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 polishe

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

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