Deformation Quantization: Quantum Mechanics Lives and Works in Phase-Space

Wigner's quasi-probability distribution function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence (eg, quantum computing); quantum chaos;"Welcher Weg"discussions; semiclassical limits. It is also of importance in signal processing. Nevertheless, a remarkable aspect of its internal logic, pioneered by Moyal, has only emerged in the last quarter-century: It furnishes a third, alternative, formulation of Quantum Mechanics, independent of the conventional Hilbert Space, or Path Integral formulations. In this logically complete and self-standing formulation, one need not choose sides--coordinate or momentum space. It works in full phase-space, accommodating the uncertainty principle. This is an introductory overview of the formulation with simple illustrations.


Introduction
There are three logically autonomous alternative paths to quantization. The first is the standard one utilizing operators in Hilbert space, developed by Heisenberg, Schrödinger, Dirac, and others in the Twenties of the past century. The second one relies on Path Integrals, and was conceived by Dirac 1 and constructed by Feynman.
The last one (the bronze medal!) is the Phase-Space formulation, based on Wigner's (1932) quasi-distribution function 2 and Weyl's (1927) correspondence 3 between quantum-mechanical operators and ordinary c-number phase-space functions. The crucial composition structure of these functions, which relies on the ⋆-product, was fully understood by Groenewold (1946) 4 , who, together with Moyal (1949) 5 , pulled the entire formulation together. Still, insights on interpretation and a full appreciation of its conceptual autonomy took some time to mature with the work of Takabayasi 6 , Baker 7 , Fairlie 8 , and others. (A brief guide to some landmark papers is provided in the Appendix.) This complete formulation is based on the Wigner Function (WF), which is a quasi-probability distribution function in phase-space, It is a special representation of the density matrix (in the Weyl correspondence, as detailed in Section 8). Alternatively, it is a generating function for all spatial autocorrelation functions of a given quantum-mechanical wavefunction ψ(x).
There are several outstanding reviews on the subject 9,10,11 . Nevertheless, the central conceit of this review is that the above input wavefunctions may ultimately be forfeited, since the WFs are determined, in principle, as the solutions of suitable (celebrated) functional equations. Connections to the Hilbert space operator formulation of quantum mechanics may thus be ignored, in principle-even though they are provided in Section 8 for pedagogy and confirmation of the formulations' equivalence. One might then envision an imaginary planet on which this formulation of Quantum Mechanics preceded the conventional formulation, and its own techniques and methods arose independently, perhaps out of generalizations of classical mechanics and statistical mechanics.
It is not only wavefunctions that are missing in this formulation. Beyond an all-important (noncommutative, associative, pseudodifferential) operation, the ⋆product, which encodes the entire quantum mechanical action, there are no operators. Observables and transition amplitudes are phase-space integrals of c-number functions (which compose through the ⋆-product), weighted by the WF, as in statistical mechanics. Consequently, even though the WF is not positive-semidefinite (it can, and usually does go negative in parts of phase-space), the computation of observables and the associated concepts are evocative of classical probability theory.
This formulation of Quantum Mechanics is useful in describing quantum transport processes in phase space, of importance in quantum optics, nuclear physics, condensed matter, and the study of semiclassical limits of mesoscopic systems 12 and the transition to classical statistical mechanics 13 . It is the natural language to study quantum chaos and decoherence 14 (of utility in, eg, quantum computing), and provides intuition in QM interference such as in "Welcher Weg" problems 15 , probability flows as negative probability backflows 16 , and measurements of atomic systems 17 . The intriguing mathematical structure of the formulation is of relevance to Lie Algebras 18 , and has recently been retrofitted into M-theory advances linked to noncommutative geometry 19 , and matrix models 20 , which apply spacetime uncertainty principles 21 reliant on the ⋆-product. (Transverse spatial dimensions act formally as momenta, and, analogously to quantum mechanics, their uncertainty is increased or decreased inversely to the uncertainty of a given direction.) As a significant aside, the WF has extensive practical applications in signal processing and engineering (time-frequency analysis), since time and energy (frequency) constitute a pair of Fourier-conjugate variables just like the x and p of phase space a . a Thus, time varying signals are best represented in a WF as time varying spectrograms, analogously For simplicity, the formulation will be mostly illustrated for one coordinate and its conjugate momentum, but generalization to arbitrary-sized phase spaces is straightforward 23 , including infinite-dimensional ones, namely scalar field theory 24,25,26,27 .

The Wigner Function
As already indicated, the quasi-probability measure in phase space is the WF, It is obviously normalized, dpdxf (x, p) = 1. In the classical limit, → 0, it would reduce to the probability density in coordinate space x, usually highly localized, multiplied by δ-functions in momentum: the classical limit is "spiky"! This expression has more x − p symmetry than is apparent, as Fourier transformation to momentum-space wavefunctions yields a completely symmetric expression with the roles of x and p reversed, and, upon rescaling of the arguments x and p, a symmetric classical limit. It is also manifestly Real b . It is also constrained by the Schwarz Inequality to be bounded, Again, this bound disappears in the spiky classical limit.
p-or x-projection leads to marginal (bona-fide) probability densities: a spacelike shadow dpf (x, p) = ρ(x), or else a momentum-space shadow dxf (x, p) = σ(p), respectively. Either is a (bona-fide) probability density, being positive semidefinite. But neither can be conditioned on the other, as the uncertainty principle is fighting back: The WF f (x, p) itself can, and most often does go negative in some areas of phase-space 2,9 , as is illustrated below, a hallmark of QM interference in this language. (In fact, the only pure state WF which is non-negative is the Gaussian 28 .) The counter-intuitive "negative probability" aspects of this quasi-probability distribution have been explored and interpreted 29,30,16 , and negative probability flows amount to legitimate probability backflows in interesting settings 16 . Nevertheless, the WF for atomic systems can still be measured in the laboratory, albeit indirectly 17 .
Smoothing f by a filter of size larger than (eg, convolving with a phase-space Gaussian) results in a positive-semidefinite function, ie it may be thought to have been coarsened to a classical distribution 31c .
to a music score, ie the changing distribution of frequencies is monitored in time 22 : even though the description is constrained and redundant, it gives an intuitive picture of the signal that a mere time profile or frequency spectrogram fails to convey. Applications abound in acoustics, speech analysis, seismic data analysis, internal combustion engine-knocking analysis, failing helicopter component vibrations, etc. b In one space dimension, by virtue of non-degeneracy, it turns out to be p-even, but this is not a property used here. c This one is called the Husimi distribution 32 , and sometimes information scientists examine it on account of its non-negative feature. Nevertheless, it comes with a heavy price, as it needs to be "dressed" back to the WF for all practical purposes when expectation values are computed with it, ie it does not serve as an immediate quasi-probability distribution with no other measure 32 .
Among real functions, the WFs comprise a rather small, highly constrained, set. When is a real function f (x, p) a bona-fide Wigner function of the form (2)? Evidently, when its Fourier transform "left-right" factorizes, ie, so g L = g R from reality. Nevertheless, as indicated, the WF is a distribution function: it provides the integration measure in phase space to yield expectation values from phase-space c-number functions. Such functions are often classical quantities, but, in general, are associated to suitably ordered operators through Weyl's correspondence rule 3 . Given an operator ordered in this prescription, the corresponding phase-space function g(x, p) (the "classical kernel of the operator") is obtained by That operator's expectation value is then a "phase-space average" 4,5 , The classical kernel g(x, p) is often the unmodified classical expression, such as a conventional hamiltonian, H = p 2 2m + V (x), ie the transition from classical mechanics is straightforward ; but it contains when there are quantum-mechanical ordering ambiguities, such as for the classical kernel of the square of the angular momentum L · L: this one contains O( 2 ) terms introduced by the Weyl ordering, beyond the classical expression L 2 . In such cases, the classical kernels may still be produced without direct consideration of operators. A more detailed discussion of the correspondence is provided in Section 8.
In this sense, expectation values of the physical observables specified by classical kernels g(x, p) are computed through integration with the WF, in close analogy to classical probability theory, but for the non-positive-definiteness of the distribution function. This operation corresponds to tracing an operator with the density matrix (Sec 8).
The negative feature of the WF is, in the last analysis, an asset, and not a liability, and provides an efficient description of "beats" 22 .

Solving for the Wigner Function
Given a specification of observables, the next step is to find the relevant WF for a given hamiltonian. Can this be done without solving for the Schrödinger wavefunctions ψ, i.e. not using Schrödinger's equation directly? The functional equations which f satisfies completely determine it.
Firstly, its dynamical evolution is specified by Moyal's equation. This is the extension of Liouville's theorem of classical mechanics, for a classical hamiltonian H(x, p), namely ∂ t f + {f, H} = 0, to quantum mechanics, in this language 2,5 : where the ⋆-product 4 is The right hand side of (8), dubbed the "Moyal Bracket" (the Weyl-correspondent of quantum commutators), is the essentially unique one-parameter ( ) associative deformation of the Poisson Brackets of classical mechanics 33 . Expansion in around 0 reveals that it consists of the Poisson Bracket corrected by terms O( ). The equation (8) also evokes Heisenberg's equation of motion for operators, except H and f here are classical functions, and it is the ⋆-product which introduces noncommutativity. This language, thus, makes the link between quantum commutators and Poisson Brackets more transparent.
Since the ⋆-product involves exponentials of derivative operators, it may be evaluated in practice through translation of function arguments ("Bopp shifts"), The equivalent Fourier representation of the ⋆-product is 7 : An alternate integral representation of this product is 34 which readily displays noncommutativity and associativity. ⋆-multiplication of c-number phase-space functions is in complete isomorphism to Hilbert-space operator multiplication 4 , The cyclic phase-space trace is directly seen in the representation (12) to reduce to a plain product, if there is only one ⋆ involved, Moyal's equation is necessary, but does not suffice to specify the WF for a system. In the conventional formulation of quantum mechanics, systematic solution of timedependent equations is usually predicated on the spectrum of stationary ones. Timeindependent pure-state Wigner functions ⋆-commute with H, but clearly not every function ⋆-commuting with H can be a bona-fide WF (eg, any ⋆-function of H will ⋆-commute with H).
Static WFs obey more powerful functional ⋆-genvalue equations 8 (also see 35 ), where E is the energy eigenvalue of Hψ = Eψ. These amount to a complete characterization of the WFs 36 .
Lemma: For real functions f (x, p), the Wigner form (2) for pure static eigenstates is equivalent to compliance with the ⋆-genvalue equations (15) (ℜ and ℑ parts). Proof: Action of the effective differential operators on ψ * turns out to be null. Symmetrically,

hep-th/0110114
Deformation Quantization where the action on ψ is now trivial. Conversely, the pair of ⋆-eigenvalue equations dictate, for f (x, p) = dy e −iypf (x, y) , Hence, Real solutions of (15) must be of the form The eqs (15) lead to spectral properties for WFs 8,36 , as in the Hilbert space formulation. For instance, projective orthogonality of the ⋆-genfunctions follows from associativity, which allows evaluation in two alternate groupings: Moreover, precluding degeneracy (which can be treated separately), choosing f = g above yields, and hence f ⋆ f must be the stargenfunction in question, Pure state f s then ⋆-project onto their space. In general, it can be shown 6,36 that The normalization matters 6 : despite linearity of the equations, it prevents superposition of solutions. (Quantum mechanical interference works differently here, in comportance with density matrix formalism.) By virtue of (14), for different ⋆-genfunctions, the above dictates that Consequently, unless there is zero overlap for all such WFs, at least one of the two must go negative someplace to offset the positive overlap 9 -an illustration of the negative values' feature, quite far from being a liability.
Further note that integrating (15) yields the expectation of the energy, NB Likewise d , integrating the above projective condition yields that is the overlap increases to a divergent result in the classical limit, as the WFs grow increasingly spiky.
In classical (non-negative) probability distribution theory, expectation values of non-negative functions are likewise non-negative, and thus result in standard constraint inequalities for the constituent pieces of such functions, such as, e.g., moments of the variables. But it was just seen that for WFs which go negative for an arbitrary function g, |g| 2 need not ≥ 0: this can be easily seen by choosing the support of g to lie mostly in those regions of phase-space where the WF f is negative.
Still, such constraints are not lost for WFs. It turns out they are replaced by and lead to the uncertainty principle 37 . In Hilbert space operator formalism, this relation would correspond to the positivity of the norm. This expression is nonnegative because it involves a real non-negative integrand for a pure state WF satisfying the above projective condition e , (28) To produce Heisenberg's uncertainty relation, one only need choose for arbitrary complex coefficients a, b, c. The resulting positive semi-definite quadratic form is then d This discussion applies to proper WFs, corresponding to pure states' density matrices. E.g., a sum of two WFs is not a pure state in general, and does not satisfy the condition (4). For such generalizations, the "impurity" is 4 dxdp(f − hf 2 ) ≥ 0, where the inequality is only saturated into an equality for a pure state. For instance, for w ≡ (fa + f b )/2 with fa ⋆ f b = 0, the impurity is nonvanishing, dxdp(w − hw 2 ) = 1/2. e Similarly, if f 1 and f 2 are pure state WFs, the transition probability between the respective states is also non-negative 38 , manifestly by the same argument 37 , namely, dpdxf 1 f 2 = (2π ) 2 dxdp |f 1 ⋆ f 2 | 2 ≥ 0.
for any a, b, c. The eigenvalues of the corresponding matrix are then non-negative, and thus so must be its determinant. Given and the usual this condition on the 3 × 3 matrix determinant amounts to and hence ∆x ∆p ≥ /2.
The entered into the moments' constraint through the action of the ⋆-product.

Illustration: the Harmonic Oscillator
To illustrate the formalism on a simple prototype problem, one may look at the harmonic oscillator. In the spirit of this picture, one can, in fact, eschew solving the Schrödinger problem and plugging the wavefunctions into (2); instead, one may solve (15) directly for H = (p 2 + x 2 )/2 (with m = 1, ω = 1), For this hamiltonian, the equation has collapsed to two simple PDEs. The first one, the Imaginary part, restricts f to depend on only one variable, the scalar in phase space, z = 4H/ = 2(x 2 + p 2 )/ . Thus the second one, the Real part, is a simple ODE, Setting f (z) = exp(−z/2)L(z) yields Laguerre's equation, It is solved by Laguerre polynomials, for n = E/ − 1/2 = 0, 1, 2, .., so the ⋆-gen-Wigner-functions are 4 f n = (−1) n π e −2H/ L n (4H/ ); But for the Gaussian ground state, they all have zeros and go negative. These functions become spiky in the classical limit → 0; e.g., the ground state Gaussian f 0 goes to a δ-function.
Their sum provides a resolution of the identity 5 , Dirac's hamiltonian factorization method for the alternate algebraic solution of the same problem carries through intact, with ⋆-multiplication supplanting operator multiplication. That is to say, This motivates definition of raising and lowering functions (not operators) where a ⋆ a † − a † ⋆ a = 1 .
The annihilation ones ⋆-annihilate the ⋆-Fock vacuum, Thus, the associativity of the ⋆-product permits the customary ladder spectrum generation 36 They are manifestly real, like the Gaussian ground state, and left-right symmetric; it is easy to see they are ⋆-orthogonal for different eigenvalues. Likewise, they can be seen by the evident algebraic normal ordering to project to themselves, since the Gaussian ground state does, f 0 ⋆ f 0 = f 0 /h. The corresponding coherent state WFs 39,40 are likewise very analogous to the conventional formulation. This type of analysis carries over well to a broader class of problems 36 with "essentially isospectral" pairs of partner potentials, connected with each other through Darboux transformations relying on Witten superpotentials W (cf the Pöschl-Teller potential). It closely parallels the standard differential equation structure of the recursive technique. That is, the pairs of related potentials and corresponding ⋆genstate Wigner functions are constructed recursively 36 through ladder operations analogous to the algebraic method outlined for the oscillator.
Beyond such recursive potentials, examples of further simple systems where the ⋆-genvalue equations can be solved on first principles include the linear potential 41,36,42 , the exponential interaction Liouville potentials, and their supersymmetric Morse generalizations 36 (also see 43 ). Further systems may be handled through the Chebyshev-polynomial numerical techniques of ref 44 .
First principles phase-space solution of the Hydrogen atom is less than straightforward and complete. The reader is referred to 11,45,46 for significant partial results.

Time Evolution
Moyal's equation (8) is formally solved by virtue of associative combinatoric operations completely analogous to Hilbert space Quantum Mechanics, through definition of a ⋆-unitary evolution operator, a "⋆-exponential" 11 for arbitrary hamiltonians. The solution to Moyal's equation, given the WF at t = 0, then, is For the variables x and p, the evolution equations collapse to classical trajectories, where the concluding member of these two equations hold for the oscillator only. Thus, for the oscillator, As a consequence, for the oscillator, the functional form of the Wigner function is preserved along classical phase-space trajectories 4 , f (x, p; t) = f (x cos t − p sin t, p cos t + x sin t; 0). (53) Any oscillator WF configuration rotates uniformly on the phase plane around the origin, essentially classically, even though it provides a complete quantum mechanical description 4,47,2,25,27,48 .
Naturally, this rigid rotation in phase-space preserves areas, and thus automatically illustrates the uncertainty principle. By contrast, in general, in the conventional formulation of quantum mechanics, this result is deprived of intuitive import, or, at the very least, simplicity: upon integration in x (or p) to yield usual probability densities, the rotation induces apparent complicated shape variations of the oscillating probability density profile, such as wavepacket spreading (as evident in the shadow projections on the x and p axes of Fig 2).
Only when (as is the case for coherent states 49 ) a Wigner function configuration has an additional axial x − p symmetry around its own center, will it possess an invariant profile upon this rotation, and hence a shape-invariant oscillating probability density 48 .
In Dirac's interaction representation, a more complicated interaction hamiltonian superposed on the oscillator one, leads to shape changes of the WF configurations placed on the above "turntable", and serves to generalize to scalar field theory 27 . A more elaborate discussion of propagators in found in 40,41 . t p x Fig. 2. Time evolution of generic WF configurations driven by an oscillator hamiltonian. The t-arrow indicates the rotation sense of x and p, and so, for fixed x and p axes, the WF shoebox configurations rotate rigidly in the opposite direction, clockwise. (The sharp angles of the WFs in the illustration are actually unphysical, and were only chosen to monitor their "spreading wavepacket" projections more conspicuously.) These x and p-projections (shadows) are meant to be intensity profiles on those axes, but are expanded on the plane to aid visualization. The circular figure represents a coherent state, which projects on either axis identically at all times, thus without shape alteration of its wavepacket through time evolution.

Canonical Transformations
Canonical transformations (x, p) → (X(x, p), P (x, p)) preserve the phase-space volume (area) element (again, take = 1) through a trivial Jacobian, ie, they preserve Poisson Brackets Upon quantization, the c-number function hamiltonian transforms "classically", H(X, P ) ≡ H(x, p), like a scalar. Does the ⋆-product remain invariant under this transformation?
Yes, for linear canonical transformations 39 , but clearly not for general canonical transformations. Still, things can be put right, by devising general covariant transformation rules for the ⋆-product 36 : The WF transforms in comportance with Dirac's quantum canonical transformation theory 1 .
In conventional quantum mechanics, for canonical transformations generated by F (x, X), the energy eigenfunctions transform in a generalization of the "representationchanging" Fourier transform 1 , (In this expression, the generating function F may contain corrections to the classical one, in general-but for several simple quantum mechanical systems it manages not to 50 .) Hence 36 , there is a transformation functional for WFs, T (x, p; X, P ), such that where T (x, p; X, P ) Moreover, it can be shown that 36 , H(x, p) ⋆ T (x, p; X, P ) = T (x, p; X, P ) ⋆ H(X, P ).
That is, if F satisfies a ⋆−genvalue equation, then f satisfies a ⋆-genvalue equation with the same eigenvalue, and vice versa. This proves useful in constructing WFs for simple systems which can be trivialized classically through canonical transformations.

Omitted Miscellany
Phase-space quantization extends in several interesting directions which are not covered in such a short introduction. Complete sets of WFs which correspond to off-diagonal elements of the density matrix (see next section), and which thus enable investigation of interference phenomena and the formulation of quantum mechanical perturbation theory are covered in 47,51,40 . The spectral theory of WFs is discussed in 11,52,40 . Spin is treated in ref 53 .
Alternate phase-space distributions, but always equivalent to the WF, are reviewed in 9,61,55,32 .

The Weyl Correspondence
This section summarizes the bridge and equivalence of phase-space quantization to the conventional formulation of Quantum Mechanics in Hilbert space.
Weyl 3 introduced an association rule mapping invertibly c-number phase-space functions g(x, p) (called classical kernels) to operators G in a given ordering prescription. Specifically, p → p, x → x, and, in general, The eponymous ordering prescription requires that an arbitrary operator, regarded as a power series in x and p, be first ordered in a completely symmetrized expression in x and p, by use of Heisenberg's commutation relations, [x, p] = i . A term with m powers of p and n powers of x will be obtained from the coefficient of τ m σ n in the expansion of (τ p + σx) m+n . It is evident how the map yields a Weyl-ordered operator from a polynomial classical kernel. It includes every possible ordering with multiplicity one, eg, In this correspondence scheme, then, Conversely 4 , the c-number phase-space kernels g(x, p) of Weyl-ordered operators G(x, p) are specified by p → p, x → x, or, more precisely, since the above trace reduces to dz e iτ σ /2 z|e −iσx e −iτ p G|z = 2π dz z − τ |G|z e iσ(τ /2−z) .
Thus, the density matrix inserted in this expression 5 yields a hermitean generalization of the Wigner function: where the ψ m (x)s are (ortho-)normalized solutions of a Schrödinger problem. (Wigner 2 mainly considered the diagonal elements of the density matrix (pure states), denoted above as f m ≡ f mm .) As a consequence, matrix elements of operators, ie, traces of them with the density matrix, are produced through mere phase-space integrals 5 , and thus expectation values follow for m = n, as utilized throughout in this review. Hence, the moment-generating functional 5 of the Wigner distribution. Products of Weyl-ordered operators are easily reordered into Weyl-ordered operators through the degenerate Campbell-Baker-Hausdorff identity. In a study of the uniqueness of the Schrödinger representation, von Neumann 34 adumbrated the composition rule of classical kernels in such operator products, appreciating that Weyl's correspondence was in fact a homomorphism. (Effectively, he arrived at a convolution representation of the star product.) Finally, Groenewold 4 neatly worked out in detail how the classical kernels f and g of two operators F and G must compose to yield the classical kernel of FG, Changing integration variables to reduces the above integral to where f ⋆ g is the expression (11).
formulation of quantum mechanics in phase space. It systematically studies all expectation values of Weyl-ordered operators, and identifies the Fourier transform of their moment-generating function (their characteristic function) to the Wigner Function. It further interprets the subtlety of the "negative probability" formalism and reconciles it with the uncertainty principle. Not least, it recasts the time evolution of the Wigner Function through a deformation of the Poisson Bracket into the Moyal Bracket (the commutator of ⋆-products, i.e., the Weyl correspondent of the Heisenberg commutator), and thus opens up the way for a systematic study of the semiclassical limit. Before publication, Dirac has already been impressed by this work, contrasting it to his own ideas on functional integration, in Bohr's Festschrift 63 . T Takabayasi (1954) 6 investigates the fundamental projective normalization condition for pure state Wigner functions, and exploits Groenewold's link to the conventional density matrix formulation. It further illuminates the diffusion of wavepackets.
G Baker (1958) 7 envisions the logical autonomy of the formulation, based on postulating the projective normalization condition. It resolves measurement subtleties in the correspondence principle and appreciates the significance of the anticommutator of the ⋆-product as well, thus shifting emphasis to the ⋆-product itself, over and above its commutator. D Fairlie (1964) 8 (also see 35 ) explores the time-independent counterpart to Moyal's evolution equation, which involves the ⋆-product, beyond mere Moyal Bracket equations, and derives (instead of postulating) the projective orthonormality conditions for the resulting Wigner functions. These now allow for a unique and full solution of the quantum system, in principle (without any reference to the conventional Hilbert-space formulation). Thus, autonomy of the formulation is fully recognized.
M Berry (1977) 10 elucidates the subtleties of the semiclassical limit, ergodicity, integrability, and the singularity structure of Wigner function evolution. F Bayen, M Flato, C Fronsdal, A Lichnerowicz, and D Sternheimer (1978) 11 analyzes systematically the deformation structure and the uniqueness of the formulation, with special emphasis on spectral theory, and consolidates it mathematically. It provides explicit solutions to standard problems and introduces influential technical innovations, such as the ⋆-exponential.
T Curtright, D Fairlie, and C Zachos (1998) 36 demonstrates more directly the equivalence of the time-independent ⋆-genvalue problem to the Hilbert space formulation, and hence its logical autonomy; formulates Darboux isospectral systems in phase space; establishes the covariant transformation rule for general nonlinear canonical transformations (with reliance on the classic work of P Dirac (1933) 1 ; and thus furnishes explicit solutions of nontrivial practical problems on first principles, without recourse to the Hilbert space formulation. Efficient techniques, e.g. for perturbation theory, are based on generating functions for complete sets of Wigner functions in T Curtright, T Uematsu, and C Zachos (2001) 40 . A self-contained