281 research outputs found
Spectral geometry, homogeneous spaces, and differential forms with finite Fourier series
Let G be a compact Lie group acting transitively on Riemannian manifolds M
and N. Let p be a G equivariant Riemannian submersion from M to N. We show that
a smooth differential form on N has finite Fourier series if and only if the
pull back has finite Fourier series on
Irreducible Scalar Many-Body Casimir Energies: Theorems and Numerical Studies
We define irreducible N-body spectral functions and Casimir energies and
consider a massless scalar quantum field interacting locally by positive
potentials with classical objects. Irreducible N-body spectral functions in
this case are shown to be conditional probabilities of random walks. The
corresponding irreducible contributions to scalar many-body Casimir energies
are finite and positive/negative for an odd/even number of objects. The force
between any two finite objects separable by a plane is always attractive in
this case. Analytical and numerical world-line results for the irreducible
four-body Casimir energy of a scalar with Dirichlet boundary conditions on a
tic-tac-toe pattern of lines are presented. Numerical results for the
irreducible three-body Casimir energy of a massless scalar satisfying Dirichlet
boundary conditions on three intersecting lines forming an isosceles triangle
are also reported. In both cases the symmetric configuration (square and
isosceles triangle) corresponds to the minimal irreducible contribution to the
Casimir energy.Comment: Writeup of talk given at QFEXT11 (Sept.18-24) in Benasque, Spain. 10
pages, 3 figure
Examples of signature (2,2) manifolds with commuting curvature operators
We exhibit Walker manifolds of signature (2,2) with various commutativity
properties for the Ricci operator, the skew-symmetric curvature operator, and
the Jacobi operator. If the Walker metric is a Riemannian extension of an
underlying affine structure A, these properties are related to the Ricci tensor
of A
Non-commutative geometry and the standard model vacuum
The space of Dirac operators for the Connes-Chamseddine spectral action for
the standard model of particle physics coupled to gravity is studied. The model
is extended by including right-handed neutrino states, and the S0-reality axiom
is not assumed. The possibility of allowing more general fluctuations than the
inner fluctuations of the vacuum is proposed. The maximal case of all possible
fluctuations is studied by considering the equations of motion for the vacuum.
Whilst there are interesting non-trivial vacua with Majorana-like mass terms
for the leptons, the conclusion is that the equations are too restrictive to
allow solutions with the standard model mass matrix.Comment: 21 pages. v2: some comments improve
The trace of the heat kernel on a compact hyperbolic 3-orbifold
The heat coefficients related to the Laplace-Beltrami operator defined on the
hyperbolic compact manifold H^3/\Ga are evaluated in the case in which the
discrete group \Ga contains elliptic and hyperbolic elements. It is shown
that while hyperbolic elements give only exponentially vanishing corrections to
the trace of the heat kernel, elliptic elements modify all coefficients of the
asymptotic expansion, but the Weyl term, which remains unchanged. Some physical
consequences are briefly discussed in the examples.Comment: 11 page
Multiple reflection expansion and heat kernel coefficients
We propose the multiple reflection expansion as a tool for the calculation of
heat kernel coefficients. As an example, we give the coefficients for a sphere
as a finite sum over reflections, obtaining as a byproduct a relation between
the coefficients for Dirichlet and Neumann boundary conditions. Further, we
calculate the heat kernel coefficients for the most general matching conditions
on the surface of a sphere, including those cases corresponding to the presence
of delta and delta prime background potentials. In the latter case, the
multiple reflection expansion is shown to be non-convergent.Comment: 21 pages, corrected for some misprint
Strong ellipticity and spectral properties of chiral bag boundary conditions
We prove strong ellipticity of chiral bag boundary conditions on even
dimensional manifolds. From a knowledge of the heat kernel in an infinite
cylinder, some basic properties of the zeta function are analyzed on
cylindrical product manifolds of arbitrary even dimension.Comment: 16 pages, LaTeX, References adde
Detection and localization of speech in the presence of competing speech signals
Presented at the 12th International Conference on Auditory Display (ICAD), London, UK, June 20-23, 2006.Auditory displays are often used to convey important information in complex operational environments. One problem with these displays is that potentially critical information can be corrupted or lost when multiple warning sounds are presented at the same time. In this experiment, we examined a listener's ability to detect and localize a target speech token in the presence of from 1 to 5 simultaneous competing speech tokens. Two conditions were examined: a condition in which all of the speech tokens were presented from the same location (the `co-located' condition) and a condition in which the speech tokens were presented from different random locations (the `spatially separated' condition). The results suggest that both detection and localization degrade as the number of competing sounds increases. However, the changes in detection performance were found to be surprisingly small and there appeared to be little or no benefit of spatial separation for detection. Localization, on the other hand, was found to degrade substantially and systematically as the number of competing speech tokens increased. Overall, these results suggest that listeners are able to extract substantial information from these speech tokens even when the target is presented with 5 competing simultaneous sounds
General Relativity in terms of Dirac Eigenvalues
The eigenvalues of the Dirac operator on a curved spacetime are
diffeomorphism-invariant functions of the geometry. They form an infinite set
of ``observables'' for general relativity. Recent work of Chamseddine and
Connes suggests that they can be taken as variables for an invariant
description of the gravitational field's dynamics. We compute the Poisson
brackets of these eigenvalues and find them in terms of the energy-momentum of
the eigenspinors and the propagator of the linearized Einstein equations. We
show that the eigenspinors' energy-momentum is the Jacobian matrix of the
change of coordinates from metric to eigenvalues. We also consider a minor
modification of the spectral action, which eliminates the disturbing huge
cosmological term and derive its equations of motion. These are satisfied if
the energy momentum of the trans Planckian eigenspinors scale linearly with the
eigenvalue; we argue that this requirement approximates the Einstein equations.Comment: 6 pages, RevTe
Vacuum polarization of massive scalar fields in the spacetime of the electrically charged nonlinear black hole
The approximate renormalized stress-energy tensor of the quantized massive
conformally coupled scalar field in the spacetime of electrically charged
nonlinear black hole is constructed. It is achieved by functional
differentiation of the lowest order of the DeWitt-Schwinger effective action
involving coincidence limit of the Hadamard-Minakshisundaram-DeWitt-Seely
coefficient The result is compared with the analogous result derived
for the Reissner-Nordstr\"om black hole. It is shown that the most important
differences occur in the vicinity of the event horizon of the black hole near
the extremality limit. The structure of the nonlinear black hole is briefly
studied by means of the Lambert functions.Comment: 22 pages, 10 figure
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