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...

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Дата:	2007
Автори:	Hallowell, K., Waldron, A.
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Мова:	English
Опубліковано:	Інститут математики НАН України 2007
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra / K. Hallowell, A. Waldron // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 17 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1472122025-02-09T21:05:56Z The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra Hallowell, K. Waldron, A. 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. This paper is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson. It is a pleasure to thank the organizers of the 2007 Midwest Geometry Conference, and especially Susanne Branson for a truly excellent meeting in honor of Tom Branson. We thank David Cherney, Stanley Deser, Rod Gover, Andrew Hodge, Greg Kuperberg, Eric Rains and Abrar Shaukat for discussions. 2007 Article The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra / K. Hallowell, A. Waldron // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 17 назв. — англ. 1815-0659 Waldron https://nasplib.isofts.kiev.ua/handle/123456789/147212 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	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.
format	Article
author	Hallowell, K. Waldron, A.
spellingShingle	Hallowell, K. Waldron, A. The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Hallowell, K. Waldron, A.
author_sort	Hallowell, K.
title	The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra
title_short	The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra
title_full	The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra
title_fullStr	The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra
title_full_unstemmed	The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra
title_sort	symmetric tensor lichnerowicz algebra and a novel associative fourier-jacobi algebra
publisher	Інститут математики НАН України
publishDate	2007
url	https://nasplib.isofts.kiev.ua/handle/123456789/147212
citation_txt	The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra / K. Hallowell, A. Waldron // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 17 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
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first_indexed	2025-11-30T19:03:38Z
last_indexed	2025-11-30T19:03:38Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 089, 12 pages The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier–Jacobi Algebra? Karl HALLOWELL and Andrew WALDRON Department of Mathematics, University of California, Davis CA 95616, USA E-mail: hallowell@math.ucdavis.edu, wally@math.ucdavis.edu Received July 21, 2007; Published online September 13, 2007 Original article is available at http://www.emis.de/journals/SIGMA/2007/089/ Abstract. 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. Key words: symmetric tensors; Fourier–Jacobi algebras; higher spins; operator orderings 2000 Mathematics Subject Classification: 51P05; 53Z05; 53B21; 70H33; 81R99; 81S10 1 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 supersym- metric quantum mechanical models [1, 2]. The supercharges of those models correspond to Dirac, and exterior derivative and codifferential operators acting on spinors and forms, respec- tively. 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 superalge- bras 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 tensors1 [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 differen- tial 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 ?This paper is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson. The full collection is available at http://www.emis.de/journals/SIGMA/MGC2007.html 1See [11] for a flat space discussion of the symmetric tensor theory and [12, 13] for its origins in higher spin theories. file:hallowell@math.ucdavis.edu file:wally@math.ucdavis.edu http://www.emis.de/journals/SIGMA/2007/089/ http://www.emis.de/journals/SIGMA/MGC2007.html 2 K. Hallowell and A. Waldron to the manifold with the tangent vector to the geodesic. These are analogous to supersymmet- ries 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 intro- duced 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 en- veloping 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 (aver- aging 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 Ũ(sp(2,R)), with relations allowing elements to be easi- ly reordered. This algebra is described and derived in detail in Section 4. The final Section discusses applications and our conclusions. 2 The classical model Let (M, gµν) be an n-dimensional (pseudo-)Riemannian manifold with an orthonormal frame em so that2 ds2 = dxµgµνdx ν = emηmne n. We consider the motion of an ant xµ(t) – as depicted in Fig. 1 – who carries a complex vector zm(t) (expressed relative to the orthonormal frame) tangent to M . (In physics nomenclature, zm 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 ∇ẋµ dt = ẋνRµ ν m nz ∗ mz n +∇µRm n r s z ∗ mz nz∗rz s, ∇zm dt = iRm n r s z nz∗rz s. (1) 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 S = ∫ dt { 1 2 ẋµgµν ẋ ν + iz∗m ∇zm dt + 1 2 Rm n r s z ∗ mz nz∗rz s } . (2) 2Although the metric signature impacts the unitarity of the quantum Hilbert space of our model, all the results presented here hold for arbitrary signature. Similarly, none of our results depend on the existence of a global orthonormal frame. The Symmetric Tensor Lichnerowicz Algebra 3 M CI x (t)µ zm(t) TM Figure 1. An ant laden with a complex tangent vector. 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 δxµ = ξẋµ, δzm = ξżm. Less trivial, are symplectic transformations of (zm, z∗m), δzm = −αzm + βz∗m, δz∗m = γzm + αz∗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 zm. Nonetheless, treating zm 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 zm with the tangent vector ẋµ to the ant’s path δxµ = i(z∗µε− zµε∗), Dzµ = ẋµε. (3) Here, D is the covariant variation and is defined by Dvµ ≡ δvµ + Γµ ρσδxρvσ 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 ∇κRµνρσ = 0, (4) 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 formulation ẋµ = πµ. This and the evolution equations (1) also follow from an action principle S(1) = ∫ dt { pµẋ µ + iz∗mż m − 1 2 πµg µνπν + 1 2 Rm n r s z ∗ mz nz∗rz s } , (5) 4 K. Hallowell and A. Waldron where the covariant and canonical momenta πµ and pµ are related by πµ = pµ − iωµ m nz ∗ mz n. Here the spin connection is determined by requiring covariant constancy of the orthonormal frame ∇ρeµ m = ∂ρeµ m − Γν ρµeν m + ωρ m neµ n = 0. From the first order action (5) we immediately read off the contact one-form pµdx µ + iz∗mdz m which is already in Darboux coordinates, so Poisson brackets follow immediately {pµ, x ν}PB = δν µ, {zm, z∗n}PB = iδm n . The Noether charges for the symmetries of the model can now be computed H = 1 2 πµg µνπν − 1 2 Rm n r s z ∗ mz nz∗rz s, f = ( z∗mz ∗m z∗mz m z∗mz m zmz m ) , v = ( iz∗µπµ izµπµ ) . (6) 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. 3 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(xµ) pµ = 1 i ∂ ∂xµ . The spinning degrees of freedom become oscillators acting on a Fock space. Rather than using the standard notation zm = am annihilating a Fock vacuum am\|0〉, we represent \|0〉 = 1 and to preempt their geometric interpretation, set z∗µ = dxµ, zµ = ∂ ∂(dxµ) . Therefore, wavefunctions become Ψ = ∞∑ s=0 ψµ1...µs(x)dx µ1 · · · dxµs , 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 〈Φ\|Ψ〉 = ∫ M ∞∑ s=0 s! √ g φµ1...µsψµ1...µs . The Symmetric Tensor Lichnerowicz Algebra 5 Furthermore, the covariant momentum corresponds to the covariant derivative iπµΨ = ∇µΨ (it is necessary to contract the open index µ with dxµ for this to hold true for subsequent applications of πµ). Next we turn to the symplectic symmetries f in (6). We call the off-diagonal charge N = dxµ ∂ ∂(dxµ) , which simply counts the number of indices of a symmetric tensor N ψµ1...µsdx µ1 · · · dxµs = sψµ1...µsdx µ1 · · · dxµs . We call the diagonal charges g = dxµgµνdx ν , tr = ∂ ∂(dxµ) gµν ∂ ∂(dxν) , as they produce new symmetric tensors by either multiplying by the metric tensor and sym- metrizing, or tracing a pair of indices g ψµ1...µsdx µ1 · · · dxµs = g(µ1µ2 ψµ3...µs+2)dx µ1 · · · dxµs+2 , tr ψµ1...µsdx µ1 · · · dxµs = s(s− 1)ψµ µµ1...µs−2dx µ1 · · · dxµs−2 . These three operators obey the sp(2,R) Lie algebra [N, tr] = −2 tr, [N,g] = 2g, [tr,g] = 4N + 2n. 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, grad = dxµ∇µ, div = ∂ ∂(dxµ) ∇µ, because they are natural generalizations to symmetric tensors of the exterior derivative and codifferential for differential forms. To be sure gradψµ1...µsdx µ1 · · · dxµs = ∇(µ1 ψµ2...µs+1)dx µ1 · · · dxµs+1 , divψµ1...µsdx µ1 · · · dxµs = s∇µψµµ1...µs−1dx µ1 · · · dxµs−1 . This pair of operators forms the defining representation of sp(2,R) [N,grad] = grad, [N,div] = −div, [tr,grad] = 2div, [div,g] = 2grad. It remains to commute the operators div and grad. The result is [div,grad] = ∆−R##, (7) where ∆ = ∇µ∇µ is the Bochner Laplacian and R## ≡ Rµ ν ρ σdxµ ∂ ∂(dxν) dxρ ∂ ∂(dxσ) . 6 K. Hallowell and A. Waldron 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 [�,g] = [�,N] = [�, tr] = 0. Moreover, whenever the symmetric space condition (4) holds, the operator � is central [�,div] = [�,grad] = 0. 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 [div,grad] = �− 2c. (8) If we include a further operator ord whose rôle is to count derivatives [ord,g] = [ord,N] = [ord, tr] = 0, [ord,grad] = grad, [ord,div] = div, [ord,�] = 2�, then, {ord,g,N, tr,grad,div,�} form a maximal parabolic subgroup of sp(4,R) up to the rank 2 deformation by the sp(2,R) Casimir in (8). On flat manifolds M , the operators {g,N, tr, grad,div,�} obey the Fourier–Jacobi Lie algebra of sp(2,R). In the next Section, we present a novel reformulation of its universal enveloping algebra based on introducing a certain square root of the Casimir operator c. 4 The Fourier–Jacobi algebra Let us first collect together the deformed Fourier–Jacobi Lie algebra built from geometric oper- ators on constant curvature spaces [tr,g] = 4N + 2n, [N, tr] = −2 tr, [N,g] = 2g, [N,grad] = grad, [N,div] = −div, [tr,grad] = 2div, [div,g] = 2grad, [div,grad] = �− 2c. 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. The Symmetric Tensor Lichnerowicz Algebra 7 tr grad ord 0 N, ord N +1 0 div +1 +2 −1 g+2 −2 Figure 2. The root diagram for the Fourier–Jacobi Lie algebra. 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 NΦ = sΦ, trΦ = 0. The highest weight module is spanned by {Φ,g Φ,g2Φ, . . .}. 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 repre- sentation it belongs to by repeatedly applying the trace operator trkΨ 6= 0 = trk+1Ψ, which implies that Ψ can be expressed in terms of a highest weight vector as Ψ = gk 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 κ ≡ N − C − 1 2 , (9) where N ≡ N + n 2 , C ≡ √( n− 2 2 )2 − c. In other words, κ measures how far the symmetric tensor Ψ is from being trace-free. The operator 4N is simply the right hand side of the tr, g commutator. More important is the 8 K. Hallowell and A. Waldron square root of the Casimir C which acts on the highest weight Φ as C Φ = [ s+ n− 2 2 ] Φ, 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 Pochhammer3 functions of (N , C) : g tr = (N − C − 1)(N + C − 1), ⇒ : (g tr)m :≡ gmtrm = 1 4m ( C − N + 1 2 ) m ( C +N − 1 2 ) m . 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 ϕµ1...µs is trace-free, then its gradient ∇(µ1 ϕµ2...µs+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 ∇(µ1 ϕµ2...µs+1) − s 2s+ n− 2 g(µ1µ2 ∇µϕµ3...µs+1)µ. We denote this operator by g̃rad. Having introduced C and N , it has the simple expression g̃rad ≡ grad− g div 1 N + C − 1 , (10) 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 d̃iv ≡ div − 1 N + C − 1 grad tr. (11) Note also that the linear operator N + C − 1 is indeed invertible since its spectrum is 2s+ 2k+ n − 2 on eigenstates gkΦ (these expressions also make sense in dimensions n = 1, 2 thanks to the operators div and grad tr). The beauty of the operators (d̃iv, g̃rad) is that they commute with the depth operator κ. This implies N d̃iv = d̃iv (N − 1), N g̃rad = g̃rad (N + 1), C d̃iv = d̃iv (C − 1), C g̃rad = g̃rad (C + 1). 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 ) g̃radg = g g̃rad N + C − 1 N + C + 1 , tr d̃iv = N + C − 1 N + C + 1 d̃iv tr. 3Recall that the Pochhammer symbol is defined as (x)m ≡ x(x + 1) · · · (x + m− 1). The Symmetric Tensor Lichnerowicz Algebra 9 These relations allow us to invert equations (10) and (11) grad = 1 2 N + C − 3 C − 1 g̃rad + 1 2C N + C − 3 N + C − 1 g d̃iv, div = 1 2 d̃iv N + C − 3 C − 1 + 1 2 g̃rad tr N + C − 3 C(N + C − 1) . In turn we can now compute relations for reordering the gradient and trace operators tr g̃rad = (N + C + 1)(N + C − 3) (N + C − 1)2 g̃rad tr, d̃iv g = g d̃iv (N + C + 1)(N + C − 3) (N + C − 1)2 . The final relation we need is for d̃iv and g̃rad. After some computations we find d̃iv g̃rad = C2(N + C + 1)(N + C − 3)2 (C + 1)(C − 1)(N + C − 1)3 g̃rad d̃iv + 2C2(N + C + 1) (C + 1)(N + C − 1)2 [ � + 2 ( C + n 2 − 1 )( C − n 2 + 1 )] . 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, g̃rad, d̃iv,�} by Ũ(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. 5 Conclusions We have presented a detailed study of symmetric tensors on curved manifolds. The key technol- ogy 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 spin- ning particle models that describe spinors, differential forms, multiforms, and indeed the most general tensor-spinor fields on a Riemannian manifold [3]. There are many applications and further research avenues. One simple question is that given the strong analogy between the theory of differential forms and the symmetric tensor one presented here, are there symmetric tensor analogs of de Rham cohomology? The answer is yes. Recall, for example, the Maxwell detour complex d δ 0 −→ Λ0M −→ Λ1M → · · · · · · → Λ1M −→ Λ0M −→ 0. \| ↑ δd (12) This is mathematical shorthand for Maxwell’s electromagnetism in curved backgrounds. The physics translation is to replace the sequence of antisymmetric tensor bundles (Λ0M, Λ1M, Λ1M , Λ0M) by the words (gauge parameters, potentials, field equations, Bianchi identities). 10 K. Hallowell and A. Waldron C N = N C tr g = (N + C + 1)(N − C + 1) g tr = (N + C − 1)(N − C − 1) N tr = tr (N − 2) N g = g (N + 2) C tr = tr C C g = g C N d̃iv = d̃iv (N − 1) N g̃rad = g̃rad (N + 1) C d̃iv = d̃iv (C − 1) C g̃rad = g̃rad (C + 1) g̃radg = g g̃rad N + C − 1 N + C + 1 tr d̃iv = N + C − 1 N + C + 1 d̃iv tr tr g̃rad = (N + C + 1)(N + C − 3) (N + C − 1)2 g̃rad tr d̃iv g = g d̃iv (N + C + 1)(N + C − 3) (N + C − 1)2 d̃iv g̃rad = C2(N + C + 1)(N + C − 3)2 (C + 1)(C − 1)(N + C − 1)3 g̃rad d̃iv + 2C2(N + C + 1) (C + 1)(N + C − 1)2 [ � + 2 ( C + n 2 − 1 )( C − n 2 + 1 )] Figure 3. Defining relations for the Ũ(sp(2,R)J) algebra. Then the fact that (12) is a complex implies that Maxwell’s equations δdA = 0 are gauge invariant because δdd = 0, and subject to a Bianchi identity as δδd = 0. An analogous complex exists for symmetric tensors although not in general backgrounds, for brevity we give the flat space result [12, 10] grad div 0 −→ SM −→ SM → · · · · · · → SM −→ SM −→ 0, \| ↑ G where G = �− graddiv + 1 2 ( grad2 tr + g div2 ) − 1 2 g ( � + 1 2 graddiv ) tr. 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 Ũ (sp(2,R)J). For example, does there exist an algebra Ũ (osp(2p\|Q)J) where osp denotes the orthosymplectic superalgebra. 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Deser S., Waldron A., Null propagation of partially massless higher spins in (A)dS and cosmological constant speculations, Phys. Lett. B 513 (2001), 137–141, hep-th/0105181. Deser S., Waldron A., Arbitrary spin representations in de Sitter from dS/CFT with applications to dS supergravity, Nuclear Phys. B 662 (2003), 379–392, hep-th/0301068. Deser S., Waldron A., Conformal invariance of partially massless higher spins, Phys. Lett. B 603 (2004), 30–34, hep-th/0408155. [16] Deser A., Waldron A., Einstein tensors for mixed symmetry higher spins, in preparation. [17] Cherney D., Hallowell D., Hodge A., Shaukat A., Waldron A., Higher rank Fourier–Jacobi algebras, in preparation. http://arxiv.org/abs/hep-th/0102166 http://arxiv.org/abs/hep-th/0103198 http://arxiv.org/abs/hep-th/0103255 http://arxiv.org/abs/hep-th/0105181 http://arxiv.org/abs/hep-th/0301068 http://arxiv.org/abs/hep-th/0408155 1 Introduction 2 The classical model 3 Quantization and geometry 4 The Fourier-Jacobi algebra 5 Conclusions References

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

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