473 research outputs found

    The N=2N=2 super W4W_4 algebra and its associated generalized KdV hierarchies

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    We construct the N=2N=2 super W4W_4 algebra as a certain reduction of the second Gel'fand-Dikii bracket on the dual of the Lie superalgebra of N=1N=1 super pseudo-differential operators. The algebra is put in manifestly N=2N=2 supersymmetric form in terms of three N=2N=2 superfields Φi(X)\Phi_i(X), with Φ1\Phi_1 being the N=2N=2 energy momentum tensor and Φ2\Phi_2 and Φ3\Phi_3 being conformal spin 22 and 33 superfields respectively. A search for integrable hierarchies of the generalized KdV variety with this algebra as Hamiltonian structure gives three solutions, exactly the same number as for the W2W_2 (super KdV) and W3W_3 (super Boussinesq) cases.Comment: 16 pages, LaTeX, UTAS-PHYS-92-3

    A possible mathematics for the unification of quantum mechanics and general relativity

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    This paper summarizes and generalizes a recently proposed mathematical framework that unifies the standard formalisms of special relativity and quantum mechanics. The framework is based on Hilbert spaces H of functions of four space-time variables x,t, furnished with an additional indefinite inner product invariant under Poincar\'e transformations, and isomorphisms of these spaces that preserve the indefinite metric. The indefinite metric is responsible for breaking the symmetry between space and time variables and for selecting a family of Hilbert subspaces that are preserved under Galileo transformations. Within these subspaces the usual quantum mechanics with Schr\"odinger evolution and t as the evolution parameter is derived. Simultaneously, the Minkowski space-time is isometrically embedded into H, Poincar\'e transformations have unique extensions to isomorphisms of H and the embedding commutes with Poincar\'e transformations. The main new result is a proof that the framework accommodates arbitrary pseudo-Riemannian space-times furnished with the action of the diffeomorphism group

    Invariant tensors and Casimir operators for simple compact Lie groups

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    The Casimir operators of a Lie algebra are in one-to-one correspondence with the symmetric invariant tensors of the algebra. There is an infinite family of Casimir operators whose members are expressible in terms of a number of primitive Casimirs equal to the rank of the underlying group. A systematic derivation is presented of a complete set of identities expressing non-primitive symmetric tensors in terms of primitive tensors. Several examples are given including an application to an exceptional Lie algebra.Comment: 11 pages, LaTeX, minor changes, version in J. Math. Phy

    Gel'fand-Zetlin Basis and Clebsch-Gordan Coefficients for Covariant Representations of the Lie superalgebra gl(m|n)

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    A Gel'fand-Zetlin basis is introduced for the irreducible covariant tensor representations of the Lie superalgebra gl(m|n). Explicit expressions for the generators of the Lie superalgebra acting on this basis are determined. Furthermore, Clebsch-Gordan coefficients corresponding to the tensor product of any covariant tensor representation of gl(m|n) with the natural representation V ([1,0,...,0]) of gl(m|n) with highest weight (1,0,. . . ,0) are computed. Both results are steps for the explicit construction of the parastatistics Fock space.Comment: 16 page

    Ladder operators for isospectral oscillators

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    We present, for the isospectral family of oscillator Hamiltonians, a systematic procedure for constructing raising and lowering operators satisfying any prescribed `distorted' Heisenberg algebra (including the qq-generalization). This is done by means of an operator transformation implemented by a shift operator. The latter is obtained by solving an appropriate partial isometry condition in the Hilbert space. Formal representations of the non-local operators concerned are given in terms of pseudo-differential operators. Using the new annihilation operators, new classes of coherent states are constructed for isospectral oscillator Hamiltonians. The corresponding Fock-Bargmann representations are also considered, with specific reference to the order of the entire function family in each case.Comment: 13 page

    Simple derivation of general Fierz-type identities

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    General Fierz-type identities are examined and their well known connection with completeness relations in matrix vector spaces is shown. In particular, I derive the chiral Fierz identities in a simple and systematic way by using a chiral basis for the complex 4×44\times4 matrices. Other completeness relations for the fundamental representations of SU(N) algebras can be extracted using the same reasoning.Comment: 9pages. Few sentences modified in introduction and in conclusion. Typos corrected. An example added in introduction. Title modifie

    Spectral analysis and an area-preserving extension of a piecewise linear intermittent map

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    We investigate spectral properties of a 1-dimensional piecewise linear intermittent map, which has not only a marginal fixed point but also a singular structure suppressing injections of the orbits into neighborhoods of the marginal fixed point. We explicitly derive generalized eigenvalues and eigenfunctions of the Frobenius--Perron operator of the map for classes of observables and piecewise constant initial densities, and it is found that the Frobenius--Perron operator has two simple real eigenvalues 1 and λd∈(−1,0)\lambda_d \in (-1,0), and a continuous spectrum on the real line [0,1][0,1]. From these spectral properties, we also found that this system exhibits power law decay of correlations. This analytical result is found to be in a good agreement with numerical simulations. Moreover, the system can be extended to an area-preserving invertible map defined on the unit square. This extended system is similar to the baker transformation, but does not satisfy hyperbolicity. A relation between this area-preserving map and a billiard system is also discussed.Comment: 12 pages, 3 figure

    Hohenberg-Kohn theorem for the lowest-energy resonance of unbound systems

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    We show that under well-defined conditions the Hohenberg-Kohn theorem (HKT) can be extended to the lowest-energy resonance of unbound systems. Using the Gel'fand Levitan theorem, the extended version of the HKT can also be applied to systems that support a finite number of bound states. The extended version of the HKT provides an adequate framework to carry out DFT calculations of negative electron affinities.Comment: 4 pages, 3 figure

    Generalized Affine Coherent States: A Natural Framework for Quantization of Metric-like Variables

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    Affine variables, which have the virtue of preserving the positive-definite character of matrix-like objects, have been suggested as replacements for the canonical variables of standard quantization schemes, especially in the context of quantum gravity. We develop the kinematics of such variables, discussing suitable coherent states, their associated resolution of unity, polarizations, and finally the realization of the coherent-state overlap function in terms of suitable path-integral formulations.Comment: 17 pages, LaTeX, no figure

    Hawking radiation from dynamical horizons

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    In completely local settings, we establish that a dynamically evolving black hole horizon can be assigned a Hawking temperature. Moreover, we calculate the Hawking flux and show that the radius of the horizon shrinks.Comment: 5 Page
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