746 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
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.
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
Scaling Laws of Stress and Strain in Brittle Fracture
A numerical realization of an elastic beam lattice is used to obtain scaling
exponents relevant to the extent of damage within the controlled, catastrophic
and total regimes of mode-I brittle fracture. The relative fraction of damage
at the onset of catastrophic rupture approaches a fixed value in the continuum
limit. This enables disorder in a real material to be quantified through its
relationship with random samples generated on the computer.Comment: 4 pages and 6 figure
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