658 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
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
Helicity, polarization, and Riemann-Silberstein vortices
Riemann-Silberstein (RS) vortices have been defined as surfaces in spacetime
where the complex form of a free electromagnetic field given by F=E+iB is null
(F.F=0), and they can indeed be interpreted as the collective history swept out
by moving vortex lines of the field. Formally, the nullity condition is similar
to the definition of "C-lines" associated with a monochromatic electric or
magnetic field, which are curves in space where the polarization ellipses
degenerate to circles. However, it was noted that RS vortices of monochromatic
fields generally oscillate at optical frequencies and are therefore
unobservable while electric and magnetic C-lines are steady. Here I show that
under the additional assumption of having definite helicity, RS vortices are
not only steady but they coincide with both sets of C-lines, electric and
magnetic. The two concepts therefore become one for waves of definite frequency
and helicity. Since the definition of RS vortices is relativistically invariant
while that of C-lines is not, it may be useful to regard the vortices as a
wideband generalization of C-lines for waves of definite helicity.Comment: 5 pages, no figures. Submitted to J of Optics A, special issue on
Singular Optics; minor changes from v.
Electromagnetic inertia, reactive energy, and energy flow velocity
In a recent paper titled "Coherent electromagnetic wavelets and their
twisting null congruences," I defined the local inertia density (I), reactive
energy density (R), and energy flow velocity (v) of an electromagnetic field.
These are the field equivalents of the mass, rest energy, and velocity of a
relativistic particle. Thus R and I are Lorentz-invariant and |v|<=c, with
equality if and only if R=0. The exceptional fields with |v|=c were called
"coherent" because their energy moves in complete harmony with the field,
leaving no inertia or reactive energy behind. Generic electromagnetic fields
become coherent only in the far zone. Elsewhere, their energy flows at speeds
|v|<c. The purpose of this paper is to confirm and clarify this statement by
studying the local energy flow in several common systems: a time-harmonic
electric dipole field, a time-dependent electric dipole field, and a standing
plane wave. For these fields, the energy current (Poynting vector) is too weak
to carry away all of the energy, thus leaving reactive energy in its wake. For
the time-dependent dipole field, we find that the energy can flow both
transversally and inwards, back to the source. Neither of these phenomena show
up in the usual computation of the energy transport velocity which considers
only averages over one period in the time-harmonic case.Comment: 20 pages, 7 figure
Hamilton's Turns for the Lorentz Group
Hamilton in the course of his studies on quaternions came up with an elegant
geometric picture for the group SU(2). In this picture the group elements are
represented by ``turns'', which are equivalence classes of directed great
circle arcs on the unit sphere , in such a manner that the rule for
composition of group elements takes the form of the familiar parallelogram law
for the Euclidean translation group. It is only recently that this construction
has been generalized to the simplest noncompact group , the double cover of SO(2,1). The present work develops a theory of
turns for , the double and universal cover of SO(3,1) and ,
rendering a geometric representation in the spirit of Hamilton available for
all low dimensional semisimple Lie groups of interest in physics. The geometric
construction is illustrated through application to polar decomposition, and to
the composition of Lorentz boosts and the resulting Wigner or Thomas rotation.Comment: 13 pages, Late
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