Can Star Products be Augmented by Classical Physics?

It has been suggested that star products in phase-space quantization may be augmented to describe additional, classical effects. That proposal is examined critically here. Two known star products that introduce classical effects are: the generalized Husimi product of coarse-grained quantization, and a non-Hermitian damped star product for the harmonic oscillator. Following these examples, we consider products related by transition differential operators to the classic Moyal star product. We restrict to Hermitian star products, avoiding problems already pointed out for the original damped product. It is shown, however, that with such star products, augmented quantization is impossible, since an appropriate classical limit does not result. For a more complete study, we then also consider generalized, or local, transition operators, that depend on the local phase-space coordinates, as well as their derivatives. In this framework, one example of possible physical interest is constructed. Because of its limited validity and complicated form, however, it cannot be concluded that augmented quantization with local transition operators is practical.


Introduction
Observables in quantum mechanics can be described by operators or by functions (and distributions) in phase space [1][2][3][4][5][6][7][8][9]. Operator and phase-space quantum mechanics are equivalent, however, as can be demonstrated through maps from one to the other. A quantization map takes the phase-space quantum mechanics to the operator version, and its inverse is known as the dequantization map. The non-commutativity of operator observables is reflected in phase space by the non-commutative star product that must be used to multiply phase-space obervables.
Quantization of a classical system is not unique either way, however. Different quantizations give rise to different star products [2,[5][6][7]10]. For example, distinct operator-ordering rules determine distinct quantization maps. The corresponding dequantizations prescribe distinct star products. Weyl operator ordering and the Moyal star product are paired, but so are standard operator ordering and the standard star product, Born-Jordan ordering and another star product, etc.
An advantage of phase space quantization is that different quantizations and corresponding star products may be related in a simple way. For example, a transition (differential) operator [6,7] can associate the phase-space observables and star products of 2 quantizations.
The same mathematical machinery can introduce physical effects distinct from quantization, however.
Coarse graining in phase space is described by the Husimi star product [9,11]. It is obtainable from the Moyal product, e.g., by a transition (differential) operator. Similarly, another transition operator produces a star product that converts the equation of motion of the harmonic oscillator into that of the damped harmonic oscillator [12,13].
Here we study star products modified to include additional classical effects not described by the original Hamiltonian. The construction of such augmented star products will be called augmented (phase-space) quantization. Taking the coarse-grained and damped star products as our guides, we consider star products related to the Moyal star product by transition operators. Our goal is to see whether or not such augmented star products can provide effective descriptions of certain quantum systems.
Our considerations will include generalized transition operators, depending not only on derivatives of phase-space coordinates, but also on the coordinates themselves. Such "local" transition operators have been considered before (see [14,15]).
The next section is a quick introduction to the elements of phase-space quantization that are relevant to our study, and to our notation. Section 3 describes the 2 star products we take as our guides: the coarse-grained Husimi [9,11] and damped Dito-Turrubiates [12,13] products. Section 4 treats augmented star products abstractly and generally: we find that transition differential operators ("global" transition operators) do not lead to classical equations of motion augmented by additional terms. The local transition operators mentioned above must be considered. The following section demonstrates that local transition operators can furnish new examples of modified star products that incorporate additional classical physics. A new damped star product is constructed that is Hermitian, in contrast with that of Dito-Turrubiates [12,13]. The final section is our conclusion.

Phase-space quantum mechanics and star products
We will restrict to 1-dimensional systems on position space R with coordinate q, and conjugate momentum p ∈ R, so that the phase-space is R 2 . It is straightforward to generalize to several degrees of freedom. Only time-independent Hamiltonians will be treated: ∂ t H = 0.
An account of quantization in phase-space will now be given. Subsection 2.2 then outlines the canonical example, involving the Weyl map, the Moyal star product, and the Wigner transform. It will be our reference quantization, so that all other examples treated here will be related to it by transition operators, which we discuss in Subsection 2.3.
2.1. Quantization in Phase-Space. Suppose that the distribution on phase-space, f = f (q, p), is a classical observable. Then the quantization map Q produces a quantum observable: Theˆindicates thatf is an operator, which is a function of position operator,q, and momentum operator,p. The inverse (dequantization map) W is given by Strictly, both Q and W should be labelled by the phase-space coordinates, so that To avoid overly cumbersome notation, however, we will follow convention and drop these labels when confusion is unlikely.
Dequantization maps operators into phase-space distributions (the symbols of the operators). Operator products become star products, homomorphically: The star product * is a bi-differential operator expressible in terms of the left derivatives ← ∂ q , ← ∂ p , and right derivatives, and in a similar manner for the right derivatives. Here I(1, 2) enacts the identifications q 1 = q 2 = q and p 1 = p 2 = p . In useful shorthand notation, (5) is As an illustration of the homomorphism of (4), consider the Heisenberg-Weyl group relation in operator quantum mechanics: where ϕ, ξ, ϕ ′ , ξ ′ ∈ R, a consequence of the simple Baker-Campbell-Hausdorff formula. Application of the dequantization map W yields This demonstrates that a phase-space quantization produces a * -realization of the Heisenberg-Weyl group.
Let D denote an arbitrary bi-differential operator (such as the star product * , or a left-or right- for examples). Transpose exchanges left-and right-derivatives: so the transpose D t satisfies for arbitrary phase-space distributions (i.e. observables) f and g. Let D and f be the complex conjugates of bi-differential operator D and phase-space distribution f . The adjoint, or Hermitian conjugate, D † of D is the complex-conjugate transpose: A bi-differential operator, such as a star product * , can be Hermitian, D † = D, or symmetric, D t = D, or real, D = D, or none of the above.
Application of W to any relation involving operator observables yields the phase-space counterpart.
With this in mind, consider the equation of motion for a quantum observablef in the Heisenberg picture, assumingĤ † =Ĥ is the Hamiltonian.
As a result of (4), commutators of operators are mapped to * -commutators, Applying the dequantization map W to (12) yieldṡ where H := W(Ĥ), and we have introduced the Moyal bracket {·, ·} * : M denotes the associated Moyal bi-differential operator.
The formal solution to (14) is where is the symbol of the propagator, and exp * indicates the * -exponential [6,7].
In the so-called classical limit → 0, the Moyal bracket reverts to a Poisson bracket, The corresponding bi-differential operators have the same relation, Here the Poisson bi-differential operator P is defined by the Poisson bracket: Therefore, in the classical limit, the equation of motion (14) reverts tȯ which is the classical equation of motion, as expected.
The quantum state of the system is described by the density operator in the Schroedinger picture,

ρ. Its equation of motion is
The dequantization map W transforms the density operatorρ into a function W (q, p; t) = W(ρ) on phase space, obeying W (q, p; t) is called a quasi-probability distribution on phase space because, although it determines expectation values as a probability distribution would: it takes negative values. The first example of such a quasi-probability distribution is the Wigner function (see the next section).
Now that we have discussed the general form of phase-space quantization, we are in a position to identify the crux of our paper. We are interested in augmenting the star product such that instead of (19), we find where the additional term δP describes extra, classical effects. We will return to this idea in Sections 4 and 5.

2.2.
Reference phase-space quantization. As our reference example, we will use the Weyl quantization map Q 0 , which can be defined by Expanding exponentials and equating terms proportional to ϕ n ξ m produces the Weyl operator-ordering Here the sum is over all permutations π ∈ S n+m of the n + m factors inq npm , and the numbers above the operators indicate their place in the product. The sum is somewhat redundant: it can be restricted is the average of distinct terms obtained by permuting the factors ofq npm . Alternate expressions, such for example, can be derived using the Heisenberg commutation relation [2].
The dequantization map W 0 , the inverse of the Weyl map (26), is also known as the Wigner transform. The homomorphism (4) between * -and operator products, along with the Heisenberg-Weyl relation (8), gives the famous (Groenewold-)Moyal star product The transpose of the Moyal star product equals its complex conjugate, * 0 : The Moyal product is associative: which follows from Here the notation of (6) is used, with * 0 (1 + 2, 3), for example, obtained from * 0 by replacing the corresponding Moyal bracket bi-differential operator (15) is In agreement with (19), we find as expected. As already stated above, we will be interested in modifying this last relation to include additional classical physics.
The reference quasi-probability distribution function is the famous Wigner function Hereρ is the density operator, and W 0 (q, p; t) = W 0 (ρ).

2.3.
Other phase-space quantizations: transition differential operators. Phase-space quantization is not unique. In many cases, however, the different quantizations can be related by a transition differential operator T = T (∂ q , ∂ p ) [6,7].
For example, an operator ordering different from the Weyl ordering of equations (26, 27) may be used. The Born-Jordan quantization map is giving the ordering prescription [2] Q BJ q n p m = 1 n + 1 n k=0q n−kp mq k .
The quantization maps Q 0 and Q BJ are related. Applying the simple Baker-Campbell-Hausdorff formula to (38) yields so that Both Weyl and Born-Jordan operator orderings are Hermitian. A non-Hermitian example is the so-called standard operator ordering, with rule [5] Q S q n p m =q np m .
The relation to Weyl ordering is encoded in Notice that this relation, for a non-Hermitian ordering, involves a complex multiplicative function.
Suppose Q, W are the quantization and dequantization maps of a phase-space quantization. We can connect these maps to our reference reference quantization of Q 0 , W 0 using an invertible differential operator T such that We show pictorially how to relate different maps in Figure 1.
As both W and W 0 are homomorphisms from operator products to * -products (see (4)), we find for any two phase-space distributions f, g. This equation determines the product * T . With F = T f and G = T g, we have Consider transition differential operators T = T (∂ q , ∂ p ), i.e. those without any dependence on q, p.
The Born-Jordan quantization discussed above provides an example: the transition operator and star product are related by (49, 50). In this case, we have a real transition operator, T BJ = T BJ and so a Hermitian star product * BJ = * †
As an illustraition of a non-Hermitian star product, consider the standard operator ordering of (42, 43). Then the relevant transition operator and star product are Similarly, (49, 50) relate this non-real transition operator and non-Hermitian star product.

Guiding examples: coarse graining and damping
Intriguingly, attempts have been made to use transition operators to introduce classical, physical effects, and not just to relate alternative quantizations. We wish to explore this technique, and see if new physical applications might be found.
Two guiding examples are discussed in the following subsections. The first classical effect is Gaussian coarse-graining in phase-space, which gives rise to a (generalized) Husimi star product and phase-space quantization [9,11]. The second case introduced damping into the simple harmonic oscillator equations of motion, and produced a modified star product depending on the damping coefficient [12,13].
3.1. Coarse-grained Husimi quantization. Consider a distribution in phase-space, f (q, p), coarse grained as follows: Here η is a classical coarse-graining scale, independent of , and s is a squeezing parameter. When η = , and f is the Wigner function W , the expressions in (53) equal the original Husimi quasi-probability distribution [9,11].
By (53), can be interpreted as a transition differential operator. As (53) is a coarse-grained Wigner function, it would be improper to say that the Husimi distribution is the result of an alternative quantization. An additional classical physical effect (coarse-graining) is introduced with the transition operator. In other words, T η converts Weyl quantization to an augmented quantization.
From (49, 50), we find the (generalized) Husimi star product * Tη =: Notice that ⊙ t η = ⊙ η , so that, from (15), Therefore the classical limit produces a multiplicative modification of the Poisson bi-differential operator: rather than an augmentation of the form (25).

Damped quantization. Consider the simple harmonic oscillator, with Hamiltonian
Replace the Poisson bi-differential operator P with [12] P γ := P − 2γm The canonical equations of motion are changed tȯ The equation of motion of a damped harmonic oscillator results: with γ as the damping parameter.
The same replacement in the Moyal star product produces a damped star product * γ : The transition operator reproduces the star product * γ = * Tγ when used in (49, 50) [12]. T γ therefore describes an augmented quantization of the simple harmonic oscillator, with classical damping introduced.
Notice, however, that the transition operator T γ is not real. This causes significant problems when the time evolution of quasi-probability distributions and observables is considered [13]. With T γ = T γ , the damped * -product is non-Hermitian as a result: The dynamics is governed by the Moyal bi-differential operator (see (14) above), but with * † γ = * γ , Therefore, the reality of an observable, such as f in (14), is not preserved in evolution because Furthermore, by (35), That is, the damping disappears in the classical limit. A quantization of the classical damped harmonic oscillator is not described after all [13]!
The hopeful substitution [13] would fix both problems. But then the dynamical equation would becomė with formal solution The last expression is problematic, however. Although * γ is associative when γ is fixed, we find for phase-space obervables a, b and c [13]. This non-associativity result follows from in the notation of (32, 33). In turn, by (62), (72) is a consequence of Can something similar work better? Consider a generalization of the damped transition operator T γ of (63): As in the generalized Husimi quantization, the parameter η has dimensions of action. It replaces i /2, so that it does not vanish in the → 0 limit. Using η ∈ R, equation (50) yields a Hermitian star product: However, the classical limit still fails, as The damped Poisson bracket P γ of (59) is not recovered. Notice this is true even if small γ is considered, While P γ differs additively from P, equation (76) describes instead a multiplicative modification of P.
A multiplicative modification of the classical limit was also found when using the Husimi transition differential operator in the previous subsection.
To progress, we need to understand what is possible in augmented quantization. For that reason, we discuss the possibilities described by an arbitrary transition differential operator T = T (∂ q , ∂ p ) in the next section.

Augmented star products: generalities
Consider star products modified by a transition (differential) operator T = T (∂ q , ∂ p ), describing their relation to the reference Moyal star product. We will restrict to real T , so that Hermitian star products result.
4.1. -dependent transition differential operators. Notice that it was the -dependence of the Dito-Turrubiates transition operator (63) that resulted in a classical limit (67) with no augmentation.
To understand this, note that the transition differential operators that describe operator orderings that differ from the Weyl ordering, such as (51), are -dependent. For them, a non-augmented classical limit is necessary if they are to describe different quantizations of the same classical system. In other words, when these transition operators are applied, additional physical effects will not appear in the classical limit.
But is this feature avoidable when we do wish to augment the classical physics?
No. Consider an arbitrary transition differential operator T with -dependence. If a non-singular classical limit is to be found, we can write with no negative powers of . As a consequence, so that An -dependent transition differential operator cannot describe augmented classical physics.
4.2. -independent transition differential operators. We therefore now consider transition differential operators with no dependence on : ∂T /∂ = 0. The general result for star products induced by differential operators T = T (∂ q , ∂ p ) is given in equations (49, 50).
The crucial observation is that As a result, the classical limit yields where we have used ∂⊙ T /∂ = 0. Therefore, The classical limit is indeed modified if the transition differential operator is independent of . However, an augmentation (25) of the desired additive form does not result -instead we find a multiplicative modification, (83).
Of course, if we can write ⊙ T ≈ 1+δ⊙ T , we recover an approximate additive augmentation of the form by (50). We will therefore not consider this possibility further.

4.3.
Local transition operators. Since transition differential operators do not yield augmented star products, we will now consider generalized transition operators that depend on the phase-space coordinates: T = T (q, p; ∂ q , ∂ p ). Such "local" transition operators, depending on the phase-space point (q, p), were treated previously in [14]. They also appeared in [15], where the gauging of global relations between phase-space quantizations was studied.
In the more general case, we conjecture that (46) is solved by where now The bi-differential operator * T is obtained from * T (1, 2) by identifying (q 1 , p 1 ) = (q 2 , p 2 ) = (q, p): Notice that when T = T (∂ q , ∂ p ) is a differential operator, the simpler result (50) is recovered. An explicit, general expression for * T , however, in terms of q, p, It is important to note that the associativity of * T follows from that of the Moyal star product * 0 , for any invertible T , whether it depends on q and p or not. One obtains by applying (45) twice to (32).

Augmented equations of motion
In this section, we will use local transition operators T = T (q, p; ∂ q , ∂ p ) to find star products that yield classical limits augmented by additional physics. We will focus on the equations of motion for the phase-space coordinates q, p.
Let x denote either q or p. Precisely, we will require thaṫ describes the augmented quantization in the classical limit.
With T = T (q, p; ∂ q , ∂ p ) a local transition operator, we have no simple, general formula for * T . It is therefore easiest to work with T directly and use (46) to rewrite (89) aṡ where we have used ∂T /∂ = 0 and (36).
We will consider augmentations that are weak, by writing with θ a bi-differential operator. Then (90) becomeṡ where the augmenting terms are all on the right-hand side. It is helpful, perhaps, to rewrite them in the notation of (6). Denoting the terms augmenting the equations of motion by A θ (x), we have From these expressions, one sees that θ = β∂ x ′ (β a constant, and x ′ = q or p) produces no augmentation. Furthermore, no term contributes that is higher order in derivatives ∂ q , ∂ p , if it is multiplied by a constant. A multiplicative function of q and p in θ is necessary -this confirms that a local transition operator is required.
Consider then the ansatz producing As an example, consider augmenting the simple harmonic oscillator by introducing weak damping, with damping coefficient γ. To produce the equations of motion (60) for the damped oscillator, we need when the Hamiltonian (58) is used in (95). We find that θ = θ 0,1 ∂ p + θ 0,2 ∂ 2 p , θ 0,1 = 2 γ ω arctan mωq p dp , yields (96). Here arctan(mωq/p) dp indicates the indefinite integral. 1 This example demonstrates that local transition operators can produce augmented star products, describing systems with additional classical efffects. A Hermitian star product incorporating weak damping is described by the local transition operator of (91, 97).

Conclusion
Our goal was to investigate the possibility of introducing additional physics, such as damping, during quantization. To do this, we analyzed the transition operator to determine if it could yield such an augmented quantization. We demonstrated two strong results: (1) an -dependent transition differential operator must result in a classical → 0 limit with no augmentation, as shown in (80); (2) an -independent differential operator produces a multiplicative augmentation, illustrated in (83).
However, we also realized that there was a possible way around the "no-go theorem": a transition operator dependent on the phase-space point, a local transition differential operator, could be used.
As a proof of principle, we constructed a real (local) transition operator (91, 97) that results in a Hermitian star product implementing a weak damping of the simple harmonic oscillator. It seems that local transition operators may be capable of producing useful effective descriptions of systems augmented by additional classical physics.
Our first example does exhibit some peculiar features, however. For example, the transition operator is specific to the harmonic oscillator. It would have to take a different form to introduce damping into a different quantum system. This contrasts sharply with the standard procedure: to incorporate an additional physical effect into different systems, an identical term is added to the different Hamiltonians.
Furthermore, the transition operator (91, 97) is quite complicated. At some point, an effective description is not practical if it is too involved. We leave it to the reader and our possible future consideration to decide whether a local transition operator can provide a useful description of the classically-augmented physics of a quantum system. so that I(1, 2) (∂ q 1 + ∂ q 2 ) m q a 1 1 q a 2 2 = m! m ℓ=0 a 1 ℓ a 2 m − ℓ q a 1 +a 2 −m .
and Vandermonde's identity is generalized (Chu-Vandermonde identity) so that m and n in equation (112) are also arbitrary real values [17].