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Original Research ARTICLE

Front. Phys., 13 November 2018 | https://doi.org/10.3389/fphy.2018.00121

On the Classical-Quantum Relation of Constants of Motion

  • 1Department of Mathematics, Universidad Católica del Norte, Antofagasta, Chile
  • 2Institute of Philosophy and Complexity Sciences, Santiago, Chile
  • 3Vrije Universiteit Brussel, Brussels, Belgium
  • 4Departamento Ciencias Biologicas, Facultad Ciencias de la Vida, Universidad Andres Bello, Santiago, Chile

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.

AMS subject classifications. 68Q25, 68R10, 68U05

1. Introduction

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 [1]. An area known as quantization has been intensely developed to understand such transition [2, 3]. Quantization is thus a process that converts the mathematical description of a classical physical system into a description of a quantum system which resembles the classical system at a structural level. The aim of quantization goes beyond pure description of the transition between classical and quantum. Indeed, the process of quantization is not only the fundamental method used to develop important theories in physics such as particle and nuclear physics, quantum optics, among others [4], but also it is of fundamental importance in the philosophy of physics [5].

Formally, a (canonical) quantization of a Poisson algebra P on a Hilbert space H is a family of linear and injective maps OpP that associate to certain elements of P selfadjoint operators on H in a way that the quantum objects associated with the quantum observables (operators) correspond to the classical objects that represent the initial classical observable (elements of P) in the limit → 0. Weyl Calculus, which we shall denote by Op, is the main example of a quantization. It quantizes the canonical phase space T*ℝn = ℝ2n. So in this case, P=C(2n) and H=L2(n). Weyl quantization was constructed so it additionally satisfies the identities Op(xj)=Qj and Op(ξj)=Pj, where xj, ξj are the position and momentum coordinates respectively, and Qj, Pj 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 [6], formal deformation quantization [79], or geometric quantization [3, 10]. In this article we are not going to consider them.

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

OpP({f,g})=1[OpP(f),OpP(g)];

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 OpP(xj)=Qj and OpP(ξj)=Pj; this result is known as Groenewold-van Howe no-go theorem (see [1, 11] or [12]). Although one would expect that Groenewold-van Howe no-go theorem is a unsurmountable problem for the theory of quantization, the existence of Dirac quantization would actually imply that, at least heuristically, classical mechanics is enough to describe quantum mechanics, and this would contradict fundamental results in quantum theory such as the Bell inequalities [13]. Along the latter line, it was shown that Weyl Calculus under general conditions satisfies that

Op({f,g})-1[Op(f),Op(g)]=O().

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 M. In the latter case, different solutions can be found in the literature; we are going to work with the solution given by N. Landsman in his book [2], which satisfies many interesting properties. For instance, Landsman solution is equivalent to Weyl Calculus when M = ℝn.

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 [14]. Although some COM correspond to intuitive properties of the system (e.g. energy), in many cases the COM of a system are not trivial [15]. A quantity f of a classical physical system with a Hamiltonian h0 is a COM if and only if {f, h0} = 0, where {·, ·} denotes the Poisson bracket. In order to establish the analogous definition of a COM for a quantum system, one considers a Hamiltonian H0 and an observable F, and define F to be a COM if and only if [H0, F] = 0, where [·, ·] is the quantum counterpart of the Poisson operator, known as commutator.

Groenwold-van Howe no-go theorem suggests that in general Weyl calculus does not preserves constants of motion for every h0. Indeed, we can only conclude that

{f,h0}=0[Op(f),H0]=O(2).    (1)

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 [16], a solution for this problem was given. We shall recall its construction in section 2. We call this novel quantization a decomposable Weyl Calculus and we denote it by Opd. Intuitively, one should think that the difference between the new (decomposable) and traditional Weyl calculus is that, instead of mapping position-momentum coordinates into position-momentum operators, decomposable Weyl Calculus satisfies a new constraint: mapping classical COM into quantum COM.

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 h0. The following theorem, proved in Belmonte [16], provides interesting examples when the equality holds.

THEOREM 1. Weyl calculus preserves constants of motion when h0 is a function of the free Hamiltonian composed with a linear canonical transformation.

More precisely, if {h0, f} = 0 then [H0,Op(f)]=0, whenever h0=φh~0S, where h~0(x,ξ)=||ξ||2, φ is a smooth regular function on (0, ∞), and S is a linear canonical transformation (i.e. a linear symplectomorphism). This result can be proved as well if we replace h0 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 H0. In other words, if [H0,Op(f)]=0, there is a field of operators {sp(H)λ[Op(f)]λ} that decomposes Op(f). The latter motivates our work from the operator theory perspective, as it is important to compute explicitly [Op(f)]λ.

The construction of the decomposable Weyl calculus, essentially, conjectures that [Op(f)]λ=Opλ(fλ), where fλ is the decomposition of f in the reduced phase space at λ, and Opλ is a quantization for that space. This conjecture is equivalent to identity Op(f)=Opd(f); if this holds true we say that we have commutation of quantization and reduction on f, CQR for short.

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 [14]. In Belmonte [16], it was shown that every function on angular momenta is a constant of motion for h0(x,ξ)=||x||2 or h0(x,ξ)=||ξ||2. Among those COM, an important role is played by the angular momenta coordinates themselves, i.e. the functions given by lij(x, ξ) = xjξixiξj. In particular, we have CQR on each lij. Thus, after quantizing, we obtain the operators Lij = iℏ(xjixij). Another important quantum COM is the operator L2:=i<jLij2. Even though the importance in quantum mechanics of L2 is well known, the relation via quantization, of L2 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 L2) to the already known cases [16]. Interestingly, the technique used to prove the main result (theorem 4) suggests that we should expect that we can obtain CQR for a broad class of COM.

It is well known that the spectral analysis of L2 is based on the fact that we can restrict such analysis to one sphere, instead of working with the whole ℝn. The latter happens because the operators acting on each sphere obtained after decomposing L2 through the diagonalization of −Δ are all unitarily equivalent. We will show that exactly the same can be done if we replace L2 by any operator of the form Op(f), where f is a COM on which we have CQR and satisfying the additional condition f(tx, t−1ξ) = f(x, ξ), for any t > 0. In particular, this result applies to any function of angular momenta on which we have CQR. The additional condition seems artificial, but we will explain later why it emerges naturally.

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 Opd for the class of COM f that satisfy f(tx, t−1ξ) = f(x, ξ). We conclude with a discussion of our results and future lines of work in section 5.

2. Quantizing the Constants of Motion

2.1. Classical Constants of Motion

In classical mechanics the phase space is described by a symplectic manifold Σ2n (in our case Σ = T*ℝn = ℝ2n), and observables are smooth functions in it. Denote by {·, ·} the corresponding Poisson bracket on C(Σ). Fix a Hamiltonian h0C(Σ) and denote by Φ its flow. Then, for each regular λ ∈ h0(Σ), the energy level submanifold Σ^λ:=h0-1(λ)Σ is invariant under Φt. It turns out that the orbits space Σλ:=Σ^λ/Φ forms a symplectic manifold of dimension 2n − 2 (this is a particular case of Marsden-Weinstein reduction or of the Jacobi-Liouville theorem, see [17]). We denote by πλ:Σ^λΣλ the canonical quotient map.

For example, in the case h0(x,ξ)=||ξ||2, it is easy to check that Φt(x, ξ) = (x − 2, ξ) and cleary Σ^λ=n×𝕊λn-1. Moreover, Σλ=T*𝕊λn-1 and the quotient map is given by πλ(x,ξ)=(ξ,x-x,ξλξ), where we are identifying Tξ*Sλn-1 with the plane tangent to ξ (literally). See Belmonte [16] for the proof of the last claim.

A classical observable fC(Σ) is a constant of motion if {f, h0} = 0. Leibniz's rule and Jacobi identity show that the set A of all constants of motion form a Poisson subalgebra of C(Σ). It is easy to show that fA if and only if f ∘ Φt = f, for each t∈ℝ. Thus, for each fA, we can consider the field of functions fλC(Σλ) given by

fλ(πλ(σ))=f(σ),

where σΣ^λ. In particular, we can consider the flow Φtλ[f] of fλ in Σλ. It is not difficult to show that Φtλ[f]πλ=πλΦt[f], where Φt[f] is the flow of f in Σ. This is a particular case of theorem 4.3.5 in [17]. For example, h itself is a constant of motion and we have

hλλ    (2)

and

Φtλ[h]=Id.

2.2. Quantum Constants of Motion

In quantum mechanics the phase space is the projective space H of a Hilbert space H (in our case H=L2(n)), and observables are selfadjoint operators acting on H.

Let H0 be a quantum Hamiltonian and sp(H0) its spectrum. Then, there is a unique Borel measure η (up to equivalence) on sp(H0), a unique η-measurable field of Hilbert spaces {sp(H0)λH(λ)} (up to η), and a unique unitary operator T:Hsp(H0)H(λ)dη(λ) such that

[TH0u](λ)=λ(Tu)(λ)uDom(H).    (3)

and

[TeitH0u](λ)=eitλ(Tu)(λ)uH,t.

T is called the diagonalization of H0. See [18] or [19] for details.

For example, in order to compute T when H0 = Δ, we introduce first

T0:L2(n)[0,)L2(𝕊λn-1)dλ,

given by

T0u(λ)=2-12λ-14u|𝕊λn-1,

where 𝕊λn-1 is endowed with its canonical measure given by the metric induced from ℝn.

Note that co-area formula implies that T0 is unitary. T0 diagonalizes the operator Q2, therefore T=T0F diagonalizes −Δ, where F is the Fourier transform.

In the literature, T0 (and therefore T) is usually presented in a slightly different way ([20], section 2). First consider the unitary operators Sλ:L2(𝕊λn-1)L2(𝕊n-1) given by

Sλu(z)=λn-14u(λz)    (4)

This unitary operators allow us to replace H(λ)=L2(𝕊λn-1) on the direct integral above by H(λ)=L2(𝕊n-1). In this way, we obtain T0:L2(n)[0,)L2(𝕊n-1)dλ and

T0u(λ,z)=2-12λn-24u(λz),

This form of T0 is usually obtained using spherical coordinates directly. However, our version of T0 is somehow more natural, because Q2 restricted to L2(𝕊λn-1) coincides by definition with the constant scalar operator λ, which is exactly what we want to happen on H(λ). Therefore, the use of the unitary operators Sλ to relate the two ways to obtain T0 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 F, is a quantum COM if F strongly commutes with H0. Notice that this is equivalent to eitH0FFeitH0 [18].

F is a constant of motion if and only if it admits a decomposition through T [18, 19], i.e. there is a measurable field of selfadjoint operators {sp(H0) ∋ λ → Fλ} such that [TFu](λ) = Fλ[Tu(λ)]. Such field of operators are the quantum counterpart of the field of classical observables fλ. Just as in the classical context, we also have decomposition of the dynamics: clearly eitF commutes with H0 and we have that [eitF]λ=eitFλ.

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 F ∈ 𝔄 then

||F||=supλ||Fλ||    (5)

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

sp(F)=λsp(Fλ)¯    (6)

2.3. The Decomposable Weyl Calculus

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 H(λ) is the natural quantum counterpart of Σλ. So, if H0=Op(h0) one could expect that we should be able to quantize Σλ into H(λ). The main problem with this assertion is that in general we do not know what is the relation between h0(Σ) and sp(H0), though in the semi-classical limit (under certain circumstances) h0(Σ) = sp(H0) [21, 22].

For example, when h0(x,ξ)=||ξ||2, we are able to quantize Σλ into H(λ). Indeed, since Σλ=T*𝕊λn-1 and H(λ)=L2(𝕊λn-1), we are able to transform smooth functions on T*𝕊λn-1 into selfadjoint operators on L2(𝕊λn-1) by applying the Weyl-Landsman quantization [2], which we shall denote from now on by Opλ.

In Belmonte [16] one of the authors built up the following quantization that by construction preserves constants of motion.

DEFINITION 2. Assume that sp(H0) ⊆ h0(Σ). Let Ad be the space of functions fA such that fλ is an admissible symbol for Opλ for almost every λ. For each fAd, we define

Opd(f):=T*[sp(H0)Opλ(fλ)dη(λ)]T,

where T* is the adjoint operator of T defined in (3).

We call Opd the h0-decomposable Weyl calculus and Ad the space of admissible symbols for Opd.

Remark:

The crucial reasons to introduce this novel quantization are

a) Contrary to Weyl calculus Op, the decomposable calculus Opd preserves constants of motion by construction.

b) When [H0, Op(f)] = 0 for some fA, the decomposable Weyl calculus construction provides the ansatz [Op(f)]λ=Opλ(fλ) to compute the field of operators decomposing Op(f).

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

When [Op(f)]λ=Opλ(fλ), which is equivalent to satisfy Op(f)=Opd(f), we will say that we have a commutation of quantization and reduction, CQR for short, on f. Some examples of CQR are studied in Belmonte [16]. In the remaining of this article, we aim at giving a new example of CQR and at discussing how the ideas behind the involved proofs can lead to more general cases on which we have CQR on f.

3. Angular Momentum in the Free Hamiltonian

Recall that we are interested in studying the case h0(x,ξ)=||ξ||2 on ℝ2n. However, we would like to note that using the method developed in Belmonte [16], we can consider as well the more general case h~0=φh0S, where φ is a smooth regular function on [0, ∞) and S is a linear canonical transformation (symplectomorphism). An example of the latter is h~0(x,ξ)=φ(||x||2).

Clearly, we have that Op(h0)=H0=-2Δ, where Δ is the Laplace operator. Moreover, since the flow of h0 is linear and its metaplectic representation coincides with eitH0, Weyl Calculus preserves COM under these conditions. In other words if {h0, f} = 0, then [H0,Op(f)]=0 (see [16], for a detailed proof).

An important example of CQR is the following: Let v be a complete vector field on ℝn and define fv(x, ξ) = 〈v(x), ξ〉 (v(x) should be interpreted as a vector on the tangent plane at x and ξ as a vector on the cotangent plane). It was proved in Belmonte [16] that, if v is tangent to each sphere, then fv is a COM and we have CQR on fv.

3.1. Functions of Angular Momenta

Consider the classical angular momenta functions

li,j(x,ξ):=xjξi-xiξj.

Then any function of the form

f(x,ξ)=ψ(l1,2(x,ξ),,ln-1,n(x,ξ))    (7)

is a COM, where ψC(n(n-1)2). We will see later that some of those functions have been already associated with differential operators in L2(𝕊λn-1), so they should form a class of COM where CQR is applicable.

Of course each lij is a COM, in fact lij = fvij, where vij is the infinitesimal generator of the rotation of the plane generated by the elements ei and ej of the canonical base of ℝn. So, we have CQR on each lij. After applying Op on each lij, we obtain the angular momenta operators Lij = iℏ(xijxji).

Another important example of a quantum constant of motion in our case is the operator L2:=i<jLij2. It will give us an important example of CQR. Its analysis is given in any basic course on quantum mechanics, but we shall recall it to explain how we obtain our case and why we expect that its spectral analysis can be performed for any operator of the form Op(f), whenever f is a classical constant of motion on which we have CQR.

3.2. The Canonical Analysis of L2

The operator L2 is of special interest in quantum theory not only because L2 is a COM, but also for solving the free particle Schrödinger equation as

Δ=1r2r(r2r)+1r2sin(θ)θ(sinθθ)+1r2sin2(θ)2ϕ2     =1r2r(r2r)-L22r2    (8)

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

L2|l,m=2l(l+1)|l,mLz|l,m=m|l,m    (9)

with 〈θ, ϕ|l, m〉 = Yl, m(θ, ϕ), where Yl, m(θ, ϕ) is the spherical harmonics with index l (orbital angular momentum number) and m (magnetic number). The latter analysis of L2 can be applied to any Hamiltonian with a radial potential [23].

Equation (9) implies that the restriction of L2 to each sphere coincides with the Laplace-Beltrami operator [24], page 164.

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 L2 involves its decompositions, but it is usually presented in a different way: Let Lλ2 be the field of operators decomposing L2. Since L2 is invariant by Fourier transform, Lλ2 is just the restriction of L2 to the corresponding sphere. It is straightforward to check that Lλ2=Sλ*L12Sλ, where Sλ is given by (4). In particular, sp(L2)=sp(L12). This is the reason why we are able to complete the spectral analysis of L2 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 L2 but might fail for other quantum constants of motion F. Namely, the operators Sλ migh not implement a unitary equivalence between the operators decomposing F. However, in section 4, we will show that such equivalence holds true for any operator of the form Op(f), when f is any function of angular momenta on which we have CQR.

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 n = 3):

 {li,lj}=ϵijklk[Li,Lj]=ik=13ϵijkLk,    (10)

with i, j, k ∈ {x, y, z}. Moreover, the commutation relations between the angular momentum operators and the total momentum operator are also preserved.

{li,l2}=[Li,L2]=0,    (11)

The latter equalities said that L2 is a Casimir element of the Lie algebra generated by the Lj (actually, it is a Casimir element of the corresponding universal enveloping algebra); we will come back to this point later.

3.3. The Decomposable Weyl Calculus on L2

The operator L2 has a natural classical counterpart:

l2(x,ξ):=i<jlij2(x,ξ)

The reader might expect that Op(l2)=L2, but this is not true. The reasons why this happens are developed in the following results.

PROPOSITION 3. Op(l2)=L2+n(n-1)42I

Proof. Using the canonical commutation relations and the well known formula to obtain the value under Op of any polynomial ([12], page 86), we get that Op(lij2)=Lij2+122I. Since there are n(n-1)2 terms in the sum defining l2, we obtain our result.

Obviously l2 commutes with h0, so it can be decomposed through each reduction. Recall that Σλ=T*𝕊λn-1, therefore lλ2C(T*𝕊λn-1). The following is the central result of this article.

THEOREM 4.

lλ2(q,p)=λ(gqλ)-1(p,p),

where gqλ is the metric on Tq𝕊λn-1 induced fromn and (gqλ)-1 is the corresponding metric on Tq*𝕊λn-1.

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.

Proof. Since the metric on Tq𝕊λn-1 is induced from ℝn, we have that (gqλ)-1(p,p)=p,p=||p||2 (once we identify Tq𝕊λn-1 with the hyperplane tangent to q on ℝn). Moreover, under the constraints ||q||2 = λ and 〈q, p〉 = 0, we have that

2l2(q,p)=i,j(qipj-qjpi)2=jpj2(iqi2)                   -2iqipi(jqjpj)+ipi2(jqj2)=2λ||p||2.

Our second proof relies on some results from Poisson geometry which we shall briefly explain. Let Jλ:T*𝕊λn-1𝔰𝔬(n)* the natural moment map, where 𝔰𝔬(n)* is the dual of the Lie algebra 𝔰𝔬(n). It is well known that Jλ is covariant, i.e. Jλ[U(q,p)]=AdU*[J(q,p)], where USO(n) (the special orthogonal group), Ad* is the coadjoint action, and U(q, p) = (Uq, (U*)−1p). Let v ∈ 𝔰𝔬(n) and define EvC(𝔰𝔬(n)*) by Ev(z) = 〈v, z〉. Also, let vij the canonical base of 𝔰𝔬(n). It is easy to show that Jλ*(Evij)=lijλ. Also notice that C:=Evij2 is Ad*-invariant (in particular, it is a Casimir element of C(𝔰𝔬(n)*)) and lλ2=Jλ*C. In particular, lλ2(U(q),(U*)-1p)=lλ2(q,p).

Proof. It is easy to show that the equality holds when q=λen, where en is the last member of the canonical base (or any of them actually). Let q𝕊λn-1 and fix USO(n) such that U(q)=λen. Since the canonical action of the orthogonal group preserves the metric, we have that

λ(gqλ)-1(p,p)=λ(gλenλ)-1((U*)-1p,(U*)-1p)                         =lλ2(U(q),(U*)-1p)=lλ2(q,p),

and this finishes the proof.

COROLLARY 5. Opλ(lλ2)=Lλ2+n(n-1)42I, in other words [Op(l2)]λ=Opλ(lλ2).

Proof. Since the Ricci scalar (or scalar curvature) of 𝕊λn-1 is n(n-1)λ, proposition 3.7.2 in [2] implies that

Opλ(lλ2)=λOpλ(gqλ)-1=λ(-2ΔS+n(n-1)24λ),

where ΔS is the Laplace-Beltrami operator on the sphere. It is well known that -2λΔS=Lλ2 and this finishes the proof.

It is well known that the principal symbol of the Laplace-Beltrami operator on any complete Riemannian manifold is (gqλ)-1(p,p) [24]. So our previous theorem comes to show that the well known relation between L2 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 (gqλ)-1(p,p) to the Laplace-Beltrami operator plus some constant times the Ricci scalar. It seems that this was first noticed by Pauli [25]. So, the fact that Op(l2)L2 turns to be natural. However, there is no agreement about what constant should multiply the Ricci scalar. In physics literature, many different constants have been proposed, as it was noticed in [26] [some further development can be found in [27]], where a proposal for such constant followed some general interesting ideas concerning the case when the Riemannian manifold is a hypersurface. Our result follows Weyl quantization and is based on the idea that the angular momenta coordinates on the cotangent bundle of the sphere should play the role of momenta coordinates on the flat Euclidean phase space.

4. Spectral Analysis and CQR

In this section we will show that the first part of the spectral analysis of L2 can be generalized to any operator of the form Op(f), where f is a constant of motion on which we have CQR and some extra symmetry is preserved. Any function of angular momenta preserves that symmetry. Let us introduce first some notation.

Let Dλ:𝕊λ0n-1𝕊λ0λn-1 be the diffeomorphism given by Dλ(z)=λz and sλ:T*𝕊λ0n-1T*𝕊λ0λn-1 the symplectomorphism given by

sλ(q,p)=(Dλ(q),(Dλ*)-1(p))=(λq,1λp).    (12)

If λ0 = 1, then sλ is the classical counterpart of the unitary operator Sλ.

Notice that Dλ (respectively sλ) is the restriction of a map defined on ℝn (respectively T*ℝn = ℝ2n) given by the same expression, and they define a multiplicative one parameter group of diffeomorphisms (respectively symplectomorphism), i.e. D1 = Id and Dλ1·λ2 = Dλ1Dλ2 (respectively s1 = Id and sλ1·λ2 = sλ1sλ2. In fact, up to composing with the exponential, Dλ correspond to the (additive) one parameter group of the gradient ∇φ, where φ(x) = ||x||2. Therefore sλ, after composing with the exponential, coincides with the Hamiltonian one parameter group of J∇φ.

Theorem 6. Let f be an admissable constant of motion such that f(tx, t−1ξ) = f(x, ξ), for any t > 0. Then

fλsλ=f1

and

Opλ(fλ)=Sλ*Op1(f1)Sλ

Proof. The first equality follows directly from the condition f(tx,1tξ)=f(x,ξ) and the definition of fλ. Using the expression for Opλ given in Belmonte [16], section 5, we obtain

Op1(f1)Sλu(z)= 1-nλn-14𝕊n-1-{-z}f^1((z+w)||z+w||,θ(z-w)||z-w||)u(λw)dμ(w)=1-nλ1-n4𝕊λn-1-{-λz}f^1(λz+w||λz+w||,θ(λz-w)||λz-w||)u(w)dμ(w)

Notice that the Fourier transform f^1 is computed in the cotangent plane of 𝕊n − 1 at the point q=λz+w||λz+w||. Let v=θ(λz-w)||λz-w||. Then

f^1(q,v)=Tq*𝕊n-1f1(q,p)eip·vdμq*(p)             =Tq*𝕊n-1f1(q,p)eiλ-12p·λ12vdμq*(p)             =λn-12Tλq*𝕊λn-1(f1sλ-1)(λq,p)eip·λvdμq*(p).

Therefore

Op1(f1)Sλu(z)=1-nλn-14𝕊λn-1-{-λz}f^λ(λ(λz+w)||λz+w||,λθ(λz-w)||λz-w||)u(w)dμ(w)=SλOpλ(fλ)u(z)

COROLLARY 7. Let f be an admissable constant of motion such that f(tx, t−1ξ) = f(x, ξ), for any t > 0.

1. Then , for any λ ∈ [0, ∞), we have that

sp(Opd(f))=sp(Opλ(fλ)).

2. If Opλ(fλ) is bounded, for some λ ∈ [0, ∞). Then Opd(f) is bounded and

||Opd(f)||=||Opλ(fλ)||.

Proof. Both identities are direct consequences of the previous theorem and the equalities (5) and (6).

COROLLARY 8. If f is a constant of motion such that f(tx, t−1ξ) = f(x, ξ), for any t > 0, and we have CQR on f, then for any λ ∈ [0, ∞) we have that

sp(Op(f))=sp(Opλ(fλ)).

Remark:

Notice that since lij(tx,t-1ξ)=lij(x,ξ), every function of angular momenta satisfies that condition and therefore we can apply our results on them. However, not every classical constant of motion satisfies that condition, for instance h0 itself does not. The constants of motion satisfying f(tx, t−1ξ) = f(x, ξ) are completely determined by their value on the set {(x, ξ) ∈ ℝ2n/||ξ|| = 1;〈x, ξ〉 = 0}. The fact that not every f is such that f(tx, t−1ξ) = f(x, ξ) is the classical counterpart of the fact that not every quantum constant of motion is completely determined by its restriction to one sphere, as we discussed in subsection 3.2.

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

5. Discussion and Future Work

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 U(𝔤) be the corresponding universal enveloping algebra. In general, Casimir elements by definition belong to the center of U(𝔤). Using the universal property of U(𝔤), we can represent it on C*(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 G, we can identify U(𝔤) with the algebra of left-invariant differential operators on G; it turns out that the operator corresponding to the quadratic Casimir is precisely the Laplace-Beltrami operator on G (endowed with its canonical Riemannian structure). Moreover, if G acts on some Riemannian manifold M and such action satisfies certain condition, then the quadratic Casimir is represented as the Laplace-Beltrami operator on M (this seems to be the case when M is a G-homogeneous space; for instance M = 𝕊n − 1 and G = SO(n)). We can also represent U(𝔤) on C(T*M) as well. So, our theorem and the proof of the corollary show that Weyl-Landsman quantization factorizes both representations up to some lower degree term, at least for the quadratic Casimir and 𝔤 = 𝔰𝔬(n). One should wonder if this holds true for any element of U(𝔤), in particular for any Casimir element. The latter would be already interesting for quartic Casimirs, since they are related with the stress-energy tensor. Heuristically, such result would mean that the representation on C(𝔤*) (and therefore on C(T*M)) can be interpreted as a dequantization of U(𝔤). For details see Barut and Raczka [28] or Fuchs and Schweigert [29].

Coming back to our context, recall that our main purpose is to understand the decomposition of the operators of the form Op(f) through the diagonalization of the Laplace operator on ℝn, where f is a classical constant of motion for h(x, ξ) = ||ξ||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 J*φ is a constant of motion, where J:ℝ2n → 𝔰𝔬*(n) is the canonical moment map and φ ∈ C(𝔰𝔬*(n)). Notice that [J*φ]λ=Jλ*φ. Therefore, studying the problem described in the previous paragraph would lead to some further interesting examples where we might have that our conjecture [Op(f)]λ=Opλ(fλ) holds.

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.

Author Contributions

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.

Funding

TV has been supported by the Fondecyt postdoctoral project 3170122.

Conflict of Interest Statement

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.

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Keywords: example, LaTeX, quantization, constants of motion, reduction, diagonalization, angular momenta

Citation: Belmonte F and Veloz T (2018) On the Classical-Quantum Relation of Constants of Motion. Front. Phys. 6:121. doi: 10.3389/fphy.2018.00121

Received: 06 August 2018; Accepted: 08 October 2018;
Published: 13 November 2018.

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

Copyright © 2018 Belmonte and Veloz. 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.

*Correspondence: Tomas Veloz, tveloz@gmail.com