^{1}

^{2}

^{3}

^{4}

^{*}

^{1}

^{2}

^{3}

^{4}

Edited by: Ignazio Licata, Institute for Scientific Methodology (ISEM), Italy

Reviewed by: Q. H. Liu, Hunan University, China; Alberto Molgado, Universidad Autónoma de San Luis Potosì, Mexico

This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Groenewold-Van Hove theorem suggest that is not always possible to transform classical observables into quantum observables (a process known as quantization) in a way that, for all Hamiltonians, the constants of motion are preserved. The latter is a strong shortcoming for the ultimate goal of quantization, as one would expect that the notion of “constants of motion” is independent of the chosen physical scheme. It has been recently developed an approach to quantization that instead of mapping every classical observable into a quantum observable, it focuses on mapping the constants of motion themselves. In this article we will discuss the relations between classical and quantum theory under the light of this new form of quantization. In particular, we will examine the mapping of a class of operators that generalizes angular momentum where quantization satisfies the usual desirable properties.

One of the deepest problems in the description of microscopic systems is the relation between their classical and quantum descriptions. Although there is no doubt in that quantum and classical descriptions perform well in their own scales of application, one would expect that a smooth transition between these two descriptions, at least at a theoretical level, is possible. However, it is well-known that this is not a trivial matter [

Formally, a (canonical) quantization of a Poisson algebra ^{*}ℝ^{n} = ℝ^{2n}. So in this case, _{j}, ξ_{j} are the position and momentum coordinates respectively, and _{j}, _{j} are position and momentum operators respectively. Moreover, it settles their ordering by imposing selfadjointness of the operators corresponding to real valued functions on phase space (Weyl Calculus is the only Shubin's τ-quantization satisfying that property). We must mention though, that there are various other frameworks developed to do quantization, such as strict deformation quantization [

When Weyl ideas about quantization were first introduced, it was expected that quantization should hold the additional property of intertwining the Poisson bracket with the commutator, that is

we call the latter a Dirac quantization. There is a famous result in the theory of quantization, first shown by Groenewold and later by van Howe, proving that there is no Dirac quantization for the canonical phase space satisfying the condition

This is an example of a semiclassical result.

For general phase spaces, the problem of finding a quantization is in general quite difficult. However, for the purposes of this article, we only need to consider the case when the phase space is the cotangent bundle of some Riemannian manifold ^{n}.

In particular, we will focus on the relation between classical and quantum constants of motion (COM) in the quantization process. A constant of motion of a classical physical system is a quantity that is preserved during the system's evolution [_{0} is a COM if and only if {_{0}} = 0, where {·, ·} denotes the Poisson bracket. In order to establish the analogous definition of a COM for a quantum system, one considers a Hamiltonian _{0} and an observable _{0},

Groenwold-van Howe no-go theorem suggests that in general Weyl calculus does not preserves constants of motion for every _{0}. Indeed, we can only conclude that

Considering the analogies between the mathematical descriptions of classical and quantum COM, it seems natural that there should be a quantization process that preserves COM. In Belmonte [

It is important to note that in some cases the right hand of (1) might be zero, in other words, it can happen that Weyl Calculus does preserve COM for some Hamiltonian _{0}. The following theorem, proved in Belmonte [

T_{0}

More precisely, if {_{0}, _{0} for the classical harmonic oscillator Hamiltonian, but we will not consider that case here.

As we will explain in section 2, every quantum COM admits a decomposition through the spectral diagonalization of _{0}. In other words, if

The construction of the decomposable Weyl calculus, essentially, conjectures that _{λ} is the decomposition of

An important COM in physics is angular momentum. Angular momentum is a constant of motion in any system where no external torque is acting. The conservation of momentum has been proven to correspond the rotational invariance by Noether's theorem [_{ij}(_{j}ξ_{i} − _{i}ξ_{j}. In particular, we have CQR on each _{ij}. Thus, after quantizing, we obtain the operators _{ij} = _{j}∂_{i} − _{i}∂_{j}). Another important quantum COM is the operator ^{2} is well known, the relation via quantization, of ^{2} as a COM, with its classical counterpart has not been clarified. In this article we will study such relation, and thus our analysis adds up a novel example of CQR (using the well known quantum decomposition of ^{2}) to the already known cases [

It is well known that the spectral analysis of ^{2} is based on the fact that we can restrict such analysis to one sphere, instead of working with the whole ℝ^{n}. The latter happens because the operators acting on each sphere obtained after decomposing ^{2} through the diagonalization of −Δ are all unitarily equivalent. We will show that exactly the same can be done if we replace ^{2} by any operator of the form ^{−1}ξ) =

This last result is an application of the decomposable Weyl calculus that provides an advantage over traditional Weyl calculus as one is able to calculate, even when CQR is not possible, the spectrum of the quantized operator directly from the quantization process.

The paper is organized as follows. In section 2 we will review the mathematical structure of COM in classical and quantum physics and introduce the decomposable Weyl calculus. In section 3 we provide examples of the decomposable Weyl calculus considering the free Hamiltonian and various functions of angular momenta. In section 4, we develop the spectral analysis of ^{−1}ξ) =

In classical mechanics the phase space is described by a symplectic manifold Σ^{2n} (in our case Σ = ^{*}ℝ^{n} = ℝ^{2n}), and observables are smooth functions in it. Denote by {·, ·} the corresponding Poisson bracket on ^{∞}(Σ). Fix a Hamiltonian _{0}(Σ), the energy level submanifold _{t}. It turns out that the orbits space

For example, in the case _{t}(

A classical observable ^{∞}(Σ) is a constant of motion if {_{0}} = 0. Leibniz's rule and Jacobi identity show that the set ^{∞}(Σ). It is easy to show that _{t} =

where _{λ} in Σ_{λ}. It is not difficult to show that _{t}[

and

In quantum mechanics the phase space is the projective space

Let _{0} be a quantum Hamiltonian and sp(_{0}) its spectrum. Then, there is a unique Borel measure η (up to equivalence) on sp(_{0}), a unique η-measurable field of Hilbert spaces

and

_{0}. See [

For example, in order to compute _{0} = Δ, we introduce first

given by

where ^{n}.

Note that co-area formula implies that _{0} is unitary. _{0} diagonalizes the operator ^{2}, therefore

In the literature, _{0} (and therefore

This unitary operators allow us to replace

This form of _{0} is usually obtained using spherical coordinates directly. However, our version of _{0} is somehow more natural, because ^{2} restricted to _{λ} to relate the two ways to obtain _{0} explained above play an important role in our construction, as we shall explain in detail in the remaining of the article.

A quantum observable, i.e. a selfadjoint operator _{0}. Notice that this is equivalent to

_{0}) ∋ λ → _{λ}} such that [_{λ}[_{λ}. Just as in the classical context, we also have decomposition of the dynamics: clearly ^{itF} commutes with _{0} and we have that

Analogously to the classical case, the set 𝔄 of bounded quantum constants of motion is also an algebra (in fact, a von Neumann algebra). We also know that, if

From the spectral theory point of view, it is important to find the field of operators decomposing a given quantum constant of motion

Identities (2) and (3) share two crucial features. They reflect that the construction of the fibers (both classical and quantum) aim at, on the one hand, making constant the Hamiltonian on each fiber, and on the other hand, making trivial the corresponding dynamics on each fiber. These two common features point to an additional (and unexpected) shared feature: for both classical and quantum Hamiltonians, their COM can be decomposed through the respective fibers.

Therefore, we claim that _{λ}. So, if _{λ} into _{0}(Σ) and sp(_{0}), though in the semi-classical limit (under certain circumstances) _{0}(Σ) = sp(_{0}) [

For example, when _{λ} into

In Belmonte [

D_{0}) ⊆ _{0}(Σ)_{λ}

^{*}

_{0}

The crucial reasons to introduce this novel quantization are

Contrary to Weyl calculus

When [_{0}, _{ℏ}(

Interestingly, when the ansatz in b) holds, we are able to compute the spectrum of

When

Recall that we are interested in studying the case ^{2n}. However, we would like to note that using the method developed in Belmonte [

Clearly, we have that _{0} is linear and its metaplectic representation coincides with _{0},

An important example of CQR is the following: Let ^{n} and define _{v}(_{v} is a COM and we have CQR on _{v}.

Consider the classical angular momenta functions

Then any function of the form

is a COM, where

Of course each _{ij} is a COM, in fact _{ij} = _{vij}, where _{ij} is the infinitesimal generator of the rotation of the plane generated by the elements _{i} and _{j} of the canonical base of ℝ^{n}. So, we have CQR on each _{ij}. After applying _{ij}, we obtain the angular momenta operators _{ij} = _{i}∂_{j} − _{j}∂_{i}).

Another important example of a quantum constant of motion in our case is the operator

The operator ^{2} is of special interest in quantum theory not only because ^{2} is a COM, but also for solving the free particle Schrödinger equation as

Therefore one is able to apply a separation of variables, and obtain that

with 〈θ, ϕ|_{l, m}(θ, ϕ), where _{l, m}(θ, ϕ) is the spherical harmonics with index ^{2} can be applied to any Hamiltonian with a radial potential [

Equation (9) implies that the restriction of ^{2} to each sphere coincides with the Laplace-Beltrami operator [

Interestingly, the fact that the separation of variables method is successful in the Schrödinger equation represented in spherical coordinates, and that such separation leaves the radial part of the equation independent of the angular part is intimately related to the fact that in the classical counterpart the fibers of the free Hamiltonian are cotangent bundles of spheres.

The canonical spectral analysis of ^{2} involves its decompositions, but it is usually presented in a different way: Let ^{2}. Since ^{2} is invariant by Fourier transform, ^{2} to the corresponding sphere. It is straightforward to check that _{λ} is given by (4). In particular, ^{2} by calculating eigenvectors on the sphere (the spherical harmonics), but usually this is not mentioned in the literature.

Notice that the latter analysis applies to ^{2} but might fail for other quantum constants of motion _{λ} migh not implement a unitary equivalence between the operators decomposing

Angular momentum poses an excellent case of study for the problem of quantization, as the commutation relations in the classical and quantum counterpart strongly resemble each other (for the canonical case

with

The latter equalities said that ^{2} is a Casimir element of the Lie algebra generated by the _{j} (actually, it is a Casimir element of the corresponding universal enveloping algebra); we will come back to this point later.

The operator ^{2} has a natural classical counterpart:

The reader might expect that

P

^{2}, we obtain our result. □

Obviously ^{2} commutes with _{0}, so it can be decomposed through each reduction. Recall that

T

^{n}

We will present two different proofs of this result. The first one is straightforward, while the second one is more sophisticated but relies on the underlying symmetry of the sphere, and we shall explain later why this reasoning is important for further developments.

^{n}, we have that ^{n}). Moreover, under the constraints ||^{2} = λ and 〈

□

Our second proof relies on some results from Poisson geometry which we shall briefly explain. Let ^{*} is the dual of the Lie algebra 𝔰𝔬(_{λ} is covariant, i.e. ^{*} is the coadjoint action, and ^{*})^{−1}_{v}(_{ij} the canonical base of 𝔰𝔬(^{*}-invariant (in particular, it is a Casimir element of ^{∞}(𝔰𝔬(^{*})) and

_{n} is the last member of the canonical base (or any of them actually). Let

and this finishes the proof. □

C

where Δ_{S} is the Laplace-Beltrami operator on the sphere. It is well known that

It is well known that the principal symbol of the Laplace-Beltrami operator on any complete Riemannian manifold is ^{2} and the Laplace-Beltrami operator on the sphere also holds true classically.

Also notice that all the quantization the authors know for the cotangent bundle of a Riemannian manifold sent

In this section we will show that the first part of the spectral analysis of ^{2} can be generalized to any operator of the form

Let

If λ_{0} = 1, then _{λ} is the classical counterpart of the unitary operator _{λ}.

Notice that _{λ} (respectively _{λ}) is the restriction of a map defined on ℝ^{n} (respectively ^{*}ℝ^{n} = ℝ^{2n}) given by the same expression, and they define a multiplicative one parameter group of diffeomorphisms (respectively symplectomorphism), i.e. _{1} = _{λ1·λ2} = _{λ1}_{λ2} (respectively _{1} = _{λ1·λ2} = _{λ1}_{λ2}. In fact, up to composing with the exponential, _{λ} correspond to the (additive) one parameter group of the gradient ∇φ, where φ(^{2}. Therefore _{λ}, after composing with the exponential, coincides with the Hamiltonian one parameter group of _{∇φ}.

T^{−1}ξ) =

_{λ}. Using the expression for

Notice that the Fourier transform ^{n − 1} at the point

Therefore

□

C^{−1}ξ) =

C^{−1}ξ) =

Notice that since _{0} itself does not. The constants of motion satisfying ^{−1}ξ) = ^{2n}/||ξ|| = 1;〈^{−1}ξ) =

The last corollary 8 gives further reasons to look for a proof to our conjecture, at least for functions of angular momenta.

The technique used to prove theorem 4 and corollary 5 are interesting on its own right and lead to some further questions; we shall briefly explain why. Let 𝔤 be a Lie algebra and let ^{*}(𝔤^{*}). In our case, we were working with the quadratic Casimir, but one can consider any symmetric homogeneous polynomial to build Casimir elements. On the other hand, if 𝔤 is the Lie algebra of a Lie group ^{n − 1} and ^{∞}(^{*}^{∞}(𝔤^{*}) (and therefore on ^{∞}(^{*}

Coming back to our context, recall that our main purpose is to understand the decomposition of the operators of the form ^{n}, where ^{2}. Also recall that functions of angular momenta are examples of such constants of motion. The latter claim is another way to say that every function of the form ^{*}φ is a constant of motion, where ^{2n} → 𝔰𝔬^{*}(^{∞}(𝔰𝔬^{*}(

From a physically and a bit more speculative perspective, we would like to state that it is remarkable that the method introduced in this paper leaves aside the requirement of mapping position and momentum coordinates, and focuses on the mapping of COM themselves. Indeed, recall that the state of a quantum system is not determined by the position and momentum coordinates, but by a complete set of commuting observables (CSCO). Since CSCO usually includes the Hamiltonian, then the observables in CSCO are COM. Therefore a quantization that preserves COM will necessarily map a classical system into a quantum system where the description of the states can be directly determined from the quantized operators. In order to develop a decomposable Weyl calculus where the latter is possible, one should apply recursively the decomposable quantization calculus for each operator in the CSCO. If that is possible, we would obtain an intuitively more appropriate quantization than the regular quantization that demands the quantization of position and momentum coordinates.

The ground of the CQR mathematical framework presented in this article was developed by FB, while for the application of CQR to the angular momentum and the discussion around it both authors contributed equally.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.