Tepper L. Gill

The Classical Electron Problem

In this paper, we construct a parallel image of the conventional Maxwell theory by replacing the observer-time by the proper-time of the source. This formulation is mathematically, but not physically, equivalent to the conventional form. The change induces a new symmetry group which is distinct from, but closely related to the Lorentz group, and fixes the clock of the source for all observers. The new wave equation contains an additional term (dissipative), which arises instantaneously with acceleration. This shows that the origin of radiation reaction is not the action of a “charge” on itself but arises from inertial resistance to changes in motion. This dissipative term is equivalent to an effective mass so that classical radiation has both a massless and a massive part. Hence, at the local level the theory is one of particles and fields but there is no self-energy divergence (nor any of the other problems). We also show that, for any closed system of particles, there is a global inertial frame and unique (invariant) global proper-time (for each observer) from which to observe the system. This global clock is intrinsically related to the proper clocks of the individual particles and provides a unique definition of simultaneity for all events associated with the system. We suggest that this clock is the historical clock of Horwitz, Piron, and Fanchi. At this level, the theory is of the action-at-a-distance type and the absorption hypothesis of Wheeler and Feynman follows from global conservation of energy.

Analytic representation of the square-root operator

In this paper, we use the theory of fractional powers of linear operators to construct a general (analytic) representation theory for the square-root energy operator of relativistic quantum theory, which is valid for all values of the spin. We focus on the spin 1/2 case, considering a few simple yet solvable and physically interesting cases, in order to understand how to interpret the operator. Our general representation is uniquely determined by the Greens function for the corresponding Schrodinger equation. We find that, in general, the operator has a representation as a nonlocal composite of (at least) three singularities. In the standard interpretation, the particle component has two negative parts and one (hard core) positive part, while the antiparticle component has two positive parts and one (hard core) negative part. This effect is confined within a Compton wavelength such that, at the point of singularity, they cancel each other providing a finite result. Furthermore, the operator looks like the identity outside a few Compton wavelengths (cut-off). To our knowledge, this is the first example of a physically relevant operator with these properties. When the magnetic field is constant, we obtain an additional singularity, which could be interpreted as particle absorption and emission. The physical picture that emerges is that, in addition to the confined singularities and the additional attractive (repulsive) term, the effective mass of the composite acquires an oscillatory behaviour. We also derive an alternative relationship between the Dirac equation (with minimal coupling) and the square-root equation that is somewhat closer than the one obtained via the Foldy–Wouthuysen method, in that there is no change in the wavefunction. This is accomplished by considering the scalar potential to be a part of the mass. This approach leads to a new Klein–Gordon equation and a new square-root equation, both of which can have the same eigenfunctions and (related) eigenvalues as the Dirac equation. Finally, we develop a perturbation theory that allows us to extend the range of our theory to include suitable spacetime-dependent potentials.

Foundations for relativistic quantum theory. I. Feynman’s operator calculus and the Dyson conjectures

In this paper, we provide a representation theory for the Feynman operator calculus. This allows us to solve the general initial-value problem and construct the Dyson series. We show that the series is asymptotic, thus proving Dyson’s second conjecture for quantum electrodynamics. In addition, we show that the expansion may be considered exact to any finite order by producing the remainder term. This implies that every nonperturbative solution has a perturbative expansion. Using a physical analysis of information from experiment versus that implied by our models, we reformulate our theory as a sum over paths. This allows us to relate our theory to Feynman’s path integral, and to prove Dyson’s first conjecture that the divergences are in part due to a violation of Heisenberg’s uncertainly relations.

Two Mathematically Equivalent Versions of Maxwell's Equations

AbstractThis 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 Dirac equations are actually two distinct spin-

Canonical Proper Time Formulation of Relativistic Particle Dynamics

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Time‐ordered operators and Feynman–Dyson algebras

particle equations.

Polarization dynamics of vector solitons in an elliptically low‐birefringent Kerr medium

A canonical (contact) transformation is performed on the time variable (in extended phase space) to reexpress relativistic dynamics in terms of proper time, leaving the generalized coordinates and canonical momentum as functions of this time variable. The form of the energy functional conjugate to this time variable is seen to be similar to that of a nonrelativistic dynamics at all values of particle momenta. The formulation is explored for single- and multiparticle classical systems. The (form) invariance of the theory is determined by a group which results from a similarity action of the contact group on the Poincaré group. One advantage of this approach is that it overcomes the no-interaction difficulties introduced by standard methods.

A new class of Banach spaces

An approach to time‐ordered operators based upon von Neumann’s infinite tensor product Hilbert spaces is used to define Feynman–Dyson algebras. This theory is used to show that a one‐to‐one correspondence exists between path integrals and semigroups, which are integral operators defined by a kernel, the reproducing property of the kernel being a consequence of the semigroup property. For path integrals constructed from two semigroups, the results are more general than those obtained by the use of the Trotter–Kato formula. Perturbation series for the Feynman–Dyson operator calculus for time evolution and scattering operators are discussed, and it is pointed out that they are ‘‘asymptotic in the sense of Poincare’’ as defined in the theory of semigroups, thereby giving a precise formulation to a well‐known conjecture of Dyson stated many years ago in the context of quantum electrodynamics. Moreover, the series converge when these operators possess suitable holomorphy properties.

Analytic representation of the Dirac equation

The polarization dynamics of vector solitons are studied in an elliptically low birefringent optical fiber for the first time within the framework of a model described by a system of coupled nonlinear Schro¨dinger equations with all the oscillating terms in the coupling between two copropagating modes. A condition for their stability has been obtained. Our analysis is based upon Hamiltonian perturbation theory.

Nonlinear pulse propagation in elliptically birefringent optical fibers

In this paper, we construct a new class of separable Banach spaces , for 1 ≤ p ≤ ∞, each of which contains all of the standard Lp spaces, as well as the space of finitely additive measures, as compact dense embeddings. Equally important is the fact that these spaces contain all Henstock–Kurzweil integrable functions and, in particular, the Feynman kernel and the Dirac measure, as norm bounded elements. As a first application, we construct the elementary path integral in the manner originally intended by Feynman. We then suggest that is a more appropriate Hilbert space for quantum theory, in that it satisfies the requirements for the Feynman, Heisenberg and Schrodinger representations, while the conventional choice only satisfies the requirements for the Heisenberg and Schrodinger representations. As a second application, we show that the mixed topology on the space of bounded continuous functions, , used to define the weak generator for a semigroup T(t), is stronger than the norm topology on . (This means that, when extended to is strongly continuous, so that the weak generator on becomes a strong generator on .)