Woodford W. Zachary

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-

Time‐ordered operators and Feynman–Dyson algebras

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

particle equations.

Number of modes governing spin‐wave turbulence

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.

Dimension of the universal attractor in spin-wave turbulence

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.

Two‐magnon bound states in Heisenberg ferromagnets

The Landau–Lifschitz equations describe the time evolution of magnetization in classical ferromagnets and are of basic importance for the understanding of magnetism. It is shown that, under quite general conditions, dissipative forms of these equations have attracting sets which are finite dimensional in a suitable sense. It follows from these results that, after an initial transient period, only a finite number of spin‐wave modes contribute to the spin‐wave instabilities responsible for the chaotic behavior recently found in ferromagnetic resonance, in both the tranverse and parallel pumping versions.

On the equality of S operators corresponding to unitarily equivalent Hamiltonians in single channel scattering

Abstract It is shown that the Landau-Lifschitz equations of classical ferromagnetism have a finite-dimensional universal attractor A, a set which describes the dynamics in the fully nonlinear regime and which describes results complementary to recent work associated with nonlinear effects related to low-lying instability thresholds. The dimension of A is estimated to be very large, of the order of 10 10 . Physical reasons for these large values are discussed.

Unitarily equivalent multiparticle Hamiltonian systems yielding equal scattering for corresponding states

A generalized theory of bound two‐magnon states in three‐dimensional isotropic Heisenberg ferromagnets is given and the passage to the limit in which the total number of spins tends to infinity is handled rigorously. Powerful methods, mostly of the trace‐inequality type, are developed for determining upper and lower bounds to the number of such bound states in the latter limit. These methods constitute the central contribution of this paper. In the latter we apply them to investigate the existence of bound two‐magnon states in body‐centered Heisenberg ferromagnets whose nonvanishing exchange interactions are those of the nearest‐neighbor type. In work reported elsewhere, we have employed these methods to study spin‐wave impurity states in Heisenberg ferromagnets. They should be useful for determining bounds on the number of localized states in solids in many cases when interactions extending over several orders of neighbors are operative.