914 research outputs found
Lightlike infinity in GCA models of Spacetime
This paper discusses a 7 dimensional conformal geometric algebra model for
spacetime based on the notion that spacelike and timelike infinities are
distinct. I show how naturally of the dimensions represents the lightlike
infinity and appears redundant in computations, yet usefull in interpretationComment: 12 page
Spin Gauge Theory of Gravity in Clifford Space
A theory in which 16-dimensional curved Clifford space (C-space) provides a
realization of Kaluza-Klein theory is investigated. No extra dimensions of
spacetime are needed: "extra dimensions" are in C-space. We explore the spin
gauge theory in C-space and show that the generalized spin connection contains
the usual 4-dimensional gravity and Yang-Mills fields of the U(1)xSU(2)xSU(3)
gauge group. The representation space for the latter group is provided by
16-component generalized spinors composed of four usual 4-component spinors,
defined geometrically as the members of four independent minimal left ideals of
Clifford algebra.Comment: 9 pages, talk presented at the QG05 conference, 12-16 September 2005,
Cala Gonone, Ital
Deformed Clifford algebra and supersymmetric quantum mechanics on a phase space with applications in quantum optics
In order to realize supersymmetric quantum mechanics methods on a four
dimensional classical phase-space, the complexified Clifford algebra of this
space is extended by deforming it with the Moyal star-product in composing the
components of Clifford forms. Two isospectral matrix Hamiltonians having a
common bosonic part but different fermionic parts depending on four real-valued
phase space functions are obtained. The Hamiltonians are doubly intertwined via
matrix-valued functions which are divisors of zero in the resulting
Moyal-Clifford algebra. Two illustrative examples corresponding to
Jaynes-Cummings-type models of quantum optics are presented as special cases of
the method. Their spectra, eigen-spinors and Wigner functions as well as their
constants of motion are also obtained within the autonomous framework of
deformation quantization.Comment: 22 pages. published versio
Geometric Algebra Model of Distributed Representations
Formalism based on GA is an alternative to distributed representation models
developed so far --- Smolensky's tensor product, Holographic Reduced
Representations (HRR) and Binary Spatter Code (BSC). Convolutions are replaced
by geometric products, interpretable in terms of geometry which seems to be the
most natural language for visualization of higher concepts. This paper recalls
the main ideas behind the GA model and investigates recognition test results
using both inner product and a clipped version of matrix representation. The
influence of accidental blade equality on recognition is also studied. Finally,
the efficiency of the GA model is compared to that of previously developed
models.Comment: 30 pages, 19 figure
Positivity and conservation of superenergy tensors
Two essential properties of energy-momentum tensors T_{\mu\nu} are their
positivity and conservation. This is mathematically formalized by,
respectively, an energy condition, as the dominant energy condition, and the
vanishing of their divergence \nabla^\mu T_{\mu\nu}=0. The classical Bel and
Bel-Robinson superenergy tensors, generated from the Riemann and Weyl tensors,
respectively, are rank-4 tensors. But they share these two properties with
energy momentum tensors: the Dominant Property (DP) and the divergence-free
property in the absence of sources (vacuum). Senovilla defined a universal
algebraic construction which generates a basic superenergy tensor T{A} from any
arbitrary tensor A. In this construction the seed tensor A is structured as an
r-fold multivector, which can always be done. The most important feature of the
basic superenergy tensors is that they satisfy automatically the DP,
independently of the generating tensor A. In a previous paper we presented a
more compact definition of T{A} using the r-fold Clifford algebra. This form
for the superenergy tensors allowed to obtain an easy proof of the DP valid for
any dimension. In this paper we include this proof. We explain which new
elements appear when we consider the tensor T{A} generated by a
non-degree-defined r-fold multivector A and how orthogonal Lorentz
transformations and bilinear observables of spinor fields are included as
particular cases of superenergy tensors. We find some sufficient conditions for
the seed tensor A, which guarantee that the generated tensor T{A} is
divergence-free. These sufficient conditions are satisfied by some physical
fields, which are presented as examples.Comment: 19 pages, no figures. Language and minor changes. Published versio
Cartoon Computation: Quantum-like computing without quantum mechanics
We present a computational framework based on geometric structures. No
quantum mechanics is involved, and yet the algorithms perform tasks analogous
to quantum computation. Tensor products and entangled states are not needed --
they are replaced by sets of basic shapes. To test the formalism we solve in
geometric terms the Deutsch-Jozsa problem, historically the first example that
demonstrated the potential power of quantum computation. Each step of the
algorithm has a clear geometric interpetation and allows for a cartoon
representation.Comment: version accepted in J. Phys.A (Letter to the Editor
Z_2-gradings of Clifford algebras and multivector structures
Let Cl(V,g) be the real Clifford algebra associated to the real vector space
V, endowed with a nondegenerate metric g. In this paper, we study the class of
Z_2-gradings of Cl(V,g) which are somehow compatible with the multivector
structure of the Grassmann algebra over V. A complete characterization for such
Z_2-gradings is obtained by classifying all the even subalgebras coming from
them. An expression relating such subalgebras to the usual even part of Cl(V,g)
is also obtained. Finally, we employ this framework to define spinor spaces,
and to parametrize all the possible signature changes on Cl(V,g) by
Z_2-gradings of this algebra.Comment: 10 pages, LaTeX; v2 accepted for publication in J. Phys.
Elastic effects of vacancies in strontium titanate: Short- and long-range strain fields, elastic dipole tensors, and chemical strain
We present a study of the local strain effects associated with vacancy
defects in strontium titanate and report the first calculations of elastic
dipole tensors and chemical strains for point defects in perovskites. The
combination of local and long-range results will enable determination of x-ray
scattering signatures that can be compared with experiments. We find that the
oxygen vacancy possesses a special property -- a highly anisotropic elastic
dipole tensor which almost vanishes upon averaging over all possible defect
orientations. Moreover, through direct comparison with experimental
measurements of chemical strain, we place constraints on the possible defects
present in oxygen-poor strontium titanate and introduce a conjecture regarding
the nature of the predominant defect in strontium-poor stoichiometries in
samples grown via pulsed laser deposition. Finally, during the review process,
we learned of recent experimental data, from strontium titanate films deposited
via molecular-beam epitaxy, that show good agreement with our calculated value
of the chemical strain associated with strontium vacancies.Comment: 14 pages, 11 figures, 4 table
Duality in Off-Shell Electromagnetism
In this paper, we examine the Dirac monopole in the framework of Off-Shell
Electromagnetism, the five dimensional U(1) gauge theory associated with
Stueckelberg-Schrodinger relativistic quantum theory. After reviewing the Dirac
model in four dimensions, we show that the structure of the five dimensional
theory prevents a natural generalization of the Dirac monopole, since the
theory is not symmetric under duality transformations. It is shown that the
duality symmetry can be restored by generalizing the electromagnetic field
strength to an element of a Clifford algebra. Nevertheless, the generalized
framework does not permit us to recover the phenomenological (or conventional)
absence of magnetic monopoles.Comment: 18 page
Quadratic Lagrangians and Topology in Gauge Theory Gravity
We consider topological contributions to the action integral in a gauge
theory formulation of gravity. Two topological invariants are found and are
shown to arise from the scalar and pseudoscalar parts of a single integral.
Neither of these action integrals contribute to the classical field equations.
An identity is found for the invariants that is valid for non-symmetric Riemann
tensors, generalizing the usual GR expression for the topological invariants.
The link with Yang-Mills instantons in Euclidean gravity is also explored. Ten
independent quadratic terms are constructed from the Riemann tensor, and the
topological invariants reduce these to eight possible independent terms for a
quadratic Lagrangian. The resulting field equations for the parity
non-violating terms are presented. Our derivations of these results are
considerably simpler that those found in the literature
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