Let $\{\alpha_{n}\}_{n=0}^{\infty } $ denote a sequence of positive real numbers and let the sequ... more Let $\{\alpha_{n}\}_{n=0}^{\infty } $ denote a sequence of positive real numbers and let the sequence $\{\beta_{n}\}_{n=0}^{\infty } $ be defined by $\beta_{0} = 1$ and $\beta_{n+1} = \prod_{j=0}^{n}\alpha_{j}$. For $0 \leq a < 1$, $0 < t < 1$, and $n$, a nonnegative integer, the inequality $0 \leq \{\xi \beta_{n} - a\} \leq t$ is studied, where $\{x\}$ denotes the
Symmetry, Integrability and Geometry: Methods and Applications, 2007
Lichnerowicz's algebra of differential geometric operators acting on symmetric tensors can be obt... more 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.
Gravitational interactions of higher spin fields are generically plagued by inconsistencies. We p... more Gravitational interactions of higher spin fields are generically plagued by inconsistencies. We present a simple framework that couples higher spins to a broad class of gravitational backgrounds (including Ricci flat and Einstein) consistently at the classical level. The model is the simplest example of a Yang-Mills detour complex, which recently has been applied in the mathematical setting of conformal geometry. An analysis of asymptotic scattering states about the trivial field theory vacuum in the simplest version of the theory yields a rich spectrum marred by negative norm excitations. The result is a theory of a physical massless graviton, scalar field, and massive vector along with a degenerate pair of zero norm photon excitations. Coherent states of the unstable sector of the model do have positive norms, but their evolution is no longer unitary and their amplitudes grow with time. The model is of considerable interest for braneworld scenarios and ghost condensation models, and invariant theory.
We present supersymmetric, curved space, quantum mechanical models based on deformations of a par... more We present supersymmetric, curved space, quantum mechanical models based on deformations of a parabolic subalgebra of osp(2p + 2|Q). The dynamics are governed by a spinning particle action whose internal coordinates are Lorentz vectors labeled by the fundamental representation of osp(2p|Q). The states of the theory are tensors or spinor-tensors on the curved background while conserved charges correspond to the various differential geometry operators acting on these. The Hamiltonian generalizes Lichnerowicz's wave/Laplace operator. It is central, and the models are supersymmetric whenever the background is a symmetric space, although there is an osp(2p|Q) superalgebra for any curved background. The lowest purely bosonic example (2p, Q) = (2, 0) corresponds to a deformed Jacobi group and describes Lichnerowicz's original algebra of constant curvature, differential geometric operators acting on symmetric tensors. The case (2p, Q) = (0, 1) is simply the N = 1 superparticle whose supercharge amounts to the Dirac operator acting on spinors. The (2p, Q) = (0, 2) model is the N = 2 supersymmetric quantum mechanics corresponding to differential forms. (This latter pair of models are supersymmetric on any Riemannian background.) When Q is odd, the models apply to spinor-tensors. The (2p, Q) = (2, 1) model is distinguished by admitting a central Lichnerowicz-Dirac operator when the background is constant curvature. The new supersymmetric models are novel in that the Hamiltonian is not just a square of super charges, but rather a sum of commutators of supercharges and commutators of bosonic charges. These models and superalgebras are a very useful tool for any study involving high rank tensors and spinors on manifolds.
Let $\{\alpha_{n}\}_{n=0}^{\infty } $ denote a sequence of positive real numbers and let the sequ... more Let $\{\alpha_{n}\}_{n=0}^{\infty } $ denote a sequence of positive real numbers and let the sequence $\{\beta_{n}\}_{n=0}^{\infty } $ be defined by $\beta_{0} = 1$ and $\beta_{n+1} = \prod_{j=0}^{n}\alpha_{j}$. For $0 \leq a < 1$, $0 < t < 1$, and $n$, a nonnegative integer, the inequality $0 \leq \{\xi \beta_{n} - a\} \leq t$ is studied, where $\{x\}$ denotes the
Symmetry, Integrability and Geometry: Methods and Applications, 2007
Lichnerowicz's algebra of differential geometric operators acting on symmetric tensors can be obt... more 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.
Gravitational interactions of higher spin fields are generically plagued by inconsistencies. We p... more Gravitational interactions of higher spin fields are generically plagued by inconsistencies. We present a simple framework that couples higher spins to a broad class of gravitational backgrounds (including Ricci flat and Einstein) consistently at the classical level. The model is the simplest example of a Yang-Mills detour complex, which recently has been applied in the mathematical setting of conformal geometry. An analysis of asymptotic scattering states about the trivial field theory vacuum in the simplest version of the theory yields a rich spectrum marred by negative norm excitations. The result is a theory of a physical massless graviton, scalar field, and massive vector along with a degenerate pair of zero norm photon excitations. Coherent states of the unstable sector of the model do have positive norms, but their evolution is no longer unitary and their amplitudes grow with time. The model is of considerable interest for braneworld scenarios and ghost condensation models, and invariant theory.
We present supersymmetric, curved space, quantum mechanical models based on deformations of a par... more We present supersymmetric, curved space, quantum mechanical models based on deformations of a parabolic subalgebra of osp(2p + 2|Q). The dynamics are governed by a spinning particle action whose internal coordinates are Lorentz vectors labeled by the fundamental representation of osp(2p|Q). The states of the theory are tensors or spinor-tensors on the curved background while conserved charges correspond to the various differential geometry operators acting on these. The Hamiltonian generalizes Lichnerowicz's wave/Laplace operator. It is central, and the models are supersymmetric whenever the background is a symmetric space, although there is an osp(2p|Q) superalgebra for any curved background. The lowest purely bosonic example (2p, Q) = (2, 0) corresponds to a deformed Jacobi group and describes Lichnerowicz's original algebra of constant curvature, differential geometric operators acting on symmetric tensors. The case (2p, Q) = (0, 1) is simply the N = 1 superparticle whose supercharge amounts to the Dirac operator acting on spinors. The (2p, Q) = (0, 2) model is the N = 2 supersymmetric quantum mechanics corresponding to differential forms. (This latter pair of models are supersymmetric on any Riemannian background.) When Q is odd, the models apply to spinor-tensors. The (2p, Q) = (2, 1) model is distinguished by admitting a central Lichnerowicz-Dirac operator when the background is constant curvature. The new supersymmetric models are novel in that the Hamiltonian is not just a square of super charges, but rather a sum of commutators of supercharges and commutators of bosonic charges. These models and superalgebras are a very useful tool for any study involving high rank tensors and spinors on manifolds.
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Papers by Karl Hallowell