Two Mathematically Equivalent Versions of Maxwell's Equations

This paper is a review of the canonical proper-time approach to relativistic mechanics and classical electrodynamics. The purpose is to provide a physically complete classical background for a new approach to relativistic quantum theory. Here, we first show that there are two versions of Maxwell's equations. The new version fixes the clock of the field source for all inertial observers. However now, the (natural definition of the effective) speed of light is no longer an invariant for all observers, but depends on the motion of the source. This approach allows us to account for radiation reaction without the Lorentz-Dirac equation, self-energy (divergence), advanced potentials or any assumptions about the structure of the source. The theory provides a new invariance group which, in general, is a nonlinear and nonlocal representation of the Lorentz group. This approach also provides a natural (and unique) definition of simultaneity for all observers. The corresponding particle theory is independent of particle number, noninvariant under time reversal (arrow of time), compatible with quantum mechanics and has a corresponding positive definite canonical Hamiltonian associated with the clock of the source. We also provide a brief review of our work on the foundational aspects of the corresponding relativistic quantum theory. Here, we show that the standard square-root and the Dirac equations are actually two distinct spin-$\tfrac{1}{2}$ particle equations.Comment: Appeared: Foundations of Physic

Projective Representations of the Inhomogeneous Hamilton Group: Noninertial Symmetry in Quantum Mechanics

Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary representations of the central extension of the group. The formulation of the inertial states of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group, and its nonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries includes the Weyl-Heisenberg group; the hermitian representations of its algebra are the Heisenberg commutation relations that are a foundation of quantum mechanics. The Weyl-Heisenberg group is a one dimensional central extension of the abelian group and its unitary representations are therefore a particular projective representation of the abelian group of translations on phase space. A theorem involving the automorphism group shows that the maximal symmetry that leaves invariant the Heisenberg commutation relations are essentially projective representations of the inhomogeneous symplectic group. In the nonrelativistic domain, we must also have invariance of Newtonian time. This reduces the symmetry group to the inhomogeneous Hamilton group that is a local noninertial symmetry of Hamilton's equations. The projective representations of these groups are calculated using the Mackey theorems for the general case of a nonabelian normal subgroup

The Theory of Kairons

In relativistic quantum mechanics wave functions of particles satisfy field equations that have initial data on a space--like hypersurface. We propose a dual field theory of ``wavicles'' that have their initial data on a time--like worldline. Propagation of such fields is superluminal, even though the Hilbert space of the solutions carries a unitary representation of the Poincare group of mass zero. We call the objects described by these field equations ``Kairons''. The paper builds the field equations in a general relativistic framework, allowing for a torsion. Kairon fields are section of a vector bundle over space-time. The bundle has infinite--dimensional fibres.Comment: Latex, 21 pages, 1 figure, several misprints from the previous version correcte

Standing waves in the Lorentz-covariant world

When Einstein formulated his special relativity, he developed his dynamics for point particles. Of course, many valiant efforts have been made to extend his relativity to rigid bodies, but this subject is forgotten in history. This is largely because of the emergence of quantum mechanics with wave-particle duality. Instead of Lorentz-boosting rigid bodies, we now boost waves and have to deal with Lorentz transformations of waves. We now have some understanding of plane waves or running waves in the covariant picture, but we do not yet have a clear picture of standing waves. In this report, we show that there is one set of standing waves which can be Lorentz-transformed while being consistent with all physical principle of quantum mechanics and relativity. It is possible to construct a representation of the Poincar\'e group using harmonic oscillator wave functions satisfying space-time boundary conditions. This set of wave functions is capable of explaining the quantum bound state for both slow and fast hadrons. In particular it can explain the quark model for hadrons at rest, and Feynman's parton model hadrons moving with a speed close to that of light.Comment: LaTex 20 pages, presented at the 2004 meeting of the International Association of Relativistic Dynamincs, to be published in the proceeding