Moreover, gives a *-algebra isomorphism between the Moyal algebra and the corresponding algebra of smooth “observables” , with ; and between and as algebra of bounded “observables”, with . Note that ; hence physical observables with , are given by real elements of the Moyal algebras. We will, however, mostly not pay much attention to the difference between and below.
We define ( -)multiplicative operators acting on the distributions by
Then, we have the simple and elegant result
which can be interpreted as the Bopp shift. The representation given through and of Equation (7) on directly extends to arbitrary functions and coincides with the product structure with described in as and as . It is the left regular representation of the functional algebra on itself, which can be extended further to all of . One can even associate the wavefunction with a operator, though the latter does not correspond to a physical observable. It remains to be seen if the operator has any particular physical meaning. Looking at a real wavefunction, or , makes more sense as the absolute phase of a quantum state has no physical meaning anyway. makes a legitimate physical observable. We are interested only in applying all these mathematical results to the Gelfand triple and that is the background on which the explicit results concerning the states are to be understood. We will see at the end that is essentially the left ideal of that carries an irreducible representation of the Moyal algebra.
The Wigner functions that describe states, pure or mixed, are to be given in terms of functions of the density operator . For a pure state , the latter is denoted by . Two different pure states give , with a non-diagonal Wigner function given as . Focusing on the set of basis coherent states, one can check that actually and . For , we have, with Equation (23),
which for reduces to
and operators on beyond this collection are not of much interest.
As is essentially the space of Hilbert-Schmidt operators on , the classical picture from the contraction limit of the latter as a representation of the Heisenberg-Weyl symmetry obviously maintains the basic notion of the Hilbert space as to be coordinated by the classical phase space variables , though a renormalization may again be necessary to trace them from the explicit original quantum . The original is really the real span of for the coherent state basis; hence it becomes the real span of the delta functions in the classical limit. That is to say, the set of classical “density matrices” fills the whole real part of . Formally, is simply the real part of . We can see further that and . More features of the classical picture obtained will be seen in Section 7, where we discuss the description of dynamics.
6. Description of Quantum Symmetries and Time Evolution
The description of the quantum symmetries in connection with the WWGM formalism has been well presented in Ref.  , from which we summarize the basic features and give explicit details for applications to our framework with particular emphasis on the elements of the relativity symmetry. Hermitian operators, as physical observables, play the role of the symmetry generators giving rise to a group of unitary flow on the Hilbert space(s), as well as an isomorphic group of automorphisms on the set of pure state density operators. Here we only focus on and the matching set of , while extending from the latter to all of in the language of the Tomita representation.
Firstly, we have on symmetries as the group of unitary and antiunitary operators factored by its closed center of phase transformations. The isomorphic automorphism group of the set of is characterized by the subgroup of the group of real unitary transformations compatible with the star product, i.e. satisfies
[or ]. is the real subspace of the Hilbert space of Hilbert-Schmidt operators. Symmetry groups represented as subgroups of can be considered. For a star-unitary transformation on wavefunctions , we have a real unitary operator on
We write a generic one parameter group of such a (star-) unitary transformation in terms of real parameter s as with as the generator. Note that the factor 2 is put in the place of , consistent with our choice of units7. For time translation, as a unitary transformation on , we have the Schrödinger equation of motion
In the Tomita representation picture, the unitary transformation on gives a corresponding unitary transformation on as the Hilbert space of square kets, which are here simply elements in , given by 
From Equation (61), one can see that this is just a fancy restatement of Equation (63) above, with now an explicit form of as an operator in terms of the real generator function . Consider the generator as defined by
, we have
where is the star product commutator, i.e. the Moyal bracket. Hence, with ,
The result is of course to be expected. When applied to the time translation as a unitary transformation, it gives exactly the Liouville equation of motion for a density matrix as the Schrödinger equation for the latter taken as a state in with Hamiltonian operator . is the operator that represents the (real) element in the algebra of observables as well as . then represents the algebra element on . It is interesting to note that by introducing the notation as a left action, we have a corresponding right action given by with . The explicit expression for the action follows from
These operators match to the right-invariant vector fields of the Heisenberg-Weyl group as and corresponds to the left-invariant ones. When is an order one or two polynomial in the variables, which covers the cases of interest here, has a very simple explicit form. Another important feature to note is that determines only up to an additive constant. This is a consequence of the fact that the density matrix is insensitive to the phase of the pure state . Constant functions in the observable algebra correspond to multiples of on which generates pure phase transformations, i.e. and .
Let us focus first on the observables and as symmetry generators on , and look at the corresponding transformations on . From Equation (8),
giving, in terms of explicit and parameters,
The factors of 2 in the translations and may look somewhat suspicious at first sight. They are actually related to the fact that the arguments of the wavefunction correspond to half the expectation values, due to our coherent state labeling. Thus, we find that and generate translations of the expectation values, which is certainly the right feature to have. To better appreciate these results, one can also think about a sort of “Heisenberg picture” for the symmetry transformations as translations of the observables instead of states giving the same transformations of the expectation values. One can see that the differential operators play an important role as operators on . We can consider the set of , and as the fundamental set of operators―functions of which essentially describe the full algebra of observables―versus the case on for which the set is given only by and . Note that the only nonzero commutators among the set are given by
The similar fundamental set of operators was long ago introduced within the Koopman-von Neumann formulation . We see here the analogous structure in the quantum setting. For a generic , the function itself (i.e. the simple multiplicative action ), , , and are all operators to be considered on , though only two among the four are linearly independent.
Consider . We have
with the explicit action
7For readers who find the factor of 2 difficult to appreciate, our results below in the next section― especially with the symmetry description in terms of the rescaled parameters in the usual units with an explicit ―should make the full picture more transparent.
where is the angular momentum operator on the Hilbert
space and (no sum over the indices) the rotated . This corresponds to the coset space action , i.e. a rotation about the
k-th direction of both the p and x as three dimensional vectors8. Together with the and (and ) parts, we have the full set of operators from the generators of the subgroup of the U(1) extended Galilean symmetry with the time translation taken out. This portion of the ten generators , is of course a commutative set. The set of represents the symmetry on , and constitute a subalgebra of the algebra of physical observables. We can easily see that the set does the same as a right action. A always commutes with a since, in general, . Explicitly, we have
and commutes with all other generators. Note that the factors of 2 are really taking the place of because of the choice of units. The upper and lower signs correspond to the and results, respectively. For the set, we can see that the set of commutators is same as that of with however the vanishing giving a vanishing . As a result, we can also see the set without as giving the symmetry without the central extension, similar to the classical case. Besides, we have
The time translation generator , needed to complete the above set of ten
into the full extended Galilean symmetry, is given by where m
is the particle mass. One can see that commutes with each generator, except for having9
Similarly, we have giving . A generic Hamil-
tonian for a particle would have to be given with an extra additive part as the potential energy . It is also of some interest to illustrate explicitly the Heisenberg equation of motion in considerations of evolution, both in and . For the time dependent operator , on
we have , while for on we have
where ; hence, we arrive at the same equation as that on . This equation can simply be written as
Taking explicitly and applying this to the observables and , we have
with vanishing third derivatives, or with , provide particularly important examples of the equation being and . The equation reduces to a form exactly the same as the one for , and , as if it is a classical observable.
7. To The Relativity Symmetry at the Classical Limit
We have presented the formulation of the classical limit of quantum mechanics from the perspective of a contraction of the relativity symmetry, and the corresponding representations, in Ref.  within the Hilbert space picture on and . In Section 2, we have presented a formulation within the WWGM setting, focusing on the key part of the Heisenberg-Weyl subgroup. We are now going to push that to the full relativity symmetry. Taking the full extended Galilean symmetry with abstract generators , , , H and I as represented on by the set of eleven above, the contraction is to be given by
, , , , and taken to the
limit. Note that setting gives the usual commutator set with an explicit (in the place of the factor 2), which can be considered as having the generators in the usual, classical system of units. Again, we take the contraction of the representation(s), with and standing in for x and p. As the whole algebra of reduces to the Poisson algebra of classical observables , all of the yield the , which all commute among themselves. The noncommutative observable algebra for the Hilbert space of pure states , upon the symmetry contraction, reduces to a commutative algebra as a result of the reduction of to the simple sum of one-dimensional subspaces of each coherent state . Each is diagonal on the resulting Hilbert space of pure states, which contains only the delta functions. How do we recover the noncommutative relativity symmetry at the classical level then, either on the observable algebra or in the Koopman-von Neumann formulation? The answer is to be found from the Tomita representation picture of the Hilbert space . The Koopman Hilbert space essentially comes from .
Take the set of , we have
and again . These results are independent of the contraction parameter k; in fact, they are independent of . The unitary operators can be written as
Similarly, if we take and (we
drop the spatial index in x and p for simplicity, similarly for above), the results vanish in the limit. This seems to create a problem, however, we are not interested in the operators generating translations in p and x. We should be looking at translations in and , i.e. rewriting and as and . Introducing generators and satisfying
we can see that
which are again independent of , and therefore independent of k. Note that and are exactly the invariant vector fields of the manifold of
corresponding to the contracted symmetry from the Heisenberg-Weyl group. To summarize, we have the set of , , , , and giving the commutators exactly as the as the old set of , , , , and , with the factors of 2 all replaced by . With taken out, the rest constitute a representation of the contracted Galilean symmetry without the U(1) central extension, which at the abstract Lie algebra level is trivialized and decoupled from the rest.
Next, take the multiplicative operators , , , and . We have the formal relation
The commutator results for the classical operators with the (with and ) set above correspond again to results in Equation (75) with 2 replaced . Thus, we recover the full algebraic structure introduced in Ref.  for the Koopman-von Neumann classical setting.
8. To the Koopman-Von Neumann Classical Dynamics
Finally, we check the explicit dynamical description obtained for the classic setting, focusing especially on the Koopman-von Neumann formulation. The Schrödinger equation, Heisenberg equation, and Liouville equation are to be cast in the following forms in the contraction limit
with the classical (antisymmetric) Poisson bracket of classical phase space coordinates . So, the Schrödinger equation on the Hilbert space of pure state fails to make sense at the contraction limit, while the Heisenberg equation and the Liouville equation give the correct classical results. The problem of the Schrödinger equation is not beyond expectations. The Hilbert space of pure states, as an irreducible unitary representation, collapses to the simple sum of one-dimensional subspaces of the coherent states, so there is no continuous evolution to described on it any more. Recall that the Heisenberg equation can be seen as one on , and hence it survives. Moreover, the Liouville equation is the Schrödinger equation on . The reducible representation is on the Hilbert space containing all the states―pure or mixed―and therefore it is not at all bothered by the fact that most of the pure quantum states become mixed states in the classical limit. Furthermore, note that the (classical) Liouville equation is insensitive to the rescaling/renormalization of to , and similarly for the classical equation of motion going from to .
In the Koopman-von Neumann formulation, a classical wavefunction is to be introduced with for each . Each describes a mixed state in general, as does . The Koopman-von Neumann Hilbert space is one of a reducible representation, A Koopman-Schrödinger equation for relating
to the classical Liouville equation can be written then as
One can rewrite the classical equation of motion in the Koopman-Heisenberg form  as
where is the (classical) Hamiltonian vector field, which gives
Recall that the multiplicative operator is just the classical limit of the [cf. Equation (30)]; hence it is a simple multiplication with on the classical Hilbert space . Taking a closer look, we see that the solution to the equation of motion (84) before taking can be written as
This equation is nothing other than the solution to the original Heisenberg equation of (78) written in terms of the rescaled classical variables. Expanding Equation (53) and keeping only the first two terms, we have
This result is obviously consistent with Equation (88), as the first exponential factor simply cancels itself out. The classical limit is taken as the limit, but the dynamics is determined by the noncommutative part of the star product; therefore it is determined by the first nontrivial term in the expansion, which is also the dominating real term. For the Schrödinger picture considerations, however, one would keep only the dominating first term. The limit is then consistent with the limiting Schrödinger equation, but both involve the diverging factor. Again, the quantum Schrödinger equation is an equation of motion for the pure states the classical limit, and there do not form a connected set in the reduced Hilbert space (except formally at the zero magnitude point). The Koopman-Schrödinger equation is exactly given by putting back into the limiting Schrödinger equation for the diverging k.
The solution to the Koopman-Schrödinger equation is given in terms of the Koopman-Schrödinger flow in Ref.  with the generator given by
The first two terms contribute a change of a complex phase for with no effect in the Heisenberg picture. The last term, and thus the whole set of , gives the Koopman-Heisenberg equation we obtained above, as well as a time translation of (the magnitude of) in the Schrödinger picture. The part is responsible for the geometric phase  , a notion which requires formulating states, quantum or classical, as sections of a principal bundle or a Hermitian line bundle    for its description. is really a covariant derivative ( a connection form) associated to the function which guarantees , i.e. the operators form a representation of same Lie algebra as the Poisson algebra. Adopting the canonical trivialization of the bundle over , coordinated by as a Kähler manifold with a Euclidean metric on ( is the symplectic form)10, can be taken as acting on the wavefunction with
. It would be interesting to see a full bundle formulation of the WWGM formalism and its contraction limit, which is however beyond the scope of this article.
We have explicitly presented a version of the WWGM formalism for quantum mechanics, which we propose as the most natural prescription unifying, the formalism most familiar to a general physicist (the one base on a Hilbert space of wavefunctions) and the abstract mathematical algebraic formalism related to noncommutative geometry. On the (pure state) Hilbert space of wavefunctions from the canonical coherent state basis, the observable algebra as a functional algebra of the and operators can be seen as both the operator (functional) algebra as well as with a Moyal star product; . We advocate the former picture and the important notion that the algebra is essentially an irreducible (cyclic) representation of the group (C*-) algebra from the relativity symmetry within which the Hilbert space is a representation for the group. The modern mathematics of noncommutative geometry    essentially says that the noncommutative algebra is to be seen as an algebra of continuous functions of a geometric/topological space with the six noncommutative coordinates and , and coordinates are of course the basic observables in terms of which all other observables can be constructed. as a C*-algebra corresponds to the set of compact operators on is a Moyal subalgebra of as given by (which is a W*-algebra). The mathematics also offers another geometric object as a kind of dual object to the C*-algebra, namely the space of pure states , which is equivalent to the (projective) Hilbert space (of ) . The projective Hilbert space is the infinite-dimensional Kähler manifold  , with a set of “six times ” homogeneous coordinates. One key purpose of the article is to help a general physicist to appreciate such a perspective. Of course such an algebraic-geometric perspective also works perfectly well with Newtonian physics for which the observable algebra is commutative and contains functions of the classical phase space coordinates. We illustrate here also how that classical limit is retrieved from the quantum one.
This geometric notion is usually considered as only about the quantum phase space. Actually, the standard description of quantum mechanics breaks the conceptual connection between the phase space, the configuration space, and physical space itself in classical mechanics―physical space is the configuration space (all possible positions) of a free particle, or of the center of mass as a degree of freedom for a closed system of particles; the configuration space is sort of like half the phase space, with the other half being the momentum space of conjugate variables. However, from both the noncommutative geometry picture and the picture (for a single quantum particle) discussed above, it certainly looks like it does not have to be the case. In particular, is a symplectic manifold and the Schrödinger equation is an infinite set of Hamiltonian equations of motion with the configuration and conjugate momentum variables taken as, say, the real and imaginary parts of at each (or those of for any set of orthonormal basis). In Ref. , we have constructed a quantum model of physical space, or the position/configuration space of a particle, along parallel lines of the coherent state phase space construction as a representation of the relativity symmetry. Moreover, we showed that the model reduces back to the Newtonian model as the classical limit formulated as a relativity symmetry contraction limit. Part of the analysis in the current article was motivated by the idea of illustrating the solid dynamical picture underlying that framework.
The quantum physical space obtained in Ref.  is actually a Hilbert space equivalent to that of the phase space. The key reason is that for the quantum relativity symmetry as the central extension of the classical Galilean symmetry , phase space representations are generally irreducible while in the classical case they may be reduced to a sum of the position/configuration space and the momentum space ones. The central charge generator, as the commutator generates, a complex phase rotation in relation to the natural complex structure in with the complex coordinates mixing the position/configuration coordinates with the momentum coordinates, the division of which would otherwise be respected by the other relativity symmetry transformations. The analysis here establishes explicitly that the group plays the full role of a relativity symmetry for quantum mechanics with the quantum model of the physical space and gives all the corresponding aspects for the Newtonian theory as an approximation to be described as a relativity symmetry contraction. The results also gives a comprehensive treatment of the classical limit of quantum mechanics, to which there are otherwise quite some confusing notions about in the literature.
The explicit analysis in Ref.  focused only on the subgroup with the time translation generator taken out, which is good enough for the mostly kinematical considerations there. Along these lines, we put strong emphasis on the (relativity) symmetry group as the starting point. The observable algebra is essentially the group ( ) algebra or an irreducible representation of it. Actually, we focus only on the Heisenberg-Weyl subgroup , and take into consideration the full relativity symmetry only as unitary transformation on the Hilbert space and as automorphisms on the observable algebra. All of this works very well because the relevant (e.g. spin zero) representation of the group algebra is contained in . This is a natural parallel to the Hilbert space of pure states as an representation of and (or ). This is more or less the physical statement that (orbital) angular momentum and Hamiltonian variables/operators are to be written in terms of the position and momentum ones. It is really a consequence of the structure of with the series of invariant subgroups
giving the following semidirect product structures:
where T denotes the one parameter group of time translations. The other relativity transformations act on H(3) as outer automorphisms and on its group algebra as inner automorphisms. Again, the Hilbert space as a group representation naturally sits inside the representation of the group algebra with the natural (noncommutative, algebraic) multiplicative actions of the latter as the operator actions.
It is also interesting that while the rotational symmetry SO(3) is naturally to be included in the mathematical picture of even just the H(3) symmetry, the Galilean time translation is not. Moreover, we have no problem describing the transformations generated by any real/Hermitian Hamiltonian function or operator as unitary transformations on the Hilbert space and automorphisms of the observable algebra, just like any Hamiltonian flow on a symplectic manifold. But then there is no reason to single out the parameter of a particular Hamiltonian flow as physical time and the generator of physical energy. We may have to look for a more natural relativity symmetry framework in order to truly understood time, for example with Lorentz symmetry incorporated. For the current authors, we are particularly interested in using the relativity symmetry as the basic key mathematical structure, and plan on pushing forward for models of quantum spacetime and its related dynamics on the deep microscopic scale based on the idea of relativity symmetry deformation/stabilization .
In summary, quantum mechanics can be, and we believe should be, seen as a theory of particle dynamics on a quantum/noncommutative model of the physical space with a picture as the infinite dimensional Kähler manifold . It has a relativity symmetry of and the observable algebra is naturally the representation of the group C*-algebra corresponding to the representation of (time-independent spin-zero) that describes the physical space. The WWGM formalism is just such a representation theory, and hence also essentially the Hilbert space theory. Dynamics is included in the Hamiltonian flows on as well as the corresponding automorphism flows on the C*-algebra. The mathematical framework is valid for any group as relativity symmetry, and a group obtained as the contraction limit of another serves as an approximation of the latter with the full theory retrievable from pushing the contraction throughout the original theory, as our illustration of obtaining the Newtonian theory from quantum mechanics. Lie group/algebra deformations in the reverse process to contraction, hence giving natural candidates for theories the quantum and classical mechanics serves as approximation. The fully deformed/stabilized (special) relativity symmetry, probably for Planckian physics, is expected to give full noncommutativity among all X and P to which quantum mechanics is the minimal case with noncommutativity. It suggests all noncommutative models of spacetime should have be phase space models; energy-momentum is much a part of the physical space only in the classical approximation to which one can consider the configuration and the momentum parts separately.
The authors are partially supported by research grant MOST 105-2112-M-008-017 and MOST 106-2112-M-008-008 from the MOST of Taiwan.
2We use F−1 instead of simply F to keep track of difference which only manifests at the classical contraction limit discussed in the next section.
3Note that though it looks like we have inconveniently made the group parameters and the coherent state expectation values differ by a factor of 2 by using instead of units, it is really results like Equation (26) that naturally prefer the convention. The parameter space for the wavefunctions can be exactly identified with that of the Moyal star functional algebra.
4We have with being the phase space parity operator of Grossmann-Royer   ; i.e. and . Note that is actually selfadjoint, besides being unitary.
5It is interesting to see the consistency of this result for the explicit case of a coherent state . The normalization condition for a wavefunction in can be written in the form
to be compared with
In terms of the new variables we have , a Gaussian centered at the expectation values with width .
6One may also consider
Actually, we have and formally ( ). It follows that
with The twisted convolution required to maintain is simply given formally by , i.e.
8Note that carries the units of , which are taken as 2. Hence, for the dimensionless parameter , with 2 standing in for , is the right dimensionless rotation operator. A rotation on the p or x vector corresponds to the same rotation on 2p or 2x as the expectation values.
9The usual presentation of the symmetry uses the Galilean boost generators in place of , which corresponds to here. We have be matched to .
10The metric is essentially the restriction of the Fubini-Study metric on the quantum phase space (the projective Hilbert space) as the Kähler manifold to the coherent state submanifold . It is hence totally compatible with the quantum description.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
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