2,484 research outputs found
Star Products and Geometric Algebra
The formalism of geometric algebra can be described as deformed super
analysis. The deformation is done with a fermionic star product, that arises
from deformation quantization of pseudoclassical mechanics. If one then extends
the deformation to the bosonic coefficient part of superanalysis one obtains
quantum mechanics for systems with spin. This approach clarifies on the one
hand the relation between Grassmann and Clifford structures in geometric
algebra and on the other hand the relation between classical mechanics and
quantum mechanics. Moreover it gives a formalism that allows to handle
classical and quantum mechanics in a consistent manner.Comment: 21 page
About Zitterbewegung and electron structure
We start from the spinning electron theory by Barut and Zanghi, which has
been recently translated into the Clifford algebra language. We "complete" such
a translation, first of all, by expressing in the Clifford formalism a
particular Barut-Zanghi (BZ) solution, which refers (at the classical limit) to
an internal helical motion with a time-like speed [and is here shown to
originate from the superposition of positive and negative frequency solutions
of the Dirac equation]. Then, we show how to construct solutions of the Dirac
equation describing helical motions with light-like speed, which meet very well
the standard interpretation of the velocity operator in the Dirac equation
theory (and agree with the solution proposed by Hestenes, on the basis
--however-- of ad-hoc assumptions that are unnecessary in the present
approach). The above results appear to support the conjecture that the
Zitterbewegung motion (a helical motion, at the classical limit) is responsible
for the electron spin.Comment: LaTeX; 11 pages; this is a corrected version of work appeared partly
in Phys. Lett. B318 (1993) 623 and partly in "Particles, Gravity and
Space-Time" (ed.by P.I.Pronin & G.A.Sardanashvily; World Scient., Singapore,
1996), p.34
Geometric Algebra Techniques for General Relativity
Geometric (Clifford) algebra provides an efficient mathematical language for
describing physical problems. We formulate general relativity in this language.
The resulting formalism combines the efficiency of differential forms with the
straightforwardness of coordinate methods. We focus our attention on
orthonormal frames and the associated connection bivector, using them to find
the Schwarzschild and Kerr solutions, along with a detailed exposition of the
Petrov types for the Weyl tensor.Comment: 34 pages, 0 figures; submitted to Annals of Physic
Geometric Algebra and Star Products on the Phase Space
Superanalysis can be deformed with a fermionic star product into a Clifford
calculus that is equivalent to geometric algebra. With this multivector
formalism it is then possible to formulate Riemannian geometry and an
inhomogeneous generalization of exterior calculus. Moreover it is shown here
how symplectic and Poisson geometry fit in this context. The application of
this formalism together with the bosonic star product formalism of deformation
quantization leads then on space and space-time to a natural appearance of spin
structures and on phase space to BRST structures that were found in the path
integral formulation of classical mechanics. Furthermore it will be shown that
Poincare and Lie-Poisson reduction can be formulated in this formalism.Comment: 35 page
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
Electron structure, Zitterbewegung, and the new non-linear Dirac-like equation
The recent literature shows a renewed interest, with various independent
approaches, in the classical theories for spin. Considering the possible
interest of those results, at least for the electron case, we purpose in this
paper to explore their physical and mathematical meaning, by the natural and
powerful language of Clifford algebras (which, incidentally, will allow us to
unify those different approaches). In such theories, the ordinary electron is
in general associated to the mean motion of a point-like "constituent" Q, whose
trajectory is a cylindrical helix. We find, in particular, that the object Q
obeys a new, non-linear Dirac-like equation, such that --when averaging over an
internal cycle (which corresponds to linearization)-- it transforms into the
ordinary Dirac equation (valid for the electron as a whole).Comment: LaTeX; 19 pages; this is a corrected version of work appeared partly
in Hadronic J. 18 (1995) 97 and partly in Phys.Lett. B318 (1993) 48
Spin-dependent Bohm trajectories for hydrogen eigenstates
The Bohm trajectories for several hydrogen atom eigenstates are determined,
taking into account the additional momentum term that arises from the Pauli
current. Unlike the original Bohmian result, the spin-dependent term yields
nonstationary trajectories. The relationship between the trajectories and the
standard visualizations of orbitals is discussed. The trajectories for a model
problem that simulates a 1s-2p transition in hydrogen are also examined.Comment: 11 pages, 3 figure
Spinors in the hyperbolic algebra
The three-dimensional universal complex Clifford algebra is used to represent
relativistic vectors in terms of paravectors. In analogy to the Hestenes
spacetime approach spinors are introduced in an algebraic form. This removes
the dependance on an explicit matrix representation of the algebra.Comment: 9 pages Latex2
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
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