The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra

Lichnerowicz's algebra of differential geometric operators acting on symmetric tensors can be obtained from generalized geodesic motion of an observer carrying a complex tangent vector. This relation is based upon quantizing the classical evolution equations, and identifying wavefunctions with sections of the symmetric tensor bundle and Noether charges with geometric operators. In general curved spaces these operators obey a deformation of the Fourier-Jacobi Lie algebra of sp(2,R). These results have already been generalized by the authors to arbitrary tensor and spinor bundles using supersymmetric quantum mechanical models and have also been applied to the theory of higher spin particles. These Proceedings review these results in their simplest, symmetric tensor setting. New results on a novel and extremely useful reformulation of the rank 2 deformation of the Fourier-Jacobi Lie algebra in terms of an associative algebra are also presented. This new algebra was originally motivated by studies of operator orderings in enveloping algebras. It provides a new method that is superior in many respects to common techniques such as Weyl or normal ordering.


Introduction
The study of geometry using first quantized particle models has a long history. Notable examples are the study of Pontryagin classes and Morse theory in terms of N = 1 and N = 2 supersymmetric quantum mechanical models [1,2]. The supercharges of those models correspond to Dirac, and exterior derivative and codifferential operators acting on spinors and forms, respectively. The model we concentrate on here describes gradient and divergence operators acting on symmetric tensors and therefore involves no supersymmetries at all. Hence, even though the symmetries of this model are analogous to supersymmetries, no knowledge of superalgebras is required to read these Proceedings. All the above models fit into a very general class of orthosymplectic spinning particle theories studied in detail by the authors in [3]. Spinors, differential forms, multiforms [4,5,6,7,8] and symmetric tensors 1 [9,10] are all fitted into a single framework in that work. Here we focus on the symmetric tensor case, both for its simplicity, and because we want to present new results on the symmetric tensor Lichnerowicz algebra developed in [10]. The underlying classical system is geodesic motion on a Riemannian manifold along with parallel transport of a complex tangent vector. This is described by a pair of ordinary differential equations to which we add further curvature couplings designed to maximize the set of constants of the motion. Of particular interest are symmetries interchanging the vector tangent to the manifold with the tangent vector to the geodesic. These are analogous to supersymmetries and correspond to gradient and divergence operators. This correspondence is achieved by quantizing the model. The complex tangent vector describes spinning degrees of freedom so that wavefunctions are sections of the symmetric tensor bundle. The Noether charges of the theory become operators on these sections. In particular, the Hamiltonian is a curvature modified Laplace operator. In fact, it is precisely the wave operator acting on symmetric tensors introduced some time ago by Lichnerowicz [9] on the basis of its algebraic properties on symmetric spaces [10]. Moreover, the set of all Noether charges obey a deformation of the Fourier-Jacobi Lie algebra sp(2, R) J . The classical model is described in Section 2, while its quantization and relation to geometry are given in Section 3.
Applications, such as higher spin theories [10,14], call for expressions in the universal enveloping algebra U(sp(2, R) J ) involving arbitrarily high powers of the generators. Manipulating these expressions requires a standard ordering, oft used examples being Weyl ordering (averaging over operator orderings) or normal ordering (based on a choice of polarization such that certain operators are moved preferentially to the right, say). In a study of partially massless higher spins [15], we found a new operator ordering scheme to be particularly advantageous [10].
The key idea is to rewrite generators of the sp(2, R) subalgebra, wherever possible, as powers of Cartan elements or the quadratic Casimir operator. Immediately, this scheme runs into a difficulty, namely that the remaining sp(2, R) J generators do not have a simple commutation relation with the quadratic sp(2, R) Casimir. This problem is solved by a trick: we introduce a certain square root of the quadratic Casimir whose rôle is to measure how far states are from being highest weight. Then we use this square root operator to construct modified versions of the remaining sp(2, R) J generators. Instead of a simple Lie algebra, we then obtain an elegant associative algebra, which we denote U(sp(2, R)), with relations allowing elements to be easily reordered. This algebra is described and derived in detail in Section 4. The final Section discusses applications and our conclusions.

The classical model
Let (M, g µν ) be an n-dimensional (pseudo-)Riemannian manifold with an orthonormal frame e m so that 2 ds 2 = dx µ g µν dx ν = e m η mn e n . We consider the motion of an ant x µ (t) -as depicted in Fig. 1 -who carries a complex vector z m (t) (expressed relative to the orthonormal frame) tangent to M . (In physics nomenclature, z m is referred to as commuting spinning degrees of freedom.) The Levi-Civita connection will be denoted by ∇. The ant determines its path and in which direction to hold the complex vector by the system of generalized geodesic ODEs The non-linear couplings to the curvature tensor on the right hand side of these equations have been carefully chosen to maximize the set of constants of the motion. They may obtained by extremizing a generalized energy integral To study constants of the motion, we look for symmetries of this action principle. The most obvious of these are translations of t → t + ξ along the parameterized path traversed by our ant. Infinitesimally this yields the invariance Less trivial, are symplectic transformations of (z m , z * m ), The parameters (α, β, γ) are real and correspond to the Lie algebra sp(2, R). The astute reader will observe that the symmetry transformation of z * m is not the complex conjugate of z m . Nonetheless, treating z m and z * m as independent variables, the above sp(2, R) transformations do leave the action invariant. This is in fact sufficient to ensure existence of corresponding constants of the motion and Noether charges. In the quantum theory, these charges will play an important geometric rôle.
The most interesting symmetries of the model interchange the complex tangent vector z m with the tangent vectorẋ µ to the ant's path Here, D is the covariant variation and is defined by Dv µ ≡ δv µ + Γ µ ρσ δx ρ v σ where Γ denotes the Christoffel symbols. It saves one from having to vary covariantly constant quantities.
The transformations (3) are not an exact symmetry for an arbitrary Riemannian manifold. In fact, the action (2) is invariant only when the locally symmetric space condition holds. Or in other words, the Riemann tensor is covariantly constant. Constant curvature spaces provide an, but by no means the only, example of such a manifold.
To compute constants of the motion we work in a first order formulatioṅ This and the evolution equations (1) also follow from an action principle where the covariant and canonical momenta π µ and p µ are related by Here the spin connection is determined by requiring covariant constancy of the orthonormal frame ∇ ρ e µ m = ∂ ρ e µ m − Γ ν ρµ e ν m + ω ρ m n e µ n = 0. From the first order action (5) we immediately read off the contact one-form p µ dx µ + iz * m dz m which is already in Darboux coordinates, so Poisson brackets follow immediately The Noether charges for the symmetries of the model can now be computed The first of these is the Hamiltonian. We have arranged the symplectic symmetry charges in a symmetric matrix f using the isomorphism between the symplectic Lie algebra and symmetric matrices. The remaining charges appear as a column vector v since they in fact form a doublet representation of sp(2, R). It is important to remember that this latter pair of charges are constants of the motion for locally symmetric spaces only.

Quantization and geometry
Quantization proceeds along usual lines replacing the Poisson brackets by quantum commutators [p, x] = −i and [z, z * ] = . We set = 1 in what follows and represent the canonical momentum as a derivative acting on wavefunctions ψ(x µ ) The spinning degrees of freedom become oscillators acting on a Fock space. Rather than using the standard notation z m = a m annihilating a Fock vacuum a m |0 , we represent |0 = 1 and to preempt their geometric interpretation, set .
or in words -sections of the symmetric tensor bundle SM over M . Therefore, we can now start relating quantum mechanical operations to differential geometry ones on symmetric tensors. Firstly, the quantum mechanical inner product yields the natural inner product for symmetric tensors Furthermore, the covariant momentum corresponds to the covariant derivative iπ µ Ψ = ∇ µ Ψ (it is necessary to contract the open index µ with dx µ for this to hold true for subsequent applications of π µ ). Next we turn to the symplectic symmetries f in (6). We call the off-diagonal charge which simply counts the number of indices of a symmetric tensor N ψ µ 1 ...µs dx µ 1 · · · dx µs = s ψ µ 1 ...µs dx µ 1 · · · dx µs .
We call the diagonal charges as they produce new symmetric tensors by either multiplying by the metric tensor and symmetrizing, or tracing a pair of indices These three operators obey the sp(2, R) Lie algebra We call its quadratic Casimir c = g tr − N(N + n − 2).
Trace-free symmetric tensors with a definite number of indices, (N − s)Ψ = 0 = trΨ, are the highest weight vectors for unitary discrete series representations of this sp(2, R) algebra. The Noether charges v in (6) are linear in momenta and therefore covariant derivatives, when acting on wavefunctions. We call them the gradient and divergence, because they are natural generalizations to symmetric tensors of the exterior derivative and codifferential for differential forms. To be sure This pair of operators forms the defining representation of sp(2, R) It remains to commute the operators div and grad. The result is where ∆ = ∇ µ ∇ µ is the Bochner Laplacian and This relation is closely analogous to that for the exterior derivative and codifferential {d, δ} = ∆ F where ∆ F is the form-Laplacian. Here, since we are dealing with symmetric tensors, the anticommutator is replaced by a commutator. A shrewd reader might sense that the second order operator on the right hand side of (7) should be related to the quantum mechanical Hamiltonian operator. This is indeed the case; calling = ∆ + R ## , we have [div, grad] = − 2R ## , where = −2H so long as an appropriate operator ordering is chosen for the Hamiltonian (a full account is given in [3]). On any manifold Moreover, whenever the symmetric space condition (4) holds, the operator is central In fact, is precisely the wave operator introduced quite some time ago by Lichnerowicz on the basis of its special algebra with gradient and divergence operators [9]. Finally, in the special case of constant curvature manifolds, choosing units in which the scalar curvature R = −n(n − 1), the curvature operator R ## equals the sp(2, R) Casimir so that If we include a further operator ord whose rôle is to count derivatives Its root diagram is given in Fig. 2. From now on we compute in the explicit realization given by its action on sections of the symmetric tensor bundle. Therefore we are working with linear operators, so any algebra we find is automatically consistent and associative. To start with we analyze the sp(2, R) Lie algebra built from {g, N, tr}. Unitary discrete series representations with respect to the adjoint involution tr † = g, N † = N are built from highest weights Φ such that The highest weight module is spanned by We can characterize this representation by the eigenvalue s of N acting on the highest weight, or alternatively by the eigenvalue −s(s + n − 2) of the Casimir c acting on any state in the module. Conversely, given a eigenstate Ψ of N and c, we can determine which discrete series representation it belongs to by repeatedly applying the trace operator tr k Ψ = 0 = tr k+1 Ψ, which implies that Ψ can be expressed in terms of a highest weight vector as Ψ = g k Φ. Our key observation is that it is highly advantageous to introduce the linear operator κ whose eigenvalue acting on Ψ is the depth k, namely where In other words, κ measures how far the symmetric tensor Ψ is from being trace-free. The operator 4 N is simply the right hand side of the tr, g commutator. More important is the square root of the Casimir C which acts on the highest weight Φ as which explains equation (9). Our claim is that for many applications involving high powers of the operators g, N and tr, rather than normal or Weyl orderings it is far more expeditious to work with functions of N , C (up to perhaps an overall power of g or tr). By way of translation, we note that a normal ordered product of g and tr can be expressed as Pochhammer 3 functions of (N , C) : This claim has little significance until we introduce the doublet (grad, div). Indeed, since the Casimir c has a rather unpleasant commutation relation with either of these operators, computing in terms of (N , C) may seem unwise. In fact this is not the case once one appropriately modifies the divergence and gradient operators.
To motivate the claim we return to symmetric tensors. Suppose ϕ µ 1 ...µs is trace-free, then its gradient ∇ (µ 1 ϕ µ 2 ...µ s+1 ) is in general not trace free (unless the divergence of ϕ happens to vanish). Since we would like to work with states diagonalizing both c and N, it is propitious to replace the regular gradient with its trace-free counterpart We denote this operator by grad. Having introduced C and N , it has the simple expression which we take to be its definition acting on any section of the symmetric tensor bundle. It is important to note that although this operator maps trace-free tensors to trace-free tensors, it is designed to maintain how far a more general tensor is from being trace-free. Therefore it does not project arbitrary tensors to trace-free ones. We also introduce a similar definition for a trace-free divergence following from the quantum mechanical adjoint grad † = −div div ≡ div − 1 N + C − 1 grad tr.
Note also that the linear operator N + C − 1 is indeed invertible since its spectrum is 2s + 2k + n − 2 on eigenstates g k Φ (these expressions also make sense in dimensions n = 1, 2 thanks to the operators div and grad tr).
The beauty of the operators ( div, grad) is that they commute with the depth operator κ. This implies Moreover, an easy computation using the definitions (10) and (11) shows that the ordering of gradient and metric operators can be interchanged at the cost of only a rational function of (C, N ) These relations allow us to invert equations (10) and (11) .
In turn we can now compute relations for reordering the gradient and trace operators The final relation we need is for div and grad. After some computations we find This result is valid in constant curvature spaces. The term in square brackets equals [ − 2c] and will be modified accordingly upon departure from constant curvature. We denote the new algebra built from {g, N , tr, C, grad, div, } by U(sp(2, R) J ) and have collected together its defining relations in Fig. 3. As it is defined by linear operators acting on symmetric tensors, associativity is assured. An interesting, yet open, question is whether it can be defined on the universal enveloping algebra of sp(2, R) J . Nonetheless, the explicit symmetric tensor representation guarantees its consistency and therefore we may study it and its representations as an abstract algebra in its own right.

Conclusions
We have presented a detailed study of symmetric tensors on curved manifolds. The key technology employed is the quantum mechanics of a bosonic spinning particle model. Also, many of our constructions were originally motivated by studies of higher spin quantum field theories [10]. The spinning particle model presented here is one of a general class of orthosymplectic spinning particle models that describe spinors, differential forms, multiforms, and indeed the most general tensor-spinor fields on a Riemannian manifold [3].
Notice that we have specified no grading on the symmetric tensor bundle SM . In fact the operator G is the generating function for the equations of motion (and actions) for massless higher spins of arbitrary degree. A very fascinating question is whether such complexes exist for the most general orthosymplectic spinning particle models -preliminary studies suggest an affirmative answer [16]. Another interesting open question is the generality of the algebra U (sp(2, R) J ). For example, does there exist an algebra U (osp(2p|Q) J ) where osp denotes the orthosymplectic superalgebra.
Since the key idea is to include the square root of the Casimir operator in the algebra, higher rank generalizations ought involve the higher order Casimir operators. A positive answer to this question would be most welcome and is under investigation [17].