Lawrence Horwitz

Relativistic quantum mechanics has been formulated as a theory of the evolution ofevents in spacetime; the wave functions are square-integrable functions on the four-dimensional spacetime, parametrized by a universal invariant world time τ. The representation of states with spin is induced with a little group that is the subgroup of O(3, 1) leaving invariant a timelike vector nμ; a positive definite invariant scalar product, for which matrix elements of tensor operators are covariant, emerges from this construction. In a previous study a second-order equation was introduced similar to the second-order Dirac equation, based on a quadratic function of two operators which are the self-adjoint parts, in this new scalar product, of γμpμ. It is shown in this paper that one of these operators, in fact the one from which the gyromagnetic moment is obtained, can be used to construct a first-order equation. The corresponding quantum theory is somewhat analogous to Dirac's spinor form; the Hamilton equations appear to describe dynamical degrees of freedom in a spacelike hyperplane orthogonal to nμ (in Dirac's theory the motion appears to be lightlike). It is shown that the integration over nμ required by unitarity results in timelike motion (as in the expectation value of Dirac's α). Explicit forms are obtained for the wave functions and currents for free motion. The general form of the theory is written for the (five-dimensional) pre-Maxwell fields required by gauge invariance

This paper examines the Stark effect, as a first order perturbation of manifestly covariant hydrogen-like bound states. These bound states are solutions to a relativistic Schrödinger equation with invariant evolution parameter, and represent mass eigenstates whose eigenvalues correspond to the well-known energy spectrum of the nonrelativistic theory. In analogy to the nonrelativistic case, the off-diagonal perturbation leads to a lifting of the degeneracy in the mass spectrum. In the covariant case, not only do the spectral lines split, but they acquire an imaginary part which is linear in the applied electric field, thus revealing induced bound state decay in first order perturbation theory. This imaginary part results from the coupling of the external field to the non-compact boost generator. In order to recover the conventional first order Stark splitting, we must include a scalar potential term. This term may be understood as a fifth gauge potential, which compensates for dependence of gauge transformations on the invariant evolution parameter

This book describes a relativistic quantum theory developed by the author starting from the E.C.G. Stueckelberg approach proposed in the early 40s. In this framework a universal invariant evolution parameter (corresponding to the time originally postulated by Newton) is introduced to describe dynamical evolution. This theory is able to provide solutions for some of the fundamental problems encountered in early attempts to construct a relativistic quantum theory. A relativistically covariant construction is given for which particle spins and angular momenta can be combined through the usual rotation group Clebsch-Gordan coefficients. Solutions are defined for both the classical and quantum two body bound state and scattering problems. The recently developed quantum Lax-Phillips theory of semigroup evolution of resonant states is described. The experiment of Lindner and coworkers on interference in time is discussed showing how the property of coherence in time provides a simple understanding of the results. The full gauge invariance of the Stueckelberg-Schroedinger equation results in a 5D generalization of the usual gauge theories. A description of this structure and some of its consequences for both Abelian and non-Abelian fields are discussed. A review of the basic foundations of relativistic classical and quantum statistical mechanics is also given. The Bekenstein-Sanders construction for imbedding Milgrom's theory of modified spacetime structure into general relativity as an alternative to dark matter is also studied.