DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 653–664

QUANTISING A RIGID BODY

RAFFAELE VITOLO

Abstract. This paper concerns the quantisation of a rigid body in the framework of the “covariant quantisation” on a curved spacetime with absolute time proposed by A. Jadczyk and M. Modugno.

We start with a spacetime for a pattern one–body mechanics, which is constituted by a 4–dimensional affine space fibred over time and equipped with a vertical Euclidean metric and an electromagnetic field. Then, we obtain the multi–spacetime for n–body mechanics by taking the n–fold fibred product of the above structure. Eventually, we obtain the spacetime for a rigid body by considering the fibred subbundle of the multi–spacetime defined by a rigid constraint.

We show that the general scheme of the “covariant quantisation” can be easily applied to the rigid body. In particular, we are concerned with the existence and classification of the inequivalent quantum structures.

Introduction

A covariant formulation of classical and quantum mechanics on a curved spacetime with absolute time based on fibred manifolds, jets, non linear connections, cosymplectic forms and Frölicher smooth spaces has been proposed by A. Jadczyk and M. Modugno [6, 7] and further developed in [1, 5, 10, 14, 15] and references therein. We shall briefly call this approach “covariant quantisation”. It presents analogies with geometric quantisation ([3, 4, 8, 13, 12, 16] and references therein), but important novelties as well. In fact, it overcomes typical difficulties of geometric quantisation such as the problem of polarisations; moreover, in the flat case it reproduces the standard quantum mechanics (hence all standard examples).

Then, it would be interesting to treat new physically relevant examples. Here, we discuss an original geometric formulation of classical and quantum mechanics for a rigid body according to the general scheme of “covariant quantisation”. The present analysis is more geometrical than physical, and has to be considered as a preliminar research, especially for what concerns the quantum theory. But we hope

1991 Mathematics Subject Classification. 53C05, 53C15, 55N05, 55N30, 58A20, 58F06, 70H40, 70E99, 81S10.

Key words and phrases. Covariant quantisation, rigid body. This paper is in final form and no version of it will be submitted for publication elsewhere.

that this geometric setting could help understanding concrete physical objects (for examples, moleculae).

We start the paper with a sketch of the essential features of the general “covariant quantisation” following [5, 7, 10, 14, 15]. The classical theory is based on a fibred manifold (“spacetime”) over time equipped with a vertical Riemannian metric (“space–like metric”), a time preserving linear connection (“gravitational connection”) and a 2–form (“electromagnetic field”). The above objects yield a 2–form on the first jet space of spacetime (“phase space”), which is assumed to be closed. This form controls the classical dynamics. The quantum theory is based on a line bundle over spacetime equipped with a Hermitian and universal connection, whose curvature is proportional to the above classical 2–form. This quantum structure yields in a natural way a Lagrangian (hence the dynamics) and quantum operators.

In view of the formulation of classical and quantum mechanics of the rigid body in the framework of the above theory, we proceed in three steps ([11]). Namely, we start with a flat spacetime for a pattern one–body mechanics. Then, we consider an n–fold fibred product of the pattern structure as multi–spacetime for the n–body mechanics. Eventually, we consider the subbundle of the multi–spacetime induced by a rigid constraint as spacetime for the rigid body mechanics. Clearly, this spacetime fits the requirements of the “covariant quantisation”; hence, the general machinery for classical and quantum mechanics can be easily applied to the rigid body. In particular, we discuss the existence and classification of the inequivalent quantum structures. In forthcoming work we shall pursue the physical analysis.

We assume manifolds and maps to be C^{∞}.

Acknowledgments. We would like to thank D. Canarutto, J. Marsden, M. Modugno and C. Tejero Prieto for useful discussions.

This work has been partially supported by MURST, GNFM of CNR and the Universities of Florence and Lecce.

1. Covariant quantisation

We start with a brief sketch of the the covariant formulation of classical and quantum mechanics on a curved spacetime with absolute time proposed by A. Jadczyk and M. Modugno [6, 7] and further developed in [1, 5, 10, 14, 15] and in several other papers. We shall briefly call this approach covariant quantisation. For further details and discussions the reader will refer to the above literature and references therein.

In order to make the independence of classical and quantum mechanics from scales explicit, we introduce the “spaces of units of measurement” [7]. Roughly speaking, a unit space has the algebraic structure of IR^{+} but has no distinguished ‘basis’. The basic objects of our theory (metric, electromagnetic field, etc.) will be valued into scaled vector bundles, that is into vector bundles multiplied tensorially with unit spaces. In this way, each tensor field carries explicit information on its physical dimensions.

Actually, we assume the following basic unit spaces:

– 𝕋, the space of time intervals;

– 𝕃, the space of lengths;

– 𝕄, the space of masses.

We set 𝕋^{−1} := 𝕋^{∗}. We shall use rational tensor powers of unit spaces. We assume the Planck’s constant ℏ ∈ 𝕋^{−1} ⊗ 𝕃^{2} ⊗ 𝕄. Moreover, a particle will be assumed to have a mass m ∈ 𝕄 and a charge q ∈ 𝕋^{−1}⊗𝕃^{3/2}⊗𝕄^{1/2}.

1.1. Covariant classical mechanics.

Assumption G.1. [5, 7] We assume an oriented manifold E of dimension 1 + n equipped with

- a fibring t : E → T over an oriented 1-dimensional affine space T (associated with the vector space 𝕋⊗IR),

- a scaled vertical Riemannian metric g : E → 𝕃^{2}⊗S^{2}V^{∗}E,

- a dt–preserving linear connection K^{♮} : T E → T^{∗}E⊗_{E}T T E,

- a scaled 2–form f : E → (𝕃^{1/2}⊗𝕄^{1/2})⊗Λ^{2}T^{∗}E.

We interpret E as spacetime, t as absolute time map, T as absolute time, g as spacelike metric, K^{♮} as gravitational field and f as electromagnetic field.

With reference to a given particle with mass m ∈ 𝕄 and charge q ∈ 𝕋^{−1} ⊗ 𝕃^{3/2}⊗𝕄^{1/2}, it is convenient to normalise the metric and electromagnetic fields by setting G:= \frac{m}{ℏ}g : E → 𝕋⊗S^{2}V^{∗}E and F := \frac{q}{ℏ}f : E → Λ^{2}T^{∗}E.

We shall refer to fibred chart (x^{0}, x^{i}) on E adapted to a time scale u_{0} ∈ 𝕋. Latin indices i, j, … and Greek indices λ, μ, … will denote space–like and spacetime coordinates, respectively. For short, we shall denote the induced dual bases of vector fields and forms by ∂_{λ} and d^{λ}. The vertical restriction will be denoted by the Checkˇ .

We have the coordinate expression G = G_{ij}u_{0} ⊗ď^{i} ⊗ď^{j}. The fact that K^{♮} is dt–preserving reads in coordinates as K^{♮}_{λ}^{0}_{μ} = 0.

A section s : T → E is said to be a motion.

We assume [5, 7] as phase space for classical mechanics the first jet space [9] of motions J_{1}E; the first jet prolongation of a motion s is said to be its velocity. We denote by (x^{λ}, x_{0}^{i}) the chart induced on J_{1}E. We shall be involved with the natural complemetary maps � : J_{1}E × 𝕋 → T E and θ : J_{1}E ×_{E} T E → V E, with expressions � = u^{0}⊗(∂_{0} + x_{0}^{i}∂_{i}) and θ = (d^{i} − x_{0}^{i}d^{0})⊗∂_{i}. We set θ^{i} ≡ d^{i} − x_{0}^{i}d^{0}.

An observer is defined to be a section o : E → J_{1}E. Let o be an observer. Then, we obtain the map ∇[o] : J_{1}E → 𝕋^{−1}⊗E : j_{e} − o(e). If s is a motion, then we define the observed velocity to be the section ∇[o] ∘ j_{1}s. We define observed kinetic energy and momentum, respectively, as the maps K[o] := \frac{1}{2} G(∇[o], ∇[o]) : J_{1}E → T^{∗}E and Q[o] := θ^{∗}∘G^{♭}(∇[o]) : J_{1}E → T^{∗}E, with coordinate expressions K[o] = \frac{1}{2}G_{ij}x_{0}^{i}x_{0}^{j} d^{0} and Q[o] = G_{ij}x_{0}^{j} θ^{i}.

The linear connection K^{♮} yields [5, 7] an affine connection Γ^{♮} on the affine bundle J_{1}E → E, with coordinate expressions Γ^{♮}_{λ}_{0}^{i}_{μ}^{0} = K^{♮}_{λ}^{i}_{μ}, and the non linear connection γ^{♮}:= � ⌟ Γ^{♮} : J_{1}E → 𝕋^{∗}⊗T J_{1}E on the fibred manifold J_{1}E → T, with coordinate expression γ^{♮} = u^{0} ⊗(∂_{0} + x_{0}^{i}∂_{i} + γ_{0}_{0}^{i}∂_{i}^{0}), where γ_{0}_{0}^{i}:= K^{♮}_{h}^{i}_{k}x_{0}^{h}x_{0}^{k} + 2K^{♮}_{h}^{i}_{0}x_{0}^{h} + K^{♮}_{0}^{i}_{0}. Moreover, Γ^{♮} yields [5, 7] the 2–form Ω^{♮}:= ν[Γ^{♮}]¯∧θ : J_{1}E → Λ^{2}T^{∗}J_{1}E, where ν[Γ^{♮}] is the vertical projection complementary to Γ^{♮} and ¯∧ is the wedge product followed by a contraction with G. We have the coordinate expression Ω^{♮} = G_{ij} (d_{0}^{i} − γ_{0}_{0}^{i}d^{0} + Γ_{h}_{0}^{i}_{h}^{0}θ^{h}) ∧ θ^{j}. The 2–form Ω is non degenerate as dt ∧ Ω ∧ … ∧ Ω is a scaled volume form of J_{1}E.

There is a natural geometric way [5, 7] to couple the gravitational and electromagnetic objects into global objects, in such a way that all mutual relations holding for gravitational objects are preserved for total objects. Later on we shall refer to such total objects and we can forget about the two component fields. In particular, we deal with the total 2–form Ω := Ω^{♮} + \frac{1}{2}F and the total connection γ = γ^{♮} + γ^{e}, where γ^{e} turns out to be the Lorentz force γ^{e} = −G^{♯} ˇ (� ⌟ F).

Assumption G.2. We assume dΩ^{♮} = 0 and dF = 0.

The first equation has several important consequences [5, 7]. In particular, it yields ∇g = 0. The second equation is the first Maxwell equation.

The closed form Ω controls the classical dynamics [5, 10, 10]. It admits local potentials, called Poincaré–Cartan forms, of the type Θ : J_{1}E → T^{∗}E, with coordinate expression Θ = −(\frac{1}{2}G_{ij}x_{0}^{i}x_{0}^{j} + A_{i}x_{0}^{i})d^{0} + (G_{ij}x_{0}^{j} + A_{i})d^{i}, where A : E → T^{∗}E is a local potential of the closed 2–form 2o^{∗}Ω : E → Λ^{2}T^{∗}E and the pullback is taken with respect to the observer o associated with the chosen chart. We define the local Lagrangian associated with a Poincaré–Cartan form Θ to be the form L:= � ⌟ Θ : J_{1}E → T^{∗}E, with coordinate expression L = (\frac{1}{2}G_{ij}x_{0}^{i}x_{0}^{j} +A_{i}x_{0}^{i} +A_{0})u^{0}. The Euler–Lagrange equation, in the unknown motion s, associated with this Lagrangian turns out to be the global equation ∇[γ]j_{1}s = 0, that is ∇[γ^{♮}]j_{1}s = −γ^{e}∘j_{1}s. This equation is just the generalised Newton equation of motion for a charged particle in a given gravitational and electromagnetic field; we assume it to be our classical equation of motion. Moreover, given an observer o, we define the Hamiltonian and the momentum as the maps H[o] := −o ⌟ Θ : J_{1}E → T^{∗}E and P[o] := −ν[o]^{∗}∘V_{E}L : J_{1}E → T^{∗}E (where ν[o] is the induced projection ν[o] : T E → V E), with coordinate expressions H[o] = (\frac{1}{2}G_{ij} x_{0}^{i}x_{0}^{j} − A_{0})d^{0} and P[o] = (G_{ij}x_{0}^{j} + A_{i})d^{i}.

1.2. Covariant quantum mechanics. A quantum bundle (over spacetime) is defined [5, 7] to be a complex line bundle Q → E, equipped with a Hermitian metric h with values in ℂ⊗Λ^{3}V^{∗}E. The choice of such a Hermitian metric allows us to avoid half–densities. A quantum section Ψ : E → Q describes a quantum particle.

Let η denote the vertical volume form induced by g. A local section b : R → 𝕃^{3/2}⊗Q, such that h(b, b) = η, is a local base, said to be normal. We denote the local complex dual base of b by z : Q → 𝕃^{−3/2} ⊗ℂ. If Ψ is a quantum section, then we write locally Ψ = ψb.

The Liouville vector field is the vector field � : Q → V Q : q ↦ (q, q), with coordinate expression � = z∂_{z}.

A family of Hermitian connections on Q → E parametrised by sections of J_{1}E → E determines a Hermitian connection on Q^{↑}:= J_{1}E ×_{E} Q → J_{1}E, called universal.

We define [5, 7] a quantum connection to be a connection � on Q^{↑} → J_{1}E, which is Hermitian, universal, and whose curvature is given by R[�] = iΩ⊗�.

We remark that the equation dΩ = 0 turns out to be just the Bianchi identity for a quantum connection �.

A pair (Q, �) is said to be a quantum structure.

Two complex line bundles Q_{1}, Q_{2} on E are said to be equivalent if there exists an isomorphism of Hermitian line bundles f : Q_{1} → Q_{2} on E (the existence of such an f is equivalent to the existence of an isomorphism of line bundles). Two quantum structures (Q_{1}, �_{1}), (Q_{2}, �_{2}), are said to be equivalent if there exists an equivalence f : Q_{1} → Q_{2} which maps �_{1} into �_{2}.

A quantum bundle is said to be admissible if it admits a quantum connection. Actually, the following theorem holds.

Let us consider the Čech cohomology H^{∗}(E, X) with values in X = IR or X = ℤ, the inclusion morphism i : ℤ → IR and the induced group morphism i^{∗} : H^{∗}(E, ℤ) → H^{∗}(E, IR).

Proposition 1.1. [14, 15] The following conditions are equivalent:

i. there exists a quantum structure on E,

ii. the class [Ω] ∈ H^{2}(E, IR) lies in the subgroup i^{2} (H^{2}(E, ℤ)) ⊂ H^{2}(E, IR). Moreover, inequivalent quantum structures are in bijection with the set

(i^{2})^{−1}([Ω]) × H^{1}(R, IR)/H^{1}(R, ℤ) .

More precisely, the first factor parametrises admissible quantum bundles and the second factor parametrises quantum connections.

The quantum theory is based on the only assumption of a quantum structure, supposing that the background spacetime admits one.

Assumption G.3. [5, 7] We assume a quantum bundle Q equipped with a quantum connection � to be given.

The quantum connection � can be locally expressed [5, 7] as � = �^{∥} + iΘ⊗�, where �^{∥} is the flat connection induced by a local trivialisation of Q^{↑} and Θ is a Poincaré–Cartan form.

All further quantum objects will be derived from the above quantum structure by natural procedures [5, 7].

We have been forced to assume � on the pullback bundle Q^{↑} because of the required link with the 2–form Ω. On the other hand, we wish to derive from � new quantum objects, which are observer independent, hence living on the quantum bundle. For this purpose we follow a successful projectability criterion.

So, by means of a projectability criterion we can exibit [5, 7] a natural Lagrangian on Q, which yields, by a standard procedure, the momentum, the Euler– Lagrange equation (generalised Schrödinger equation) and a conserved form (probability current). The coordinate expression of the Schrödinger equation is

i ((∂_{0} − iA_{0} + \frac{∂_{0}\sqrt{|G|}}{\sqrt{|G|}}) + \frac{1}{2}Δ^{o}_{0})ψ = 0 ,

where Δ^{o}_{0} := G^{hk}(∂_{h} − iA_{h})(∂_{k} − iA_{k}) + ^{\frac{∂_{h}(G^{hk}\sqrt{|G|})}{\sqrt{|G|}}}(∂_{k} − iA_{k}).

Next, we sketch the formulation of quantum operators.

By means of a projectability criterion we can exhibit [5, 7] on the classical phase space a distinguished Lie algebra Q(J_{1}E) ⊂ C^{∞}(J_{1}E) of functions (observables), which are polynomial of second degree with respect to the affine fibring J_{1}E → E and whose second fibre derivative is proportional to the metric G. Functions of the above type are said to be quantisable and have coordinate expression of the type f = \frac{1}{2}f^{0} G_{ij} x_{0}^{i}x_{0}^{j} +f_{i} x_{0}^{i} +f_{0}, where f^{0}, f_{i}, f_{0} are functions of E. The bracket of this algebra is defined in terms of the Poisson bracket and the connection γ.

Then, by classifying [5, 7] the vector fields on Q^{↑}, which preserve the quantum structure and are projectable on Q, we see that their projections on the quantum bundle constitute a Lie algebra of vectors fields, which is isomorphic to the Lie algebra of quantisable functions. These fields can be regarded as pre–quantum operators Z[f] acting on quantum sections.

The quantum bundle over time is defined [5, 7] to be the bundle S → T, whose fibres S_{τ} are constituted by smooth quantum sections at the time τ with compact support. This infinite dimensional complex vector bundle turns out to be F–smooth in the sense of Frölicher [2] and inherits a pre–Hilbert structure via integration on the fibres. A Hilbert bundle can be obtained by completion.

We can prove [5, 7] that there is a unique linear connection χ on S → T, such that ∇[χ] is proportional to the Schrödinger operator.

Eventually, a natural procedure yields [5, 7] the symmetric quantum operator f̂ on the pre–Hilbert bundle associated with every quantisable function f, as a linear combination of the corresponding pre–quantum operator Z[f] and ∇[χ]. We have the coordinate expression

f̂(Ψ) = (f_{0} − i\frac{1}{2}∂_{h}f^{h} − if^{h}(∂_{h} − i A_{h}) − \frac{1}{2} f^{0}Δ^{o}_{0})ψb .

For example, we have

x̂^{α}(Ψ) = x^{α}Ψ , P̂_{j}(Ψ) = −i(∂_{j} + \frac{1}{2} \frac{∂_{j}\sqrt{|G|}}{\sqrt{|G|}})ψb , Ĥ_{0}(Ψ) = (\frac{1}{2}Δ^{o}_{0} − A_{0})ψb .

2. Rigid body

Now, we consider a rigid body and show how it can be quantised according to the scheme of the obove general theory. The procedure consists in three steps:

- we start with a flat “pattern spacetime” for the formulation of one–body classical and quantum mechanics;

- then, we consider the n–fold fibred product of the flat pattern spacetime and related structures as the framework for n–body classical and quantum mechanics;

- eventually, we consider the rigid constrained fibred submanifold of the above n–fold fibred product along with the induced structures as the framework for rigid–body classical and quantum mechanics.

Each of the above steps fits the general setting of the “covariant quantisation” sketched in the above section. So, that scheme can be applied to these specific cases.

2.1. Rigid body classical mechanics.

Pattern one–body mechanics. Following the general scheme, we start by assuming [11] as spacetime t : E → T for one–body mechanics a flat spacetime, which is called pattern spacetime. All objects related to this pattern spacetime are called pattern objects and labelled by the subscript p.

Assumption R.1. We assume as pattern spacetime a 4–dimensional affine space E_{p}, equipped with an affine map t_{p} as time map, the connection K_{p}^{♮} induced by the affine structure as gravitational connection, a space–like metric g_{p} and an electromagnetic field f_{p}. Moreover, we assume dΩ_{p}^{♮} = 0, df_{p} = 0.

A motion s_{p} : T → E_{p} and an observer o : E → J_{1}E are said to be inertial if they are affine maps.

The affine structure induces some trivialisations. We denote by Ē_{p} the vector space associated with E_{p}. Let us consider the maps Dt_{p} : Ē_{p} → 𝕋 ⊗ IR and id_{𝕋^{−1}}⊗Dt_{p} : Ē_{p} → (𝕋^{−1} ⊗𝕋)⊗IR and the induced vector and affine subspaces S_{p}:= Dt_{p}^{−1}(0) ⊂ Ē_{p} and U_{p}:=(id⊗Dt_{p})^{−1}(1) ⊂ 𝕋^{−1}⊗Ē_{p}.

Then, t_{p} : E_{p} → T turns out to be a principal bundle associated with the abelian group S_{p}. Moreover, we have the natural isomorphisms V E_{p} ≃ E_{p} × S_{p} and J_{1}E_{p} ≃ E_{p} × U_{p}.

The axiom dΩ_{p}^{♮} = 0 implies ∇g_{p} = 0; hence, g_{p} can be regarded as a Euclidean metric on S_{p}. Conversely, starting with a Euclidean metric on S_{p}, we would get dΩ_{p}^{♮} = 0.

We stress, that, because of the affine structure of spacetime, Ω is globally exact; in particular, for each inertial observer o, Θ_{p}^{♮}[o] := −K[o] + Q[o] is a distinguished global potential of Ω_{p}^{♮}.

Multi-body mechanics. Then, following the general scheme, we continue by assuming [11] as spacetime t : E → T for n–body mechanics the n–fold fibred product of the pattern spacetime, which is called multi–spacetime and denoted by t_{m} : E_{m} → T. All objects related to this multi–spacetime are called multi–objects and labelled by the subscript m.

Thus, we consider a system on n particles with masses m_{1}, …, m_{n} ∈ 𝕄 and charges q_{1}, …, q_{n} ∈ 𝕋^{−1}⊗𝕃^{3/2}⊗𝕄^{1/2}.

We define m_{0} := ∑_{i} m_{i} ∈ 𝕄 to be the total mass and μ_{i}:= m_{i}/m_{0} ∈ IR^{+} the ith weights. For i = 1, …, n, we introduce n identical copies of the pattern objects E_{i}:= E_{p}, S_{i}:= S_{p}, U_{i}:= U_{p}, g_{i}:= g_{p}, f_{i}:= f_{p}.

Assumption R.2. We assume as multi–spacetime the affine fibred product

E_{m}:= E_{1} ×_{T} ⋯ ×_{T} E_{n} ,

equipped with the associated projection t_{m} as time map, the connection K_{m}^{♮} induced by the affine structure as gravitational connection, g_{m}:=(μ_{1}g_{1}, …, μ_{n}g_{n}) as space–like metric and F_{m}:=(\frac{q_{1}}{ℏ} f_{1}, …, \frac{q_{n}}{ℏ} f_{n}) as normalised electromagnetic field.

We define the normalised metric as G_{m} = \frac{m_{0}}{ℏ} g_{m}.

Of course, we obtain dΩ_{m}^{♮} = 0 and dF_{m} = 0; then, Ω_{r} is globally exact.

Thus, the structure of multi–spacetime is analogous to that of pattern spacetime. The different dimension of the fibre in the two cases has no importance in many respects; hence, most concepts and results can be straightfowardly translated from the first case to the second one. So, we can formulate the classical and quantum mechanics of an n-body analogously to that of a one–body equipped with the total mass and effected by the given multi–metric, multi–gravitational field and normalised multi–electromagnetic field.

On the other hand, the multi–spacetime is equipped with the projections of the fibred product, which provide the information concerning each particle.

Moreover, due to the affine structure, the multi–spacetime is equipped [11] with another important splitting, which is related to the center of mass. Namely, the multi–spacetime splits naturally into the 3–dimensional affine subspace of center of mass and the (3n − 3)–dimensional space of distances relative to the center of mass. This splitting will effect all geometrical, kinematical and dynamical structures, including the equation of motion.

In view of the definition of the center of mass and in order to emphasise its role, we consider a spacetime structure constituted by the copies E_{0} := E_{p} of the pattern spacetime, g_{0} := g_{p} of the pattern metric and K_{0}^{♮}:= K_{p}^{♮} of the pattern gravitational connection. Moreover, we refer to the total mass m_{0} := ∑_{i} m_{i}, the induced normalised metric G_{0} = \frac{m_{0}}{ℏ} g_{0} and the total electromagnetic field F_{0} := ∑_{i} F_{i}. All objects related to this spacetime structure are labelled by the subscript 0.

For each τ ∈ T, we define the center of mass of e ∈ E_{0τ} to be the unique element e_{0} ∈ E_{0τ} such that ∑_{i} μ_{i}(e_{i} − e_{0}) = 0. We say that E_{0} is the space of centers of mass and denote the associated projection by π_{0} : E → E_{0}.

Moreover, we call S_{d}:={v ∈ S_{m} | ∑_{i} μ_{i}v_{i} = 0} ⊂ S_{m} the difference space. Then, we have the affine splitting E → E_{0} × S_{d} : e ↦ (e_{0}, v_{d}) :=(π_{0}(e), e − e_{0}), which is orthogonal with respect to the multi–metric G_{m}.

This splitting is reflected on most multi–objects, whose components are accordingly labelled by the subscripts 0 and d.

Rigid body mechanics. Eventually, we achieve the scheme for a rigid body in the framework of the “covariant quantisation” by considering a rigid constraint on the multi–spacetime and assuming as spacetime t : E → T for the rigid body the constrained subbundle of the multi–spacetime, which is called rigid–body spacetime and denoted by t : R → T. All objects related to this rigid–spacetime are called rigid–body objects and labelled by the subscript r.

In order to avoid trivialities, we suppose throughout the rest of the paper that n ≥ 2.

We consider a set {l_{ij} ∈ 𝕃^{2} | i, j = 1, …, n, i ≠ j, l_{ij} = l_{ji}, l_{ik} ≤ l_{ij} + l_{jk}} and define the subsets

i_{r} : R :={e ∈ E | ∥e_{i} − e_{j}∥ = l_{ij}, 1 ≤ i < j ≤ n} ↪ E ,
i_{a} : R_{a}:={v ∈ S_{d} | ∥v_{i} − v_{j}∥ = l_{ij} , 1 ≤ i < j ≤ n} ↪ S_{d} .

The affine splitting E_{m} → E_{0} ×S_{d} restricts to a splitting R → E_{0} ×R_{a}. Thus, we obtain a curved fibred manifold t : R → T consisting of the cartesian product of the affine bundle t : E_{0} → T with the (spacelike) submanifold R_{a} ⊂ S_{d}.

The metric g_{m} yields an orthogonal projection π_{a} : T S_{d} → T R_{a}. The connection K_{d}^{♮} yields a connection K_{a}^{♮} on the bundle T × R_{a} → T via pullback i_{r}^{∗} and projection π_{a}.

Assumption R.3. We assume as rigid–body spacetime the fibred manifold t : R → T equipped with the induced spacelike metric g_{r}:= i_{r}^{∗}g_{m}, the gravitational connection K_{r}^{♮}:= K_{0}^{♮} × K_{a}^{♮} and the normalised electromagnetic field F_{r}:= i_{r}^{∗}F_{m}.

Regarding the rigid body as a one–body with mass m_{0}, we refer to the normalised metric G_{r} = \frac{m_{0}}{ℏ} g_{r}.

The connection γ_{d}^{♮} yields a connection K_{a}^{♮} on the bundle R_{a} → T via pullback i_{a}^{∗} and projection π_{a}. The connection γ_{r}^{♮} on the bundle J_{1}R → T induced by K_{r}^{♮} turns out to be γ_{r}^{♮} = γ_{0}^{♮} × γ_{a}^{♮}.

The tensor γ_{d}^{e} yields a tensor γ_{a}^{e} on the bundle J_{1}R_{a} via pullback i_{a}^{∗} and projection π_{a}. The tensor γ_{r}^{e} on J_{1}R induced by F_{r} and G_{r} coincides with γ_{r}^{e}.

We obtain Ω_{r} = i_{r}^{∗}Ω_{m}. Hence, dΩ_{r} = 0 and Ω_{r} is globally exact.

The geometry of R_{a} depends on the initial mutual positions of particles and is time independent. In particular, particles can either lie on a straight line, or lie on a plane, or “fill” the whole space. We say R to be degenerate in the first case, weakly non degenerate in the second case, non degenerate in the third case. Of course, if n = 2, then R is degenerate; if n = 3, then R can be degenerate or weakly non degenerate. It can be easily proved [11] that the choice of a point r ∈ R yields the following diffeomorphisms:

– if R is degenerate, then R_{a} ≃ S^{2}, where S^{2} is the unit sphere in IR^{2};

– if R is weakly non degenerate, then R_{a} ≃ SO(3, S_{p});

– if R is non degenerate, then R_{a} ≃ O(3, S_{p}).

Henceforth, in this section, we shall refer only to the most interesting (weakly) non degenerate case. Moreover, we shall refer only to one of the two connected components of R_{a} (which is diffeomorphic to SO(3, S_{p})), for continuity reasons.

The velocity space of R splits as J_{1}R ≃ (E_{0} × U_{0}) × (𝕋^{−1} ⊗T R_{a}). So, to understand the geometry of J_{1}R we shall concentrate on the space T R_{a}.

According to the classical formula for the velocity of a rigid body [11], for each (d_{1}, …, d_{n}; v_{1}, …, v_{n}) ∈ T R_{a} ⊂ T S_{d}, there is a unique ω ∈ 𝕃^{−1} ⊗S_{p} such that v_{i} = ω × d_{i}, where × is the cross product induced by the metric g_{p}. Hence, we obtain the fibred linear isomorphism over R_{a}

T R_{a} → R_{a} × (𝕃^{−1}⊗S_{p}) : (d_{1}, …, d_{n}; v_{1}, …, v_{n}) → (d_{1}, …, d_{n}; ω) .

This result can also been interpreted in terms of the classical parallelisation of the tangent space of a Lie group, by recalling the Lie algebra so(3, S_{p}) ≃ 𝕃^{2} ⊗Λ^{2}S_{p}^{∗} and the Hodge isomorphism ∗ : 𝕃^{2}⊗Λ^{2}S_{p}^{∗} → 𝕃^{−1}⊗S_{p}.

In virtue of the above parallelisation, the metric G_{a} can be regarded as the inertia tensor G_{a} : R_{a} × ((𝕃^{−1}⊗S_{a}) × (𝕃^{−1}⊗S_{a})) → 𝕋⊗IR given by (Ω, Ω′) ↦ ∑_{i}(G_{i}(d_{i}, d_{i})g_{a}(ω, ω′) − G_{i}(d_{i}, ω)G_{i}(d_{i}, ω′)).

Just to compare the above scheme to the standard dynamics of a rigid body, we can prove [11] that the the equation of motion becomes the Euler system of equations

∇[γ_{0}^{♮}]j_{1}s_{0} = γ_{0}^{e}∘j_{1}s_{r} , ∇[γ_{a}^{♮}]j_{1}s_{a} = γ_{a}^{e}∘j_{1}s_{a} ,

where ∇[γ_{0}^{♮}]j_{1}s_{0} is the acceleration of the center of mass, ∇[γ_{a}^{♮}]j_{1}s_{a} is the angular acceleration, γ_{0}^{e}∘j_{1}s_{r} turns out to be the total force and γ_{a}^{e}∘j_{1}s_{a} turns out to be the total momentum of forces.

2.2. Rigid body quantum mechanics. The setting of the above section shows that we can apply straightfowardly the “covariant quantisation” scheme to the rigid body.

First, we analyse the existence and classification of quantum structures according to Proposition 1.1.

The form Ω_{r} is exact, hence the existence condition is fulfilled. So, we have just to compute all possible inequivalent quantum structures. We recall that [Ω_{r}] = 0 ∈ H^{2}(R, IR) due to the exactness of Ω_{r}.

If R is (weakly) non degenerate, then the universal covering π : SU(2) → R_{a} is a principal bundle whose group is π_{1}(R_{a}) = ℤ_{2}. This group acts on the right on SU(2) via deck transformations, which, in this case, turn out to be multiplication by the identity matrix I or by −I. It also acts on the left on ℂ with two representations ρ_{f} and ρ_{s}, respectively the trivial representation and the representation via multiplication by +1 or −1. Accordingly, we have two bundles associated with π, namely

Q_{f} = SU(2) ×_{ρ_{0}} ℂ  and  Q_{s} = SU(2) ×_{ρ_{1}} ℂ .

Of course, Q_{f} = R_{a} × ℂ → R_{a} is the trivial bundle and Q_{s} → R_{a} is a non trivial bundle whose cocycle is the same as the cocycle of π. We believe that the second bundle might be related to spin.

Moreover, both Q_{f} and Q_{s} have a natural flat connection, which we denote by �_{f}^{∥}, �_{s}^{∥}, respectively.

Theorem 2.1. (Classification I) Let R be (weakly) non degenerate. Then, i. the only inequivalent admissible quantum bundles on R are Q_{f} and Q_{s}; ii. the only inequivalent quantum structures are (Q_{f}, �_{f}) and (Q_{s}, �_{s}), with

�_{f} = �_{f}^{∥} + iΘ_{f} ⊗� , �_{s} = �_{s}^{∥} + iΘ_{s}⊗� ,

where Θ_{f} and Θ_{s} are two global potentials of Ω. Proof. In fact, we have

(i^{2})^{−1}([Ω_{r}]) = ker i^{2} ≃ H^{2}(R, ℤ) ≃ H^{2}(SO(3), ℤ) = ℤ_{2} ;

hence we have only the trivial bundle and a non trivial bundle as representatives of the two equivalence classes of admissible quantum bundles. These bundles coincide with Q_{f} and Q_{s}.

The second statement comes from the fact that there is exactly one equivalence class of quantum connections on Q_{f} and Q_{s} (see Proposition 1.1).

Theorem 2.2. (Classification II) Let R be degenerate. Then,

i. the only (inequivalent) admissible quantum bundle on R is the trivial bundle Q:= R × ℂ;

ii. the only (inequivalent) quantum structure on R is (Q, �), with

� = �^{∥} + iΘ⊗� ,

where �^{∥} is the natural flat connection on Q → R and Θ is a global potential of Ω_{r}. Proof. It comes from H^{2}(R, X) = H^{2}(S^{2}, X) = X, with X = IR or X = ℤ, H^{1}(S^{2}, IR) = H^{1}(S^{2}, ℤ) = 0, and i^{2} : ℤ → ℝ the inclusion map.

Then, we can choose each one of the above quantum structures and apply the machinery of the “covariant quantisation”. In particular, we can easily write the Schrödinger equation and the Hamiltonian operator.

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Department of Mathematics “E. De Giorgi”, University of Lecce Via per Arnesano, 73100 Lecce, ITALY E-mail: vitolo@ilenic.unile.it