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Quantum mechanics of a generalised rigid body

We consider the quantum version of Arnold's generalisation of a rigid body in classical mechanics. Thus, we quantise the motion on an arbitrary Lie group manifold of a particle whose classical trajectories correspond to the geodesics of any one-sided-invariant metric. We show how the derivation of the spectrum of energy eigenstates can be simplified by making use of automorphisms of the Lie algebra and (for groups of Type I) by methods of harmonic analysis. We show how the method can be extended to cosets, generalising the linear rigid rotor. As examples, we consider all connected and simply-connected Lie groups up to dimension 3. This includes the universal cover of the archetypical rigid body, along with a number of new exactly-solvable models. We also discuss a possible application to the topical problem of quantising a perfect fluid.


Introduction
Euler showed long ago that the classical mechanics of a free, rigid body moving about a fixed point (e.g. its centre of mass) simplifies, in that the motion of the angular momentum relative to the body depends (via the Euler equations) only on its own initial value and not on the position of the body in space. Moreover, the magnitude of this angular momentum is conserved.
Rigid body motion can be thought of as the geodesic motion (with respect to a onesided-invariant metric) of a particle on the Lie group manifold SO (3). In 1966, Arnold [1] showed how Euler's simplifications generalise to geodesic motion of a particle on any Lie group manifold G. In a nutshell, the space of angular momenta of the rigid body is replaced by the dual of the Lie algebra, the motion in that space is governed by a set of generalised Euler-Arnold equations, and the motion is restricted to an orbit of the coadjoint representation of G (generalising the spheres of constant magnitude of angular momentum for the rigid body).
Arnold even went so far as to generalise to infinite-dimensional Lie groups, such as the group, Diff(M ), of diffeomorphisms of a manifold M . 1 For the subgroup of volumepreserving diffeomorphisms, the Arnold-Euler equations describe the motion of an incompressible fluid on M ; for the particular case where M is the two-torus, Arnold obtained a celebrated result on the stability of fluid flows and thence an estimate of the unreliability of long-term weather forecasting.
The quantum mechanics of a rigid body has also been known for some time (see, e.g., [3]). Here we attempt to carry out the generalisation to arbitrary G, as well as to coset spaces G/H, where H is a proper subgroup of G. We then study, as illustrative examples, all connected and simply-connected Lie groups up to dimension 3. 2 It is too much to hope that these quantum-mechanical systems are all exactly solvable, because the archetypical case of an asymmetric rigid body already provides a counterexample: there we can, at best, reduce the problem of finding the energy spectrum to diagonalization of finite-dimensional matrices. But in many cases, we encounter systems that are exactly solvable, using standard results in representation theory and harmonic analysis. 3 In this way, we uncover a number of new, exactly-solvable models in quantum mechanics, generalising a rigid body. We hope that some of these will turn out to be of physical interest. We also hope that our results may provide some guidance towards the solution of a problem that is of some topical interest, namely the quantisation of incompressible fluids. The quantisation of a compressible fluid has been carried out using standard methods of quantum field theory in [4,5]. Classically, an incompressible fluid is easily obtained as a limit of a compressible fluid, by restricting to flows in which the fluid velocity is much lower than the fluid's sound speed. One cannot impose such a restriction at the quantum level. 4 Nevertheless, a quantum incompressible fluid may exist sui generis, and a generalisation to infinite-dimensional Lie groups of the approach we develop here may provide a means to describe it.
The outline is as follows. In the next Section, we discuss the formalism. In §3 we quickly dispatch the 1-dimensional Lie group R, where the dynamics is that of a free particle in 1-d and generalise to R n . In §4, we solve the unique non-Abelian group in 2 dimensions, which is isomorphic to the group of orientation-preserving affine transformations of R. There are two proper subgroups isomorphic to R, up to conjugation, and we are able to formulate and solve the quantum mechanics problem on the corresponding coset space for one of these. In §5, we study the groups that arise in dimension 3, ending with our first compact example, the universal cover SU (2) of the rigid body group, which is perhaps where we ought to have begun.

Formalism
Before we can attempt to solve the quantum mechanics of a free particle on a Lie group G or coset space G/H, we must first define it. A proper definition of quantum mechanics is still lacking (and occupies much learned discussion in the literature) and so we shall simply follow our noses, in an attempt to come up with something suitable. We begin by discussing the the dynamics on a group G and then generalise to a coset space G/H.

Quantum mechanics on G
What are the necessary ingredients for the quantum mechanics of a free particle moving on a Lie group manifold G?
Firstly, we need a Hilbert space of physical states. Every Lie group admits a unique (up to a multiplication by a constant) left (say) invariant measure µ and so an obvious candidate for the Hilbert space for the dynamics on G is the space L 2 (G, µ) of functions f : G → C that are square-integrable w.r.t. µ. 5 Secondly, we need some operators that act on the Hilbert space L 2 (G, µ) and that play the rôle of observables. The obvious candidates are the group elements themselves, which have an action on f ∈ L 2 (G, µ) by left multiplication of the argument of the function. But these are not good candidates for observables when the dynamics is that of a free particle! Indeed, consider the simplest example of free-particle motion on the Lie group R, with co-ordinate x ∈ R. The classical action is dt 1 2ẋ 2 . If we attempt to compute the 2-point correlation function of x using the path integral formalism in Fourier space, we get x(t)x(t ) = dω ω 2 e iωt ; because of the pole at ω = 0, the Fourier transform is undefined. In the language of physicists, the 2-point function is infra-red divergent, and so x cannot be an observable. 6 So, we need to find some other operators to play the rôle of observables. Now, rightinvariant vector fields send some dense subspace of smooth functions in L 2 (G, µ) into themselves and thus we take a subspace of the observables to be the space of right-invariant 5 In fact, from a physical point of view, we also wish to include states that do not belong to L 2 (G, µ).
These generalised eigenvectors (also known as non-normalisable modes or scattering states) can be thought of as living in the dual space to the maximal subspace of L 2 (G, µ) on which the operators corresponding to observables are defined. For G = R, for example, the generalised eigenfunctions of the operators x and ∂x, correspond to the Dirac delta function and e ikx , respectively. The corresponding eigenvalues are included automatically in the functional-analytic definition of the spectrum as the complement in C of the resolvent set.
6 A similar argument also shows that we cannot study the quantum mechanics on a Lie group G by assuming that the energy eigenstates are localised near some g ∈ G and then using the fact that G is locally like R n to approximate the path integral as a Gaussian integral. As we shall see in explicit examples later on, the energy eigenstates correspond to wavefunctions that are completely delocalised in G.
vector fields on G, otherwise known as the Lie algebra, g, of G. 7 The quantum commutators of the theory, which are defined by composition of the operators corresponding to observables, are then just given by the Lie brackets. We can enlarge the space of observables further by passing to the universal enveloping algebra. This enables us to obtain our third and final ingredient, which is a prescription for the dynamics, in the form of a Hamiltonian. Classically, this is easy to do. A Lie group is equipped with an infinite family of metrics, namely the one-sided-invariant metrics. Picking one such metric, a Hamiltonian (which is just the kinetic energy in this free-particle case) is given by plugging the velocity vectorġ corresponding to a classical trajectory g(t) ∈ G into the metric.
Quantum mechanically, the corresponding Hamiltonian should be some operator in our space of observables. Now the one-sided-invariant metrics on a group G of dimension n are in 1-to-1 correspondence with the positive-definite, symmetric, n × n, real matrices. We thus take the Hamiltonian to be the quadratic element of the universal enveloping algebra given by where α ij are the entries of such a matrix and where {X R i } is a basis for the Lie algebra of right-invariant vector fields. This Hamiltonian is nothing but the Laplace-Beltrami operator corresponding to the given metric.

Quantum mechanics on G/H
In the case of a coset space, we can proceed in much the same way, although there are obstacles in that we may not be able to find a G-invariant measure or metric on G/H.
As regards the measure, a simple necessary and sufficient condition for existence may be formulated as follows [6]. 8 Let G and H have dimension n and k, respectively and let {ω 1 , . . . , ω n−k } be a set of linearly-independent left-invariant 1-forms on G whose restriction to T e (H) vanishes. The (n−k)-form ω ≡ ω 1 ∧ω 2 · · ·∧ω n−k is unique up to multiplication by a constant and G/H has a G-invariant volume form Ω (and hence a G-invariant measure) iff. dω = 0. If this holds, ω is given by pulling back Ω using the natural projection π : G → G/H. We can define observables by taking the right-invariant vector fields on G and pushing them forward to G/H using π. This yields well-defined vector fields on G/H, since right invariance of the vector fields on G guarantees an identical outcome when pushing forward the vectors at all points in the preimage π −1 (p) ⊂ G of a point p ∈ G/H. 9 To define a classical Hamiltonian, we need a metric on G/H. For our methods to be applicable, we need this metric to be G invariant. (Such a metric is not guaranteed to exist, since if it did, we could use it to make a G-invariant volume form.) It turns out that there is a unique such metric on G/H for every metric on G (given by the quotient metric) that is invariant under left translations in G and right translations in H. For the quantum Hamiltonian, this amounts to requiring that the Hamiltonian on G built out of right-invariant vector fields on G, has a vanishing Lie bracket with the right-invariant vector fields in H. Pushing this Hamiltonian forward to G/H then yields the quantum Hamiltonian on G/H.

Method of solution
We now describe our strategy for solving these quantum mechanical systems. The spectrum of energy eigenstates, for example, is given by the spectrum of the Laplace-Beltrami operator corresponding to the given metric and so we could attempt to solve this partial differential eigenvalue equation directly. But as we shall see, these PDEs are rather uninviting in their appearance and are not always amenable to solution by humble methods such as separation of variables.
A more sure-fire approach is to try to exploit the group-theoretic and geometric structure underlying the dynamics. Indeed, the Hilbert space carries a highly reducible, unitary representation of G (the left regular representation) and so an obvious step is to try to reduce this representation into its unitary, irreducible representations ('unirreps', henceforth). The action of the Hamiltonian, which is an element of the universal enveloping algebra, then decomposes into actions on the irreducible subspaces, which we can attack individually. 10 The decomposition of functions on G/H generalises the Fourier transform on R and is a standard tool of harmonic analysis. Let us describe it schematically for the simplest case, where H = {e}. Let the unirreps of G be D λ , where λ labels the equivalence classes.
Given a function f in L 2 (G, µ g ) (where we put a subscript g on µ to remind ourselves that it is a measure on G), define a representation-valued integral transform bỹ where D denotes the Hermitian conjugate of D. The Plancherel formula then says that there exists a measure µ λ on the space of equivalence classes of unirreps, such that the transform may be inverted, to obtain and π * (X g )f = X g f (π(g )) = X g f (π(g)) = X gh f (π(g)). These are equal by right-invariance. 10 This decomposition corresponds to the fact that the classical motion is confined to a coadjoint orbit of G. Each coadjoint orbit has an associated reduced phase space and mathematicians have used the presumed existence of a corresponding reduced quantum-mechanical system to find the unirreps of a variety of Lie groups. See [8] for details.
(As an example, for the case G = R with x ∈ R, let the invariant measure be µ x = dx. The irreps are 1 dimensional, D λ (x) = e iλx and the representation-valuedf (λ) is just the usual C-valued Fourier transform. By the Parseval theorem, we have that the Plancherel measure is µ λ = dλ 2π .) The inversion is thus an explicit decomposition of L 2 (G, µ g ) into subspaces carrying the irreps D λ , with measure µ λ . Notice that the decomposition involves, in general, a direct integral rather than a direct sum. For non-compact groups, some set of unirreps may have zero Plancherel measure, meaning that they do not appear in the decomposition.
At this point, a triumvirate of problems appear. The first is that, even when G is non-compact (as we shall see explicitly in the dimension two example), we will need to know all the unirreps of G to decompose the left regular reps on G/H for arbitrary H. 11 The classification of the unirreps of a general Lie group is an unsolved problem 12 and we will only be able to make progress in this way in cases where the irreps appearing in the decomposition are known. The second is that the decomposition is not unique for certain Lie groups, known as Type II groups; it is unclear how one should deal with these. The third is that the unirreps of non-Abelian groups are often multi-dimensional (or even infinite dimensional in the non-compact case). Thus, even if we manage to effect the decomposition, we may be left with a matrix or differential operator representing the action of the Hamiltonian whose eigenvalues we are unable to find. This is the obstruction that occurs in the rigid body case and we shall see that it arises in several other cases in dimension three.
Nevertheless, we shall see that many cases are tractable. Two simplifying observations reduce the burden. The first is that while a generic one-sided-invariant Hamiltonian has n(n + 1)/2 arbitrary coefficients, we may be able to simplify its form, w. l. o. g., using automorphisms of the Lie algebra g. These are just the invertible linear maps from g to itself that preserve the structure constants of the algebra (in some basis) and we may use them to simplify the Hamiltonian without changing the form of the operators representing the algebra elements. Unfortunately, finding the automorphisms (which may be outer) generally requires a brute-force approach. 13 The second observation is that there is a group, K, of residual automorphisms that preserve the form of the simplified Hamiltonian. If non-trivial, K provides further constraints on the form of the spectrum. Consider, for example, the maximal compact subgroup, K ⊂ K. We can simplify the problem of finding the spectrum by decomposing the Hilbert space into irreps of K, because Schur's lemmata then imply that the Hamiltonian H acts as the identity on the irreps of K and will not mix inequivalent irreps appearing in the 11 For H compact, L 2 (G/H) is isomorphic to the subspace of L 2 (G) defined by the functions that are constant within the cosets. So all the unirreps in L 2 (G/H) already appear in L 2 (G). 12 Examples of general cases where the classification is known are when G is either a compact [9] or an exponential Lie group [10]. An exponential Lie group is a special case of a solvable group, for which the exponential map is a diffeomorphism. 13 Some simplifying tricks may be found in [11], which classifies the automorphism groups for all Lie algebras up to dimension five. This work also makes it clear that there is no simple way to treat direct product groups, because one cannot use the automorphisms to block-diagonalize the Hamiltonian into operators acting only on the separate factors.
decomposition (it can mix equivalent irreps, however). In fact, it is sometimes convenient to consider an even larger subgroup of automorphisms, namely those that preserve the form of the simplified Hamiltonian up to a permutation of the latter's coefficients. We call this the group of permuting automorphisms and denote it by J. Evidently, there is a homomorphism from J to the permutation group of the coefficients with kernel K and so K is a normal subgroup of J. To give an example of how J may be useful, given one energy eigenvalue, corresponding to a state carrying an irrep of J, one can find other energy eigenvalues of states in the irrep by effecting the corresponding permutations of the coefficients.

Dimension one and R n
In one dimension, the only Lie algebra is [X, X] = 0, and the corresponding (connected and simply-connected) group is isomorphic to R. We might as well do R n , n ∈ N, while we are at it. The non-trivial automorphisms of the algebra are all outer and consist of all invertible linear maps from R n to itself, viz. GL(n, R); a transformation in SO(n, R) ⊂ GL(n, R) can be used to write the most general Hamiltonian in diagonal form and then a diagonal matrix in GL(n, R) can be used to write the Hamiltonian in the form where {X 1 , . . . , X n } is some basis for the Lie algebra. The unirreps of the Abelian group R n are one-dimensional, with action on C given by multiplication by e i k· x , with k ∈ R n . The decomposition of the left regular representation (in which the Lie algebra acts as X i = ∂ ∂x i ) into irreps is achieved by means of the Fourier transform The energy eigenvalues are thus given by E = k · k ≥ 0, with Plancherel measure d n k (2π) n . In this case, then, we are easily able to find the spectrum without recourse to the group, O(n, R), of residual automorphisms of (3.1). We could, of course, also decompose the left regular rep into irreps of the group O(n, R), by passing to hyperspherical polar coordinates, whereupon the decomposition reduces to that of functions on the hypersphere. This is hardly a simplification, since the only invariant subspace of both R n and O(n, R) is spanned by the constant function. There is one simplifying consequence, however, which is that the invariant subspaces of R n that are carried into one another by the action of O(n, R), namely all those with equal values of k · k, must be degenerate in energy, as indeed they are.

Dimension two
We now illustrate the decomposition of the left regular rep on f ∈ L 2 (G, µ) into irreps, and the subsequent derivation of the spectrum, in the case where G is a Lie group of dimension 2. There is a unique non-Abelian 2-d Lie algebra (up to isomorphism), with Lie bracket [X, Y ] = Y in some basis {X, Y }. The corresponding connected and simply-connected group is the group of orientation-preserving affine transformations of the real line. We denote this by G = Aff(R) + , also known as the 'ax + b group' with a > 0. This group is non-unimodular and so we must be careful to pick the left-invariant Haar measure for µ.
In fact, it turns out to be much easier to consider the right regular rep, where the spectrum can be derived, fortuitously, without the heavy machinery of harmonic analysis. However, we feel that it is helpful to see how this machinery works in this simplest nontrivial case and so we postpone discussion of the right regular rep until the end of this Section.

Aff(R) + group properties and irreps
Aff(R) + is isomorphic to the group of matrices under matrix multiplication. Its algebra is isomorphic to and the left-invariant vector fields are Each of these yields the desired algebra [X, Y ] = Y , with the Lie brackets of left-invariant fields with right-invariant fields vanishing. The group is both non-Abelian and non-compact and has 2 infinite-dimensional unirreps There is also a family of 1 dimensional unirreps, labelled by c ∈ R, which act by multiplication on the Hilbert space C [12].

Decomposition of the left regular representation into irreps
By analogy with the similarity transform that block diagonalizes a finite dimensional reducible matrix representation, we seek an invertible linear map between the vectors f ∈ L 2 (G, µ) and vectors in the Hilbert spaces of the irreps. Following (2.2), we first define integral transforms 1415f where p ∈ {±} and D now denotes the complex conjugate of D. Note that the desired group transformations of thef s are fixed by the properties of the representation matrices D(g) and the Haar measure µ g . Schematically, the action T (g ) of the group element g is: and thus thef p (and similarly thef c ) indeed transform under an irrep of G.
It is then the inversion of the above integral transforms -the reconstruction of the original f ( a b ) from thef p (s, t) andf c -that determines the weights with which each irrep appears in the left regular representation. To this end we observe that, using the counting measure p=± over the two infinite dimensional irreps, it is possible to obtain a resolution of the identity on the group coordinates: 16 and we can therefore write the inversion as p=± is the Plancherel measure on the irreps, and is notably zero on the 1-d c irreps, which do not feature in the decomposition of the left regular representation. Again, we 14 The integral transform in (2.2) is representation-valued; here we transform the 'representation matrix elements', with 'indices' s, t instead. 15 There are some subtleties in the harmonic analysis of this and other non-unimodular groups [13,14], which we gloss over here. 16 The equivalent relation on the irreps is just the Schur orthogonality relation dadb explicitly compute the group action on our formula. Under left translation by g transform under the irrep action (4.5).

The spectrum of the left regular representation
Following the rubric of §2, the quantum Hamiltonian is given by to put the Hamiltonian in the simplified form H = A(X 2 + Y 2 ), A > 0, which we will assume hereafter.
We are now in a position to solve for the spectrum of the left regular representation, which is the set of eigenvalues −E of the Hamiltonian 17 This PDE looks hard to solve (in particular, it is not separable in the coordinates (a, b)), but its solution is made simple by using the group structure of (4.7) and integrating by 17 We write −E as we have used the mathematicians' convention of skew-Hermitian group generators, leading to a negative definite spectrum. We expect E ≥ 0.
The resulting ODE is a modified Bessel equation, whose solutions are well-known [15]. Selecting from these only generalised eigenfunctions of L 2 (R + , ds s ) (which in turn yields generalised eigenfunctions of L 2 (Aff(R) + , dadb a 2 ) after recomposition), the eigenvalues and eigenfunctions are We have explicitly checked that this is an eigenfunction of the Laplacian, with eigenvalue −Aν 2 = −E.

The spectrum of the coset representation
We now consider quantum mechanics on the quotients of Aff(R) + by subgroups isomorphic to R. There are two such subgroups up to conjugation, namely H = { 1 b 0 1 }, 18 One way of doing this would be to integrate over the delta function in D p (s, t; a b ) in the decomposition (4.7) whence, for example, f p(1,0) denoting a derivative with respect to the first argument. are a → g a a for G/H and b → g a b + g b for G/H . To formulate quantum mechanics we require a G-invariant measure. The modular function on G is ∆(a, b) = 1/a. Both H and H are unimodular, so only H (with a = 1) yields an invariant measure on G/H, given by da a . Imbued with a group action, the functions f (Σ) ∈ L 2 (G/H, dµ) form a representation of G, which may similarly be decomposed into irreps, following the same logic as before. We first define an integral transform of functions in this spacẽ Now the appropriate decomposition of the identity is impling the inversion theorem From this we read off that the coset representation on L 2 (G/H, da a ) decomposes into a direct integral of the 1-d irreps labelled by c, with Plancherel measure dc. 19 Note that, as expected from the arguments of §2, the pushforwards of right-invariant fields, given by π * (X R ) = −a∂a, π * (Y R ) = 0 and π * (X R ) = −b∂ b , π * (Y R ) = −∂ b , are well-defined on G/H and G/H , respectively, but that the pushforward π * (Y L ) = a∂ b is not.
Note that the irreps appearing in the coset rep on G/H are distinct from those appearing in the left regular rep on G (where the Plancherel measure is given by the counting measure on the irreps ±). As we remarked earlier, this can only happen because H is non-compact.
The eigenvalue equation reduces to: with a spectrum E = A(2πc) 2 ∈ [0, ∞) and uniform measure on c.

The spectrum of the right regular representation
In fact, it is much easier to solve for the spectrum of the right regular representation on G, which is unitarily equivalent and so should yield identical results. For this, we do not need any of the machinery of harmonic analysis. Indeed, the left-invariant vector fields in where E > 0 and λ ∈ R is the separation constant. Notice that we obtain precisely the spectrum we had for the left regular representation, namely two copies (labelled by ±) of an infinite number of copies (labelled by λ) of each E > 0.

Dimension three
Up to isomorphism, there are nine classes, I-IX, of 3-d Lie algebras (two of which contain infinitely many algebras), which were first catalogued by Bianchi [16,17]. Bianchi I is isomorphic to R 3 , cf. §3. Bianchi II-VII all take the form of a semi-direct product group R 2 R and can be treated in a unified way; we treat the Heisenberg group (Bianchi II) in detail below, in analogue to our treatment of Aff(R) + in §4. Within the treatment of the Heisenberg case, references are made to Tables 1 and 2, from which the reader may substitute in the corresponding results for his or her favourite Bianchi algebra II-VII. Many of these results come from [18] (though our conventions differ by minus signs), in which the unitary duals are obtained by induction via the Mackey machine and the Plancherel measures are derived. Ref. [18] also contains a result on the spectrum of the Laplace operator corresponding to an arbitrary one-sided-invariant Riemannian metric, which would correspond to the spectrum of our quantum-mechanical Hamiltonian. Unfortunately, the result given is incorrect. The erroneous result is that (in our sign convention) the spectrum is given by [−c, ∞), where c < 0 is the constant scalar Riemann curvature of the metric. The result is obtained using a result of Donnelly [19] that applies only to manifolds whose sectional curvatures are all non-positive. Unfortunately, it is a theorem [20] that every unimodular Lie group has a strictly positive sectional curvature (unless all of its curvatures vanish). The Bianchi groups discussed in [18] are all unimodular with non-vanishing scalar curvature and so the result never applies. Indeed, it is easy to furnish a counterexample: the Bianchi II group has Haar measure dxdydz in our co-ordinates and the Laplace operator is purely derivative. Thus, the constant function on the group is a generalised eigenfunction, and the spectrum includes 0 < −c. Many further counterexamples appear below.

Bianchi II-VII
Denote by H the Heisenberg group formed by the set of unit determinant upper triangular 3 × 3 real matrices under matrix multiplication. Note that the group composition law is a semi-direct product of those of (x, y) ∈ R 2 and z ∈ R: (x, y, z) • (x , y , z ) = (x + x , y + y + zx , z + z ) ( Table 1a). The corresponding algebra is then: with the Lie bracket [Z, X] = Y . The group is unimodular and has Haar measure dxdydz (generally det F (−z)dxdydz, see Table 1a).
H has a continuum of inequivalent infinite-dimensional irreps labelled by k ∈ R/{0} D k (s, t; x y z ) = e ik(xs+y) δ(s + z − t), (5.1) which act on functions φ(t) ∈ L 2 (R, dt) (Table 1b). There are also one dimensional irreps, labelled by (µ, ν) ∈ R 2 , D µ,ν ( which act by multiplication on the Hilbert space C [12]. As before, to decompose L 2 (H) we require completeness relations on both the irrep (or unitary dual) coordinates, and the group coordinates. The former comes from the Schur orthogonality relation: The 4π 2 |k| suggests that in the latter we should use the Plancherel measure |k| 4π 2 (Table 1c) dkdsdt |k| 4π 2 D k (s, t; x y z )D k (s, t; x y to achieve the correct resolution of the identity w.r.t the Haar measure dxdydz. Note again that the 1d (µ, ν) irreps do not feature in the decomposition of L 2 (H), which takes the form: 20f The Hamiltonian in the left regular rep is a quadratic form a ij X i X j of the Lie algebra components X j = (X, Y, Z) T , which may be simplified by the automorphism M ij , where Without loss of generality, we may then write H = X 2 + AY 2 + Z 2 (Table 1d,f). All automorphisms M ij of the algebra induce an action (also an automorphism) on the group coordinates e α i X i → e α i M ji X j , and, through the D matrices, also an action on the irrep coordinates, whereby unirreps are mixed. Unlike Aff(R) + , there is a non-trivial group of residual automorphisms, locally isomorphic to SO(2, R), that preserve the simplified form of the Heisenberg Hamiltonian, viz. (Table 1e), These only mix unirreps with identical spectra, identifying a possible degeneracy. These are the pairs of irreps labelled by k and −k, as we will see presently.
The actions of the Lie algebra elements are X R f ( . This yields an eigenvalue equation: The ODE (Table 2a), which is just the Schrödinger equation for the simple harmonic oscillator, is solved by parabolic cylinder functions [15], of which the normalisable ones 20 Generally,f ν(k, k2) being the Plancherel measure of Table 1c.
H n are the Hermite polynomials, as seen in the wavefunction of the harmonic oscillator. Summing over the possible values of k, the spectrum of the left regular representation is The spectra of the left regular representations of the other Bianchi groups are obtained in same way. When decomposed into their actions on the irreps, the Laplacians give second order ODEs in the form of a Schrödinger equation, f + ( − V (s))f = 0 for some positive potential V (s), and where ∝ E. Only the ODEs of Bianchi II, III and V are analytically solvable, but we can still find the spectrum of each Bianchi group as follows. 21 The potentials of Bianchi IV, VI (q < 0) and VII tend to zero as one of either s → ∞ or s → −∞; the scattering theory of such one sided potentials predicts a continuum of solutions for > 0 [22], and their degeneracies are given by the respective Plancherel measure. For Bianchi VI (q > 0), V (s) → ∞ as s → ±∞, and we expect a infinite discrete set of 'bound' solutions, the energies of which may be approximated by the WKB method, and are seen to tend smoothly to zero as k → 0. 22 Thus for all Bianchi groups, the set of all eigenvalues in the left regular representation is a continuum E > 0 or E ≥ 0, 23 and we know the density of these states in all but one case (Bianchi VI, q > 0).

Bianchi IX -SL(2, R)
The Bianchi IX algebra is semi-simple and isomorphic to the Lie algebra of the group SL(2, R). The latter group has fundamental group π 1 (SL(2, R)) = Z and its universal cover is peculiar in that it is not a matrix Lie group, having no finite-dimensional faithful representations. The unirreps are given and the Plancherel theorem derived in [23] and we simply quote the results here, in the notation of [24]. The irreps are given by the principal series C (τ ) q , with 0 ≤ τ < 1 and q > 1 4 , the two discrete series D ± l , with l > 0, and those in the exceptional domain, E (τ ) q with 0 ≤ τ < 1 and τ (1 − τ ) < q ≤ 1 4 . Denoting σ = q − 1 4 > 0 the Plancherel decomposition is Note that only the l irreps with l > 1 2 appear. 21 In all cases, insisting that the Hamiltonian commute with one of the right-invariant vector fields results in an exactly-solvable model. 22 The nature of the WKB method as an asymptotic expansion of the exact energy eigenvalues means that, as the first term in the expansion for Bianchi VI (q > 0) is an elliptic integral, the true values cannot be expressed in terms of simple functions. 23 E = 0 is included if there is a point in the dual space coordinates that allows a flat potential.
Unfortunately the spectrum is not obtainable analytically, in general, as we can see by focussing our attention on the irreps with τ ∈ {0, 1 2 } (which are, in fact, the principal series irreps appearing in the Plancherel measure on SL(2, R)). The Lie algebra ([X − Y, H] = −2(X + Y ), [H, X + Y ] = 2(X − Y ) and [X + Y, X − Y ] = −2H, written in the SU (1, 1) basis) admits SO(2, 1) automorphisms, where the SO(2) subgroup is {H, X + Y }. This simplifies the Hamiltonian to H = A(X − Y ) 2 + B(X + Y ) 2 + CH 2 , which on the principal irreps have action (A + e + g cos(4φ))f − g(iσ + 2) sin(4φ)f + (1 + σ 2 )(g cos(4φ) − e)f = −Ef where e = 1 2 (B + C), g = 1 2 (B − C) and f (φ) ∈ L 2 (S 1 , dφ) and is odd in φ if τ = 1 2 , and even otherwise. φ ∈ (−π, π]. When the ODE is put in normal form, the first order term in the WKB approximation to E reduces to an elliptic integral, implying that the ('all-order') exact result is not a simple function of exponentials, logarithms and powers. However, in the symmetric limit g → 0 (i.e. B → C), the ODE and spectrum for these irreps tend to those of a free particle on S 1 . (2) Finally, we turn to the archetypical rigid body, starting our discussion with the more familiar case of SO(3), before addressing its universal cover, SU (2). Examples abound in nature of systems whose configuration is described by an arbitrary rotation, and whose Hamiltonian is also rotation invariant. Indeed, via spinning tops and the rotational spectroscopy of gases, both the classical and quantum dynamics of the rigid body have been well studied. We calculate the spectrum of the quantum mechanical rigid body using the formalism of the previous Sections, and show it to be consistent with known results.
The Wigner D-matrices form, respectively, a Schur orthogonality relation on the irreps, and a completeness relation on the group coordinates: where j,m,k = ∞ j=0 +j m=−j +j k=−j . Using the same notation, one may decompose the left regular representation via the decomposition of functions f ∈ L 2 (SO(3)): Note that, for a given value of j, there are (2j + 1) vectorsf j mk (labelled by k ∈ {−j, −j + 1, . . . , j}) in the decomposition of f which, under the group action, transform under the jth irrep of SO(3). This is a manifestation of a general result for compact groups, the Peter-Weyl theorem [9,26], where an irrep of dimension d features d times in the Hilbert space of the left regular representation. The Plancherel measure is thus 2j + 1 times the counting measure on j ∈ {0, 1, 2, . . . }. These 2j + 1 copies are degenerate in energy in the rotationally-invariant systems we consider, but are easily observed, e.g. in the rotation spectra of molecules in the presence of an electric field [27].
Having obtained the decomposition of the regular representation, we again proceed to construct the Hamiltonian. The structure constants of the Lie algebra are ijk and the automorphism group is immediately seen to be SO(3). We may thus pick the special orthogonal transformation necessary to diagonalize the real symmetric matrix of the Hamiltonian coefficients, leaving H = AJ 2 1 + BJ 2 2 + CJ 2 3 for some (positive) eigenvalues A, B and C. 24 The residual automorphisms are the group of order 4 generated by {e iπJ i }, isomorphic to the Klein group, whose effect is to change the sign of two of the three J i . The Klein group is Abelian, with 4, 1-d irreps. The permuting automorphisms are the group of order 24 generated by {e i π 2 J i }, isomorphic to the symmetry group of a cube, with 2 1-d, 1 2-d, and 2 3-d irreps.
The action of H on the left regular representation is via the self-adjoint right-invariant vectors Making use of the properties we find the action of H on the irreps to be 24 In other words, we can pick principal axes with respect to which the moment of inertia tensor is diagonal.
where J j i is the canonical form of the angular momentum generator in the jth matrix rep. Finding the spectrum of the left regular rep then amounts to diagonalizing a real symmetric matrix, whose elements are linear combinations of A, B and C, for each j. This diagonalization is simplified by first decomposing with respect to the Klein group of residual automorphisms. Thus, the 3-d j = 1 irrep decomposes into one copy of each of the non-trivial irreps of the Klein group and the spectrum is found to be E = A+B 2 , B+C 2 , or C+A 2 , with a 3-fold degeneracy of each. 25 For j = 2, the 5-d irrep decomposes into one copy of each of the non-trivial irreps of the Klein group and two copies of the trivial irrep. By Schur's lemmata, the latter two can mix and finding the spectrum requires us to diagonalize a 2 × 2 matrix. Unfortunately, no closed form is available for the spectrum at arbitrary j. However, thus computed, these energies (and indeed the degeneracies, measured by the strength of spectral features) are precisely those observed in molecular microwave spectra [27,28], where A, B and C are reciprocals of twice the principal moments of inertia of the molecule.
The group SO(3) furnishes us also with the archetype of a coset representation. This is seen, for example, in a linear molecule (such as CO or CO 2 , but not H 2 O) where the system's configurations under a rotation about a given axis are equivalent. The orientation of the linear molecule is then instead given by the two angles of SO(3)/SO(2) ∼ = S 2 .
We choose β and γ to be the two remaining coordinates, whereas α parametrises the coset SO(2). We can make a closed, SO(3)-invariant volume form on the group, namely i(− sin αdβ + sin β cos αdγ) ∧ i(cos αdβ + sin β sin αdγ) = sin βdβ ∧ dγ, from the two rightinvariant one forms corresponding to J R 1 and J R 2 . Let the measure be sin βdβdγ The coset representation decomposes thus: Note that the extra degeneracy of the higher dimension irreps is gone. The coset representation contains each j irrep exactly once and the Plancherel measure is now just the counting measure on j ∈ {0, 1, 2, . . . }.
The Hamiltonian on the coset space is formed from one on SO(3). We require the latter to be both SO(3) right-invariant (i.e. it commutes with all J L i ), namely H = 25 The j = 1 irrep 'decomposes' into one of the 3-d irreps of the cube group of permuting automorphisms and so we find that we can obtain all of the energy eigenvalues from any one by permutations of A, B, and C. 26 In the simplest case m = 0, D l 0k ( 2l+1 Y l,−k (β, γ), the usual spherical harmonic functions.  (2)), we use the pushforward π * of the simple map π( α β γ ) = ( β γ ), which gives π * (∂ α ) = 0, π * (∂ β ) = ∂ β and π * (∂ γ ) = ∂ γ . The result, is the Laplacian in spherical polar coordinates and the decomposition contains a piece which simply multiplies each vectorf j 0k in the decomposition by Aj(j + 1). The action on thef j mk , m = 0 is more complicated, but does not yield eigenfunctions. Thus the possible states have energy Aj(j + 1) and degeneracy 2j + 1 for each j ∈ {0, 1, 2, . . . }, as seen in the heat capacity of molecular hydrogen.
The results for the case of SU (2) are analogous and easily obtained. The irreps of SU (2) contain, in addition to those of SO (3), the irreps with j ∈ { 1 2 , 3 2 , . . . }. The Plancherel measure is 2j + 1 times the counting measure on j ∈ {0, 1 2 , 1, 3 2 , . . . }, by Peter-Weyl. The groups of residual and permuting automorphisms are the preimages under the projection map π : SU (2) → SO(3) of the corresponding groups for SO (3). The residual group of automorphisms of SU (2) is isomorphic to the (non-Abelian) quaternion group of order 8 [29] and features, in addition to the 4 1-d irreps of the Klein group, a 2-d irrep. 27 The j = 1 2 irrep of SU (2) decomposes into the 2-d irrep. The energy eigenvalue A+B+C 4 is 4-fold degenerate, with a 2-fold degeneracy coming from rotational invariance via the Plancherel measure and another 2-fold degeneracy that can be understood as a Kramers degeneracy arising from time-reversal invariance.

Discussion
We have given a formulation of the quantum mechanics of a particle on a Lie group manifold G, whose classical motions follow geodesics of an arbitrary one-sided-invariant metric. We have also shown how this formulation can sometimes be extended to motion on a coset space G/H. In many cases, the system turns out to be exactly solvable using standard methods of harmonic analysis and representation theory; we have also described the various obstacles to doing so.
We have studied all Lie algebras up to dimension three as examples. The prototypical example of a rigid body is somewhat atypical, since it is the only case in d ≤ 3 that is simplyconnected and also compact, with finite-dimensional unitary, irreducible representations. In non-compact cases, the spectrum is continuous and (at least in the cases where we are able to compute it) takes the simple form E ≥ 0. This does not mean, of course, that the physics of these different systems is the same. Indeed, what is important for the physics is not just the spectrum of eigenvalues, but also the degeneracy of those eigenvaluesthe density of states, in physicists' language. The degeneracies, given by the Plancherel measure in each case, are markedly different and we expect that they will lead to markedly different physics.
It would be of interest to try to reproduce our results using path integral methods. 28 We expect this to be non-trivial, even in the case where G is compact, where the quantum mechanics on G/H can be described by the non-linear sigma model in 0+1-d with target space G/H. Indeed, just as for the quantum mechanics of a free particle on R n (where the generalised eigenfunctions e ik·x correspond to a uniform probability density for finding the particle anywhere in R n ), the energy eigenstates for general G/H are delocalised (in the sense that the moduli-squared of the generalised eigenfunctions do not tend to 0 at infinity) in all cases where we have obtained their explicit form. This is in accord with our earlier observation that attempting to evaluate the path integral by assuming that the particle is localised leads to a contradiction. So, evaluation of the path integral will need to take account of global considerations.
Whilst these exactly-solvable, free-particle models may seem somewhat esoteric, one should bear in mind that the notion of dynamics on a manifold with invariance under a Lie group action is a rather general one in physics. We anticipate that that the exactly-solvable models provide a useful starting point for doing quantum-mechanical perturbation theory for dynamical systems of this type, generalising the use of the harmonic oscillator and free particle. Indeed, the rigid body can be used as the starting point for a perturbative study centrifugal distortions of polyatomic molecules [27].
The quantum mechanical systems of a rigid body and a free particle, corresponding to G = SO(3) and G = R n are, of course, ubiquitous in physics and chemistry. Do there exist dynamical systems to which we can apply the results for other G? To give just one example, it has been suggested that the dynamics of a certain ellipsoidal vortex solution in a fluid can, under certain restrictions, be modelled as a system moving on SO(2, 1) [31].
The most interesting application, from our point of view, is to the quantisation of a perfect, incompressible fluid moving on a manifold M . This corresponds to G = SDiff(M ), which is infinite dimensional. Two ways to proceed suggest themselves. The first is to consider an M which admits a complex structure, for which one can consider a simplified toy model of a fluid, in which one insists that the fluid maps be not just smooth, but rather be analytic. SDiff(M ) is then replaced by the group of (volume-preserving) conformal holomorphisms of M , which is finite-dimensional and can be quantised using the methods described here. These simplified systems (which we propose to call 'analytic fluids') may present a useful intermediate step on the road to quantising a genuine incompressible fluid.
The second approach is to try to define the infinite-dimensional group SDiff(M ), for some M , as a limit of a sequence of finite-dimensional groups. For the case where M is either a 2-sphere or a 2-torus (which bore so much fruit for Arnold at the classical level), it has been observed [32,33] that the structure constants of the Lie algebra of SDiff(M ) can 28 For the rigid body case, see [30]. be obtained (in a specific basis) as a convergent sequence of those of SU (N ), as N → ∞. Unfortunately, it is known [34] that these algebras are not isomorphic, so it is unclear whether this observation can be used in a mathematically-sound way.  (a) F (z) defines the group composition (x, y, z) • (x , y , z ) = (A x , A y , z + z ), where (b) The irreps of the group, labelled by k and sometimes also the discrete k 2 . D k(,k2) (s, t; All real values of the parameters are permitted, provided the matrix is non-singular. See also [21].   (c) The analytic solutions f (s) of the ODE (where possible) in the notation of the corresponding DLMF reference.f k(,k 2 ) (s, t) = f (s)g(t), where g(t) ∈ L 2 (R, det F (t)dt) is arbitrary.