·Z�t�. �68�

In Ref. 15 we proved that the noncommutative exponential may be expanded in the form

e�B = �1 + �2A2 + �4A4 + ¯ �eB, �69�

where A2 ,A4 , . . . depend only on B�z� and its derivatives. For instance,

A2�B� = − JikJjl��i� jB�� 1

16��k�lB� +

1

24��kB���lB� . �70�

It should be noted that B may also depend upon �.At this point an important remark is in order. The dependence on the variables z= �q , p� seems

to have vanished. However, this is only apparently so. Indeed, in Eq. �68� the function Z�t� in theexponent depends on time, as well as on z, as it is subject to the initial condition,

Z�t = t0� = �Q�t = t0�, P�t = t0�� = �q,p� = z .

We should thus effectively write Z=Z�z , t� instead of Z=Z�t�. However, we shall only keep theexplicit time dependence, in order not to overburden the notation. Nevertheless, this dependenceshould be kept in mind. In particular, the noncommutative exponential in Eq. �69� is a function ofz, where B�z�� i� ·Z�z , t�.

If we expand the Moyal trajectories

Z�t� = Z�0��t� + �2Z�2��t� + �4Z�4��t� + ¯ , �71�

as well as the coefficients of the exponential �notice that these coefficients should not be seen asa function of Z, but rather as a “functional” of the function Z�z , t� and its derivatives �iZ,�i� jZ , . . .as illustrated in Eq. �70�� �Eq. �69��,

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A2k�Z� = A2k�0��Z�0�� + �2A2k

�2��Z�0�,Z�2�� + �4A2k�4��Z�0�,Z�2�,Z�4�� + ¯ , k = 1,2, . . . , �72�

we get from Eq. �68�,

Z�0��t� + �2Z�2��t� + �4Z�4��t� + �6Z�6��t� + ¯

= d�f��� ei�·�Z�0�+�2Z�2�+�4Z�4�+�6Z�6�+¯�

+ �2�A2�0��Z�0�� + �2A2

�2��Z�0�,Z�2�� + �4A2�4��Z�0�,Z�2�,Z�4�� + ¯ �ei�·�Z�0�+�2Z�2�+�4Z�4�+¯�

+ �4�A4�0��Z�0�� + �2A4

�2��Z�0�,Z�2�� + ¯ �ei�·�Z�0�+�2Z�2�+¯� + �6�A6�0��Z�0�� + ¯ �ei�·�Z�0�+¯� + ¯ � .

�73�

If we expand the right-hand side of the previous equation to order O��6�, we obtain

d�f���ei�·Z�0� 1 + �2��i� · Z�2�� + A2�0�� + �4��i� · Z�4�� + 1

2 �i� · Z�2��2 + A2�0��i� · Z�2�� + A2

�2� + A4�0��

+ �6��i� · Z�6�� + �i� · Z�2���i� · Z�4�� + 112�i� · Z�2��3 + A2

�0��i� · Z�4�� + 12A2

�0��i� · Z�2��2 + A2�2�

��i� · Z�2�� + A2�4� + A4

�0��i� · Z�2�� + A4�2� + A6

�0��� .

Equating the terms order by order in � with those in the left-hand side of Eq. �73�, we get

Zr�0��t� = Fr�Z�0��t�� , �74�

Zr�2��t� = Zi

�2��t��

�Zi�0�Fr�Z�0��t�� + d�fr���ei�·Z�0��t�A2

�0��Z�0��t�� , �75�

Zr�4��t� = Zi

�4��t��

�Zi�0�Fr�Z�0��t�� +

1

2Zi

�2��t�Zj�2��t�

�

�Zi�0�

�

�Zj�0�Fr�Z�0��t��

+ d�fr���ei�·Z�0��t� A2�0��Z�0��t���i� · Z�2��t�� + A2

�2��Z�0��t�,Z�2��t�� + A4�0��Z�0��t��� ,

�76�

Zr�6��t� = Zi

�6��t��

�Zi�0�Fr�Z�0��t�� + Zi

�2��t�Zj�4��t�

�

�Zi�0�

�

�Zj�0�Fr�Z�0��t��

+1

12Zi

�2��t�Zj�2��t�Zk

�2��t��

�Zi�0�

�

�Zj�0�

�

�Zk�0�Fr�Z�0��t��

+ d�fr���ei�·Z�0��t��A2�0��Z�0��t���i� · Z�4��t�� +

1

2A2

�0��Z�0��t���i� · Z�2��t��2

+ A2�2��Z�0��t�,Z�2��t���i� · Z�2��t�� + A2

�4��Z�0��t�,Z�2��t�,Z�4��t��

+ A4�0��Z�0��t���i� · Z�2��t�� + A4

�2��Z�0��t�,Z�2��t�� + A6�0��Z�0��t���¯ , �77�

where, obviously,

012109-14 N. C. Dias and J. N. Prata J. Math. Phys. 48, 012109 �2007�

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F�Z�0��t�� = d�f���ei�·Z�0��t�. �78�

The zeroth order equation �Eq. �74�� is a closed equation for Z�0�. In view of Eq. �66� these are justthe classical Hamilton equations, which constitute a system of first order ordinary differentialequations for Z�0��t�, subject to the initial condition,

Z�0��t = 0� = z . �79�

Let us now proceed to the second order equation �Eq. �75��. The integral term on the right-handside is easily evaluated using Eq. �70�,

d�f���ei�·Z�0�A2

�0��Z�0�� = − d�f���ei�·Z�0�JikJjl��i� j�i� · Z�0���

� � 1

16�k�l�i� · Z�0�� +

1

24�k�i� · Z�0���l�i� · Z�0��

= − JikJjl��i� jZa�0�� � � 1

16��k�lZb

�0���

�Zb�0� +

1

24��kZb

�0��

���lZc�0��

�2

�Zb�0�Zc

�0� �

�Za�0� d�f���ei�·Z�0�

. �80�

Altogether we may then write

Zr�2��t� = Zi

�2��t��Fr

�Zi�0� �Z

�0�� −1

16JikJjl��i� jZa

�0����k�lZb�0��

�2Fr

�Za�0��Zb

�0� �Z�0��

−1

24JikJjl��i� jZa

�0����kZb�0����lZc

�0���3Fr

�Za�0��Zb

�0��Zc�0� �Z

�0�� . �81�

This equation could be equally derived using the method of Sec. III C for analytic Hamiltonians�cf. Eq. �63��. Now, if we look at the zeroth and second order equations �Eqs. �74� and �81��, theyconstitute a closed system of four partial differential equations for Z�0� and Z�2� with initial con-ditions �Eq. �79�� and

Z�2��t = 0� = 0. �82�

This system decouples and reduces to a � hierarchy of first order ordinary differential equations.We first solve the zeroth order classical equations of motion �Eqs. �74� and �79�� which areordinary differential equations for Z�0�. Then Eq. �81� for Z�2�, which originally is a partial differ-ential equation, can be turned into an inhomogeneous ordinary differential equation if we realizedthat the only partial derivatives �with respect to z� involved in Eq. �81� are those of Z�0�. Moreprecisely, the second and third terms on the right-hand side of Eq. �81� can be written compactlyas Br

�2��z , t� once Eqs. �74� and �79� have been solved for Z�0�. This is an inhomogeneous termwhich is a known function of t and z. Similarly, we may write Ari�z , t���Fr /�Zi

�0��Z�0��. Again,this is a known function of z and t. Consequently, Eqs. �81� and �82� may be cast in the form

Zr�2��z,t� = Ari�z,t�Zi

�2��z,t� + Br�2��z,t�, Zr

�2��z,t = 0� = 0, �83�

which constitutes a system of two ordinary differential equations for Z�2� subject to the initialconditions �Eq. �82��. The phase-space variables z= �q , p� only play the role of fixed parameters.This equation is explicitly solved in Sec. IV, Remark 4.2, for a specific example.

Proceeding inductively, let us assume that we have solved the equations to order �2n−2�, sothat we have the explicit functions Z�0� ,Z�2� , . . . ,Z�2n−2�. By inspection �see Eqs. �74�–�77��, weconclude that the equation for Z�2n� may be written as

012109-15 Features of Moyal trajectories J. Math. Phys. 48, 012109 �2007�

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Zr�2n��t� = d�fr���ei�·Z�0�

�i� · Z�2n� + ¯ �

= Zi�2n��t� ·

�Fr

�Zi�0� �Z

�0��t�� + Gr�2n�� Z�0��, Z�2��, . . . , Z�2n−2��� , �84�

where Gr�2n�� Z�0�� , Z�2�� , . . . , Z�2n−2��� denotes an expression which depends only on the functions

Z�0� ,Z�2� , . . . ,Z�2n−2� and their partial derivatives �of several orders� with respect to z. Moreover,Z�2n� is subject to the initial condition,

Z�2n��t = 0� = 0. �85�

Under the assumption that the functions Z�0� , . . . ,Z�2n−2� are already known, Gr�2n� can be computed

exactly and we obtain a set of functions Br�2n��z , t� of z and t. Equation �84� can then be written as

Zr�2n��z,t� = Ari�z,t�Zi

�2n��z,t� + Br�2n��z,t�, Zr

�2n��z,t = 0� = 0. �86�

which �like Eq. �83�� is an inhomogeneous ordinary differential equation for Z�2n� with z playingthe role of a fixed parameter. �To a given order �2n� we may thus solve the system �74�–�77� asordinary differential equations by following this hierarchical method. Alternatively, it may beadvantageous from a numerical point of view, to solve a system of equations for the partialderivatives appearing in, e.g., Eq. �81� directly. For a discussion of this strategy, we refer toSec. VD of a recent paper.48�

As advertised, the system of partial differential equations �Eqs. �74�–�77�� have beenconverted into a � hierarchy of first order ordinary differential equations of the form �86�. Theclaims of this theorem are thoroughly illustrated to order O��2� in the example below �cf. Remark4.2�. �

Remark 3.3: The theorem is stated in the space of formal series, which means that questionsof convergence are ignored �this was already the case in Secs. III A and III B�.

IV. EXAMPLE

The following example shows explicitly that Moyal and classical evolutions are different.Moreover it will illustrate the formalism developed in Sec. III B. We shall assume t0=0 forsimplicity and we start by computing the classical evolution generated by the following Hamil-tonian:

H�q,p� = q2p2. �87�

The Hamilton equations read

QC = 2QC2 PC, PC = − 2QCPC

2 . �88�

We conclude that QC · PC is a constant of motion,

QC�q,p,t� · PC�q,p,t� = q · p, ∀ t . �89�

Substituting this expression in the first equation �Eq. �88�� we get

QC�t� = 2qpQC�t� , �90�

and so, from Eq. �7�

QC�q,p,t� = qe2qpt, ∀ t � R . �91�

Upon substitution in Eq. �89�, we get

012109-16 N. C. Dias and J. N. Prata J. Math. Phys. 48, 012109 �2007�

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PC�q,p,t� = pe−2qpt, ∀ t � R . �92�

By construction this is a symplectomorphism at all times. However, this is not a canonical trans-formation. To check this, let us first compute QC�PC using the kernel representation �Eq. �2��. Ifwe Wick rotate t→ i�, we get after some integrations and an integration by parts

QC�q,p,t���q,p�PC�q,p,t� =qp + i�/2

�1 + ��t�2�2 =q��q,p�p

�1 + ��t�2�2 . �93�

It then follows that

�QC�q,p,t�,PC�q,p,t���q,p� =2

�Im�Q��q,p�P� = �1 + ��t�2�−2 � 1, ∀ t � 0, �94�

and so this is not a canonical transformation. From Eq. �31� we conclude that the Moyal evolutionmust differ from the classical solution. The Moyal evolution for an arbitrary observable A isdictated by the equation

AM = �4qp�q�

�q− p

�

�p� +

�2

2�q

�3

�q2�p− p

�3

�q�p2�AM . �95�

Now this can be solved for �QM , PM� with the initial conditions �Eq. �7��. This is feasible in thiscase. But, in general, it is the fact that we are dealing with a partial differential equation thatmakes Moyal evolution difficult to solve. We shall follow the alternative route presented in Sec.III B. This approach is reminiscent of the classical solution presented above. We shall then solvean ordinary differential equation in the noncommutative space of � functions. The equations ofmotion at time t=0 read �cf. Eq. �8��

Q = 2q2p, P = − 2qp2. �96�

These equations are valid only at time t=0. To write the corresponding ordinary differentialequations at another time t we will have to write the right-hand sides in terms of � products. Weresort to Lemma 3.1 and write Eqs. �96� in the SAS order,

Q = q�q�p + p�q�q, P = − q�p�p − p�p�q . �97�

If U�t� is as in Eq. �27�, then if we act with U�t�� on the left and with �U�−t� on the right of theseequations, we get

QM = QM�QM�PM + PM�QM�QM, PM = − QM�PM�PM − PM�PM�QM . �98�

The important thing to remark is that, contrary to Eqs. �97�, the previous expressions hold for allt� �−� ,�� with � 0 �we show below that �=� /2��. We may thus regard Eqs. �98� as ordinarydifferential equations for �QM , PM� in the noncommutative algebra of � functions. Notice that thisterminology is potentially misleading. It is crucial that we bear in mind the fact that we have tofind solutions in the space of � functions. Since �QM�t� , PM�t��=1 at any time t, we have

d

dt�QM�PM� = QM�PM + QM�PM

= PM�QM�QM�PM − QM�PM�PM�QM

= PM�QM�QM�PM − PM�QM�QM�PM + i�PM�QM − i�PM�QM = 0, �99�

and so, as in the classical case, QM�PM is a constant of motion,

012109-17 Features of Moyal trajectories J. Math. Phys. 48, 012109 �2007�

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QM�q,p,t���q,p�PM�q,p,t� = q�p = qp +i�

2. �100�

Substituting Eq. �100� in the first equation �Eq. �98��,

QM�t� = QM�t���qp� + �qp��QM�t� . �101�

The solution of this equation subject to the initial condition �Eq. �7�� is

QM�q,p,t� = e�tqp�q�e�

tqp. �102�

Similarly, we get for PM,

PM�q,p,t� = e�−tqp�p�e�

−tqp. �103�

By construction this is a unitary transformation. To finish our calculation, we just need to computeexplicitly the � products in Eqs. �102� and �103�. The noncommutative exponentials have beencomputed for quadratic functions in Refs. 9 and 15. The result is

±�t,q,p� � e�±tqp = sec2��t

2�exp�±

2

�qp tan��t

2�, �t� �

�

�, �104�

and so

QM�q,p,t� = +�q�+ = �1 + i tan��t

2�+��q+� = sec2��t�q exp� 2

�qp tan��t�, �t� �

�

2�,

�105�

where the last � product was again evaluated using the kernel representation �Eq. �2��. A similarcalculation leads to

PM�q,p,t� = sec2��t�p exp�−2

�qp tan��t�, �t� �

�

2�. �106�

Remark 4.1: The mapping �q , p�→ �QM�t� , PM�t�� for this system constitutes another exampleof a transformation which is canonical but not a symplectomorphism. Indeed we have

QM�q,p,t�,PM�q,p,t���q,p� = sec4��t� � 1, ∀ t � −�

2�,

�

2�� \ 0� . �107�

Remark 4.2: As expected in the limit �→0, the previous solutions coincide with the classicalones �QC�q , p , t� , PC�q , p , t�� �cf. Eqs. �91� and �92��. Moreover, we have checked explicitly thatthe solutions to order �2�Q�2��t� , P�2��t�� satisfy the ordinary differential equations �Eqs. �81� and�83��. Indeed, since Fq=2q2p and Fp=−2qp2, we get after some algebra from Eq. �81�,

Q�2� − 4qpQ�2� − 2q2e4qptP�2� = 2tqe2qpt�1 − tqp� ,

P�2� + 2p2e−4qptQ�2� + 4qpP�2� = 2tpe−2qpt�1 + tqp� .

Using these equations, it can be shown that

d

dt�Q�0�P�2� + Q�2�P�0�� = 4qpt .

If we impose the initial conditions Q�2��t=0�= P�2��t=0�=0, we obtain

012109-18 N. C. Dias and J. N. Prata J. Math. Phys. 48, 012109 �2007�

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qe2qptP�2� + pe−2qptQ�2� = 2qpt2.

Solving this equation for P�2� and substituting in the equation for Q�2�, we obtain

Q�2� − 2qpQ�2� = 2q2pt2e2qpt + 2qte2qpt.

This is a linear, inhomogeneous ordinary differential equation with constant coefficients. It istrivial to solve. If we impose again the initial condition Q�2��t=0�=0, the solution for Q�2� and P�2�

reads

Q�2��q,p,t� = qt2e2qpt�1 +2

3qpt�, P�2��q,p,t� = pt2e−2qpt�1 −

2

3qpt� .

If we expand our exact solution �Eqs. �105� and �106�� to order �2, we easily recover the previousexpressions. This example illustrates that, as claimed in Sec. III D, the variables �q , p� are every-where present, notwithstanding the fact that they do not play any role in solving the equations toeach order in �. They are only fixed parameters and the equations to be solved are ordinarydifferential equations with respect to time.

Remark 4.3: Finding solutions in the noncommutative space of � functions might be morerestrictive than in classical mechanics. In our example, the Moyal solutions are only valid on thebounded interval t� �−� /2� ,� /2��, whereas the classical solutions are valid everywhere on R.Perhaps it should be stressed even more emphatically that the Moyal solutions diverge within afinite time interval in sharp contrast with the classical solutions. This is a very important point. Ithighlights the danger of relying too heavily upon � expansions. Indeed, if we consider, say, thesecond order solution for the position,

QM�2��q,p,t� = QC�q,p,t�t2�1 +

2tqp

3� ,

we realize that it is a smooth function everywhere on T*M �R2 and for all t�R. The same can besaid about all the remaining orders. However, our exact solution �Eq. �105�� is singular at �t�=� /2�.

Remark 4.4: This example illustrates the claim that Moyal evolution does not act as a coor-dinate �local� transformation. Let us consider the phase-space variable,

A0�q,p� = q�p = qp +i�

2. �108�

Since this is a constant of motion, we have

AM�q,p,t� = U�t��A0�U�− t� = QM�t��PM�t� = qp +i�

2. �109�

Inverting Eqs. �105� and �106� we get

q�QM,PM,t� = cos2��t�QM exp�−1

�QMPM sin�2�t�cos2��t� ,

p�QM,PM,t� = cos2��t�PM exp� 1

�QMPM sin�2�t�cos2��t� . �110�

Upon substitution in Eq. �109� we obtain

012109-19 Features of Moyal trajectories J. Math. Phys. 48, 012109 �2007�

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AM = cos4��t�QMPM +i�

2. �111�

If Moyal evolution were a coordinate transformation we ought to have �cf. Eq. �16��

AM�t� = A0�QM�t�,PM�t�� = QMPM +i�

2. �112�

Notice also that Eq. �110� is not the time reversal of Eqs. �105� and �106� �i.e., cannot be obtainedby making t→−t� as it would be the case if Moyal evolution were a coordinate transformation.

V. HAMILTONIANS OF THE FORM H= „p2 /2m…+V„q…

The previous example is admittedly somewhat contrived. It has the virtue of being exactlysolvable and of yielding distinct classical and Moyal evolutions. However, the claim that theseevolutions are different in general remains valid even if we consider more realistic systems. ForHamiltonians which are polynomials of q’s and p’s of degree higher than 2, there will generallyappear discrepancies between the classical and Moyal evolutions of order O��2�. Indeed, let usconsider a Hamiltonian of the form

H�q,p� =p2

2m+ V�q� , �113�

and define �z= �q , p��

�1z�z� � �z,H�z��z, �2

z�q,p� � ��1z�z�,H�z��z, . . . ,

�1z�z� � z,H�z��z, �2

z�z� � �1z�z�,H�z��z, . . . . �114�

A simple calculation reveals that

�iq�z� = �i

q�z�, i = 1, . . . ,5, �6q�z� = �6

q�z� −�2

4m4V�3��q�V�4��q� , �115�

where V�n��q� is the nth derivative of V�q�. Likewise for the momentum we get

�ip�z� = �i

p�z�, i = 1, . . . ,4, �5p�z� = �5

p�z� −�2

4m3V�3��q�V�4��q� . �116�

The solutions of the classical �ZC�z , t�� and of the Moyal �ZM�z , t�� equations of motion admit theformal expansions,

ZC�z,t� = z + �n=1

�t

n!�n

z�z�, ZM�z,t� = z + �n=1

�t

n!�n

z�z� . �117�

By virtue of Eqs. �115�–�117�, we conclude that QM diverges from QC at order O�t6�,

QM�q,p,t� = QC�q,p,t� −�2t6

4 · 6!m4V�3��q�V�4��q� + O�t7� , �118�

and PM diverges from PC at order O�t5�,

PM�q,p,t� = PC�q,p,t� −�2t5

4 · 5!m3V�3��q�V�4��q� + O�t6� . �119�

For the quartic anharmonic oscillator V�q�= 12m2q2+ �� /4!�q4�� 0� and thus the previous equa-

tions yield

012109-20 N. C. Dias and J. N. Prata J. Math. Phys. 48, 012109 �2007�

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QM�q,p,t� = QC�q,p,t� −�2�2t6

4 · 6!m4q + O�t7�, PM�q,p,t� = PC�q,p,t� −�2�2t5

4 · 5!m3q + O�t6� .

�120�

It may appear from this analysis that a cubic potential generates a classical trajectory in Moyalmechanics. However, even this is not true. If we study the momentum to order O�t7� we obtain aterm of the form ��5�2t7� / �4·7!m4���V�3��q��3.

VI. CONCLUSIONS AND OUTLOOK

We have shown explicitly with various counterexamples that, in general, Moyal and classicaldynamics differ for the canonical variables of position and momentum. We argued that this stemsfrom the fact that Hamilton’s equations are only valid at some initial time for the Moyal evolution.This is a consequence of two related concepts, namely, that Moyal evolution is neither a symplec-tomorphism nor a coordinate �local� transformation. A necessary condition for both evolutions tocoincide is that Moyal �and classical� evolution be both a canonical transformation and symplec-tomorphism. Whether this is also sufficient deserves further investigation. Indeed if this conditionwas sufficient for coincidence of the two evolutions, then in a first instance it would suffice toanalyze the classical solution only in order to infer beforehand whether the Moyal evolution isequal to its classical counterpart or not. There are a few comments that can be made in this respect.Let us assume for a while that there exist Moyal trajectories which are simultaneously a symplec-tomorphism and distinct from the classical counterparts. Let us consider Eqs. �40�,

Q = Fq�q,p,t�, P = Fp�q,p,t� , �121�

where Fi�z , t�=U�t�� �zi ,H��U*�t�. The solutions are such that Q=Q�q , p , t� and P= P�q , p , t�. Wemay invert these relations and get q=q�Q , P , t� and p= p�Q , P , t�. Substituting these equations inEq. �121� we get

Q = Gq�Q,P,t�, P = Gp�Q,P,t� , �122�

where Gi�Q , P , t�=Fi�q�Q , P , t� , p�Q , P , t� , t�. In classical mechanics one would expectGq�Q , P , t�=Gq�Q , P ,0�=Fq�Q , P ,0�= �� /�P�H�Q , P�, etc. If the Moyal evolution is a symplec-tomorphism then there should exist some function HE�Q , P , t� �an effective Hamiltonian� such that

Gq�Q,P,t� =�HE

�P, Gp�Q,P,t� = −

�HE

�Q. �123�

Then the Moyal equations would be equivalent to

Q =�HE

�P, P = −

�HE

�Q. �124�

Since at t= t0 we have HE�Q , P , t= t0�=H�q , p�, we conclude that if HE is not explicitly timedependent, then, in view of Eq. �124�, Moyal and classical evolutions would have to coincide. Atthis point it is not clear to us whether there exist Moyal trajectories which are a symplectomor-phism and different from the classical trajectories generated by the same Hamiltonian. What canbe said from the previous analysis is that if such trajectories do exist then they are equivalentlygenerated by a classical flow with a time dependent Hamiltonian.

Another by-product of our analysis is the following speculation: whenever a classical systemcan be solved exactly in the space of smooth phase-space functions with the usual commutativeproduct �·� and Poisson bracket ,�, then the corresponding Moyal flow can presumably be solvedin the space of � functions with noncommutative � product and Moyal bracket �,�. One just has todeform the classical solution by introducing judiciously � products �cf. Eqs. �91� and �92� and Eqs.�105� and �106��. However, it is still unclear what is the precise prescription for substituting

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standard products by � products in this context. And even if one does find such a prescription thefinal result in terms of � functions is obviously not explicit: it is expressed in terms of � productswhich may be difficult �or impossible� to evaluate. The obvious question is whether one gainsanything by doing so. Indeed, one may just as well express the solution in terms of � products asin Eq. �27�. However, by inspection of our example of Sec. IV, we realize two important facts.First of all in our solution �Eqs. �105� and �106�� one has to evaluate the noncommutative expo-nential ± �104� for a quadratic variable qp, which is well known,9,15 whereas in the universalsolution �Eq. �27�� one needs to compute the noncommutative exponential for a quartic variableH=q2p2 �i.e., the formalism of � functions considerably simplified the resolution of the problem�.Secondly, if one is not capable of evaluating all the � products in the solution, then our prescrip-tion is more apt for performing the � expansions order by order. Indeed, for the formal solution�Eq. �27��, when one computes the � expansion, one has to bear in mind the fact that the Weylsymbol of the evolution operator U�t� may be singular in �.

Another fact that our example of Sec. IV illustrates is that the classical and Moyal trajectoriesmay diverge dramatically during a finite lapse of time and that the perturbative � expansions tendto obscure the global behavior of the trajectories. If we consider the ordinary linear differentialequations �Eqs. �63� and �74�–�77�� derived in Sec. III, the usual existence and uniqueness theo-rems apply. Typically the range of validity of the solutions found at each order will be lessrestrictive than that of the global solution. This is obviously connected with questions of conver-gence of the series, when one resums all the terms. We feel that the suggested method of solvingthe equations first in the noncommutative algebra U��A� may be a better starting point for ana-lyzing the global aspects of Moyal trajectories.

Finally, let us point out that there are many issues, related to the subject of Moyal trajectories,which remained unaddressed. Of special interest are further investigations on the mathematicalstructure of Moyal dynamics �for instance, what is the translation to the Moyal dynamics ofwell-known theorems of classical mechanics, e.g., what is the behavior of classical periodic orbitsunder Moyal evolution?� and of course the applications of Moyal dynamics to traditional problems�for instance, the quantum behavior of classically chaotic systems�. Of slightly different status isthe deeper problem of whether �or in what regimes� Moyal dynamics provides an accurate cor-rection to classical dynamics.

Note added. After this work was completed we have learned about the paper �Ref. 47� Someof the results of that paper are coincident with ours, although the two approaches seem to becomplementary. In addition, we present several examples.

ACKNOWLEDGMENT

This work was partially supported by Grant Nos. POCTI/MAT/45306/2002 and POCTI/0208/2003 of the Portuguese Science Foundation.

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