893 research outputs found

### Neoclassical Theory of Elementary Charges with Spin of 1/2

We advance here our neoclassical theory of elementary charges by integrating
into it the concept of spin of 1/2. The developed spinorial version of our
theory has many important features identical to those of the Dirac theory such
as the gyromagnetic ratio, expressions for currents including the spin current,
and antimatter states. In our theory the concepts of charge and anticharge
relate naturally to their "spin" in its rest frame in two opposite directions.
An important difference with the Dirac theory is that both the charge and
anticharge energies are positive whereas their frequencies have opposite signs

### 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

### Decoherence of spin echoes

We define a quantity, the so-called purity fidelity, which measures the rate
of dynamical irreversibility due to decoherence, observed e.g in echo
experiments, in the presence of an arbitrary small perturbation of the total
(system + environment) Hamiltonian. We derive a linear response formula for the
purity fidelity in terms of integrated time correlation functions of the
perturbation. Our relation predicts, similarly to the case of fidelity decay,
faster decay of purity fidelity the slower decay of time correlations is. In
particular, we find exponential decay in quantum mixing regime and faster,
initially quadratic and later typically gaussian decay in the regime of
non-ergodic, e.g. integrable quantum dynamics. We illustrate our approach by an
analytical calculation and numerical experiments in the Ising spin 1/2 chain
kicked with tilted homogeneous magnetic field where part of the chain is
interpreted as a system under observation and part as an environment.Comment: 22 pages, 10 figure

### The hidden geometric character of relativistic quantum mechanics

The presentation makes use of geometric algebra, also known as Clifford
algebra, in 5-dimensional spacetime. The choice of this space is given the
character of first principle, justified solely by the consequences that can be
derived from such choice and their consistency with experimental results. Given
a metric space of any dimension, one can define monogenic functions, the
natural extension of analytic functions to higher dimensions; such functions
have null vector derivative and have previously been shown by other authors to
play a decisive role in lower dimensional spaces. All monogenic functions have
null Laplacian by consequence; in an hyperbolic space this fact leads
inevitably to a wave equation with plane-like solutions. This is also true for
5-dimensional spacetime and we will explore those solutions, establishing a
parallel with the solutions of the Dirac equation. For this purpose we will
invoke the isomorphism between the complex algebra of 4x4 matrices, also known
as Dirac's matrices. There is one problem with this isomorphism, because the
solutions to Dirac's equation are usually known as spinors (column matrices)
that don't belong to the 4x4 matrix algebra and as such are excluded from the
isomorphism. We will show that a solution in terms of Dirac spinors is
equivalent to a plane wave solution. Just as one finds in the standard
formulation, monogenic functions can be naturally split into positive/negative
energy together with left/right ones. This split is provided by geometric
projectors and we will show that there is a second set of projectors providing
an alternate 4-fold split. The possible implications of this alternate split
are not yet fully understood and are presently the subject of profound
research.Comment: 29 pages. Small changes in V3 suggested by refere

### Discrete versions of some Dirac type equations and plane wave solutions

A discrete version of the plane wave solution to some discrete Dirac type
equations in the spacetime algebra is established. The conditions under which a
discrete analogue of the plane wave solution satisfies the discrete Hestenes
equation are briefly discussed.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1609.0459

### 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

### 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

### Multivector functions

AbstractIn a previous paper [1], the fundamentals of differential and integral calculus on Euclidean n-space were expressed in terms of multivector algebra. The theory is used here to derive some powerful theorems which generalize well-known theorems of potential theory and the theory of functions of a complex variable. Analytic multivector functions on Îµn are defined and shown to be appropriate generalizations of analytic functions of a complex variable. Some of their basic properties are pointed out. These results have important applications to physics which will be discussed in detail elsewhere

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