1,193 research outputs found
Relativity in Introductory Physics
A century after its formulation by Einstein, it is time to incorporate
special relativity early in the physics curriculum. The approach advocated here
employs a simple algebraic extension of vector formalism that generates
Minkowski spacetime, displays covariant symmetries, and enables calculations of
boosts and spatial rotations without matrices or tensors. The approach is part
of a comprehensive geometric algebra with applications in many areas of
physics, but only an intuitive subset is needed at the introductory level. The
approach and some of its extensions are given here and illustrated with
insights into the geometry of spacetime.Comment: 29 pages, 5 figures, several typos corrected, some discussion
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Overcoming the su(2^n) sufficient condition for the coherent control of n-qubit systems
We study quantum systems with even numbers N of levels that are completely
state-controlled by unitary transformations generated by Lie algebras
isomorphic to sp(N) of dimension N(N+1)/2. These Lie algebras are smaller than
the respective su(N) with dimension N^2-1. We show that this reduction
constrains the Hamiltonian to have symmetric energy levels. An example of such
a system is an n-qubit system. Using a geometric representation for the quantum
wave function of a finite system, we present an explicit example that shows a
two-qubit system can be controlled by the elements of the Lie algebra sp(4)
(isomorphic to spin(5) and so(5)) with dimension ten rather than su(4) with
dimension fifteen. These results enable one to envision more efficient
algorithms for the design of fields for quantum-state engineering, and they
provide more insight into the fundamental structure of quantum control.Comment: 13 pp., 2 figure
Effects of Nitrogen Quenching Gas on Spin-Exchange Optical Pumping of He-3
We consider the degree of conservation of nuclear spin polarization in the
process of optical pumping under typical spin-exchange optical pumping
conditions. Previous analyses have assumed that negligible nuclear spin
precession occurs in the brief periods of time the alkali-metal atoms are in
the excited state after absorbing photons and before undergoing quenching
collisions with nitrogen molecules. We include excited-state hyperfine
interactions, electronic spin relaxation in collisions with He and N_2,
spontaneous emission, quenching collisions, and a simplified treatment of
radiation trapping
A geometric basis for the standard-model gauge group
A geometric approach to the standard model in terms of the Clifford algebra
Cl_7 is advanced. A key feature of the model is its use of an algebraic spinor
for one generation of leptons and quarks. Spinor transformations separate into
left-sided ("exterior") and right-sided ("interior") types. By definition,
Poincare transformations are exterior ones. We consider all rotations in the
seven-dimensional space that (1) conserve the spacetime components of the
particle and antiparticle currents and (2) do not couple the right-chiral
neutrino. These rotations comprise additional exterior transformations that
commute with the Poincare group and form the group SU(2)_L, interior ones that
constitute SU(3)_C, and a unique group of coupled double-sided rotations with
U(1)_Y symmetry. The spinor mediates a physical coupling of Poincare and
isotopic symmetries within the restrictions of the Coleman--Mandula theorem.
The four extra spacelike dimensions in the model form a basis for the Higgs
isodoublet field, whose symmetry requires the chirality of SU(2). The charge
assignments of both the fundamental fermions and the Higgs boson are produced
exactly.Comment: 17 pages, LaTeX requires iopart. Accepted for publication in J. Phys.
A: Math. Gen. 9 Mar 2001. Typos correcte
A new view on relativity: Part 2. Relativistic dynamics
The Lorentz transformations are represented on the ball of relativistically
admissible velocities by Einstein velocity addition and rotations. This
representation is by projective maps. The relativistic dynamic equation can be
derived by introducing a new principle which is analogous to the Einstein's
Equivalence Principle, but can be applied for any force. By this principle, the
relativistic dynamic equation is defined by an element of the Lie algebra of
the above representation. If we introduce a new dynamic variable, called
symmetric velocity, the above representation becomes a representation by
conformal, instead of projective maps. In this variable, the relativistic
dynamic equation for systems with an invariant plane, becomes a non-linear
analytic equation in one complex variable. We obtain explicit solutions for the
motion of a charge in uniform, mutually perpendicular electric and magnetic
fields. By the above principle, we show that the relativistic dynamic equation
for the four-velocity leads to an analog of the electromagnetic tensor. This
indicates that force in special relativity is described by a differential
two-form
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
On the Solutions of the Lorentz-Dirac Equation
We discuss the unstable character of the solutions of the Lorentz-Dirac
equation and stress the need of methods like order reduction to derive a
physically acceptable equation of motion. The discussion is illustrated with
the paradigmatic example of the non-relativistic harmonic oscillator with
radiation reaction. We also illustrate removal of the noncasual
pre-acceleration with the introduction of a small correction in the
Lorentz-Dirac equation.Comment: 4 eps figs. to be published in GR
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