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About 40 years ago, since no viable candidate for "dark matter" was discovered, M. Milgrom and J. Bekenstein introduced a non-covariant modfication of gravitational theory ((MOND) to account for the anomalpous rotation curves of galaxies. Bekenstein and Sanders then developed a relativistically covariant form of the theory, called TeVeS, involvimg scalar and tensor fields which accounts for the rotation currves as well asfor the observed lensing of background radiation around galaxies without the introduction of "dark matter". The dynamical behavior of a particle in such a gravitational theory has recently been discussed by Horwitz, Gershon and Schiffer. In this paper we study the dynamics of the $N \geq 2$ particle problem in the framework of the TeVeS theory., Comment: 5 pages, no figures

We show that in a relativistically covariant formulation of the two-body bound state problem, the bound state spectrum is in agreement, up to relativistic corrections, with the non-relativistic bound state spectrum. The solution is achieved by solving the problem with support of the wave function in an O(2,1) invariant submanifold of the Minkowski spacetime. The O(3,1) invariance of the differential equation requires, however, that the solutions provide a representation of O(3,1). Such solutions are obtained by means of the method of induced representations, providing a basic insight into the subject of the symmetries of relativistic dynamics., Comment: 21 pages, no figures. Based on earlier work of refs. 18,19

A proof is given for the Fourier transform for functions in a quantum mechanical Hilbert space on a non-compact manifold in general relativity. In the (configuration space) Newton-Wigner representation we discuss the spectral decomposition of the canonical operators and give a proof of the Parseval-Plancherel relation and the Born rule for linear superposition. We then discuss the representations of pure quantum states and their dual vectors, and construct the Fock space and the associated quantum field theory for Bose-Einstein and Fermi-Dirac statistics., Comment: 13 pages, no figures. Revision including the discussion of the Dirac form of the quantum theory, pure states and their dual vectors, and the construction of the Fock space and quantum field theory. Added remark on covariance

A consistent canonical classical and quantum dynamics in the framework of special relativity was formulated by Stueckelberg in 1941, and generalized to many body theory by Horwitz and Piron in 1973 (SHP). In this paper, this theory is embedded into the framework of general relativity (GR), here denoted by SHPGR. The canonical Poisson brackets of the SHP theory remain valid (invariant under local coordinate transformations) on the manifold of GR , and provide the basis for formulating a canonical quantum theory; the result (here defined as SHPGR) is generalized to many-body theory. A scalar product is defined for constructing the Hilbert space and a Hermitian momentum operator defined. The Fourier transform is defined, connecting momentum and coordinate representations. The potential which may occur in the SHP theory emerges as a spacetime scalar mass distribution in GR, and electromagnetism corresponds to a gauge field on the quantum mechanical GR Hilbert space in both the single particle and many-body theory. A diffeomorphism covariant form of Newton's law is found as an immediate consequence of the canonical formulation of SHPGR. We compute the classical evolution of the off shell mass on the orbit of a particle and the force on a particle and its energy at the Schwarzschild horizon. The propagator for evolution of the one body state is studied and a scattering theory on the manifold is worked out., Comment: 38 pages, no figures. Results for the dynamics of a particle in the neighborhood of the black hole horizon and at large r are computed, and a scattering theory is worked out. Fourier transform techniques are developed and a Hermitian momentum operator is defined which becomes an explicit generator of translation under a Foldy-Wouthuysen type transformation. Misprints on previous version corrected

Using the methods of symplectic geometry, we establish the existence of a canonical transformation from potential model Hamiltonians of standard form in a Euclidean space to an equivalent geometrical form on a manifold, where the corresponding motions are along geodesic curves. The advantage of this representation is that it admits the computation of geometric deviation as a test for local stability, shown in previous studies to be a very effective criterion for the stability of the orbits generated by the potential model Hamiltonian. We describe here an algorithm for finding the generating function for the canonical transformation and describe some of the properties of this mapping under local diffeomorphisms. We give a convergence proof for this algorithm for the one-dimensional case, and provide a precise geometric formulation of geodesic deviation which relates the stability of the motion in the geometric form to that of the Hamiltonian standard form. We discuss the relation of bounded domains in the two representations for which Morse theory would be applicable. Numerical computations for some interesting examples will be presented in forthcoming papers., Comment: 20 pages. Additional correction

We study classically the problem of two relativistic particles with an invariant Duffing-like potential which reduces to the usual Duffing form in the nonrelativistic limit. We use a special relativistic generalization (RGEM) of the geometric method (GEM) developed for the analysis of nonrelativistic Hamiltonian systems to study the local stability of a relativistic Duffing oscillator. Poincar'e plots of the simulated motion are consistent with the RGEM. We found a threshold for the external driving force required for chaotic behavior in the Minkowski spacetime., Comment: 11 pages. 18 figures

The kinematics of pre-Maxwell electrodynamics is examined and interpretations of these fields is found through an examination of the associated Lorentz force and the structure of the energy-momentum tensor., Comment: An updated version of an old paper, posted to arXiv for accessability

We show that there exists an underlying manifold with a conformal metric and compatible connection form, and a metric type Hamiltonian (which we call the geometrical picture) that can be put into correspondence with the usual Hamilton-Lagrange mechanics. The requirement of dynamical equivalence of the two types of Hamiltonians, that the momenta generated by the two pictures be equal for all times, is sufficient to determine an expansion of the conformal factor, defined on the geometrical coordinate representation, in its domain of analyticity with coefficients to all orders determined by functions of the potential of the Hamilton-Lagrange picture, defined on the Hamilton-Lagrange coordinate representation, and its derivatives. Conversely, if the conformal function is known, the potential of a Hamilton-Lagrange picture can be determined in a similar way. We show that arbitrary local variations of the orbits in the Hamilton-Lagrange picture can be generated by variations along geodesics in the geometrical picture and establish a correspondence which provides a basis for understanding how the instability in the geometrical picture is manifested in the instability of the original Hamiltonian motion., Comment: 17 pages, titles of references given, keywords added

We use an one dimensional model of a square barrier embedded in an infinite potential well to demonstrate that tunneling leads to a complex behavior of the wave function and that the degree of complexity may be quantified by use of the spatial entropy function defined by S = -\int |\Psi(x,t)|^2 ln |\Psi(x,t)|^2 dx. There is no classical counterpart to tunneling, but a decrease in the tunneling in a short time interval may be interpreted as an approach of a quantum system to a classical system. We show that changing the square barrier by increasing the height/width do not only decrease the tunneling but also slows down the rapid rise of the entropy function, indicating that the entropy growth is an essentially quantum effect., Comment: 14 pages, 8 figures

We show the existence of Lorentz invariant Berry phases generated, in the Stueckleberg-Horwitz-Piron manifestly covariant quantum theory (SHP), by a perturbed four dimensional harmonic oscillator. These phases are associated with a fractional perturbation of the azimuthal symmetry of the oscillator. They are computed numerically by using time independent perturbation theory and the definition of the Berry phase generalized to the framework of SHP relativistic quantum theory.

A non-existence theorem of classical electrodynamics in odd-dimensional spacetimes is shown to be invalid. The source of the error is pointed out, and is then demonstrated during the derivation of the fields generated by a uniformly moving point source.

The Stueckelberg formulation of a manifestly covariant relativistic classical and quantum mechanics is briefly reviewed and it is shown that in this framework a simple (semiclassical) model exists for the description of neutrino oscillations. The model is shown to be consistent with the field equations and the Lorentz force (developed here without and with spin by canonical methods) for Glashow-Salam-Weinberg type non-Abelian fields interacting with the leptons. We discuss a possible fundamental mechanism, in the context of a relativistic theory of spin for (first quantized) quantum mechanical systems, for CP violation. The model also predicts a possibly small "pull back", i.e., early arrival of a neutrino beam, for which the neutrino motion is almost everywhere within the light cone, a result which may emerge from future long baseline experiments designed to investigate neutrino transit times with significantly higher accuracy than presently available., Comment: Revised to take into account the more recent experimental results and a discussion of a mechanism for CP violation intrinsic to the theory

An investigation of dynamical properties of solutions of toy model of interacting Pomerons with triple vertex in zero transverse dimension is performed. Stable points and corresponding solutions at the limit of large rapidity are studied in the framework of given model. A presence of closed cycles in solutions is discussed as well as an application of obtained results for the case of interacting QCD Pomerons., Comment: 16 pages

A necessary condition for the emergence of chaos is given. It is well known that the emergence of chaos requires a positive exponent which entails diverging trajectories. Here we show that this is not enough. An additional necessary condition for the emergence of chaos in the region where the trajectory of the system goes through, is that the product of the maximal positive exponent times the duration in which the system configuration point stays in the unstable region should exceed unity. We give a theoretical analysis justifying this result and a few examples., Comment: 5 pages, 6 figures

Offshell electrodynamics based on a manifestly covariant off-shell relativistic dynamics of Stueckelberg, Horwitz and Piron, is five-dimensional. In this paper, we study the problem of radiation reaction of a particle in motion in this framework. In particular, the case of above-mass-shell is studied in detail, where the renormalization of the Lorentz force leads to a system of non-linear differential equations for 3 Lorentz scalars. The system is then solved numerically, where it is shown that the mass-shell deviation scalar $\ve$ either smoothly falls down to 0 (this result provides a mechanism for the mass stability of the off-shell theory), or strongly diverges under more extreme conditions. In both cases, no runaway motion is observed. Stability analysis indicates that the system seems to have chaotic behavior in the divergent case. It is also shown that, although a motion under which the mass-shell deviation $\ve$ is constant but not-zero, is indeed possible, but, it is unstable, and eventually it either decays to 0 or diverges., Comment: 45 pages, 18 figures

We establish the relation between the Wigner-Weisskopf theory for the description of an unstable system and the theory of coupling to an environment. According to the Wigner-Weisskopf general approach, even within the pole approximation (neglecting the background contribution) the evolution of a total system subspace is not an exact semigroup for the multi-channel decay, unless the projectors into eigesntates of the reduced evolution generator $W(z)$ are orthogonal. In this case these projectors must be evaluated at different pole locations $z_\alpha\neq z_\beta$. Since the orthogonality relation does not generally hold at different values of $z$, for example, when there is symmetry breaking, the semigroup evolution is a poor approximation for the multi-channel decay, even for a very weak coupling. Nevertheless, there exists a possibility not only to ensure the orthogonality of the $W(z)$ projectors regardless the number of the poles, but also to simultaneously suppress the effect of the background contribution. This possibility arises when the theory is generalized to take into account interactions with an environment. In this case $W(z)$, and hence its eigenvectors as well, are {\it independent} of $z$, which corresponds to a structure of the coupling to the continuum spectrum associated with the Markovian limit., Comment: 9 pages, 3 figures

In previous paper derivations of the Green function have been given for 5D off-shell electrodynamics in the framework of the manifestly covariant relativistic dynamics of Stueckelberg (with invariant evolution parameter $\tau$). In this paper, we reconcile these derivations resulting in different explicit forms, and relate our results to the conventional fundamental solutions of linear 5D wave equations published in the mathematical literature. We give physical arguments for the choice of the Green function retarded in the fifth variable $\tau$., Comment: 16 pages, 1 figure

We show that the existence of the family of self-adjoint Lyapunov operators introduced in [J. Math. Phys. 51, 022104 (2010)] allows for the decomposition of the state of a quantum mechanical system into two parts: A past time asymptote, which is asymptotic to the state of the system at t goes to minus infinity and vanishes at t goes to plus infinity, and a future time asymptote, which is asymptotic to the state of the system at t goes to plus infinity and vanishes at t goes to minus infinity. We demonstrate the usefulness of this decomposition for the description of resonance phenomena by considering the resonance scattering of a particle off a square barrier potential. We show that the past time asymptote captures the behavior of the resonance. In particular, it exhibits the expected exponential decay law and spatial probability distribution., Comment: Accepted for publication in Int. J. Theor. Phys

In [J. Math. Phys. 51 (2010) 022104] a self-adjoint operator was introduced that has the property that it indicates the direction of time within the framework of standard quantum mechanics, in the sense that as a function of time its expectation value decreases monotonically for any initial state. In this paper we study some of this operator's properties. In particular, we derive its spectrum and generalized eigenstates, and treat the example of the free particle., Comment: This paper presents a mathematically rigorous reworking of some of the results appearing in arXiv:0802.2448 [quant-ph]

We wish to study an application of Stueckelberg's relativistic quantum theory in the framework of general relativity. We study the form of the wave equation of a massive body in the presence of a Schwarzschild gravitational field. We treat the mathematical behavior of the wavefunction also around and beyond the horizon (r=2M). Classically, within the horizon, the time component of the metric becomes spacelike and distance from the origin singularity becomes timelike, suggesting an inevitable propagation of all matter within the horizon to a total collapse at r=0. However, the quantum description of the wave function provides a different understanding of the behavior of matter within the horizon. We find that a test particle can almost never be found at the origin and is more probable to be found at the horizon. Matter outside the horizon has a very small wave length and therefore interference effects can be found only on a very small atomic scale. However, within the horizon, matter becomes totally "tachionic" and is potentially "spread" over all space. Small location uncertainties on the atomic scale become large around the horizon, and different mass components of the wave function can therefore interfere on a stellar scale. This interference phenomenon, where the probability of finding matter decreases as a function of the distance from the horizon, appears as an effective gravitational repulsion., Comment: 20 pages, 6 figures

Gauge fields associated with the manifestly covariant dynamics of particles in $(3,1)$ spacetime are five-dimensional. In this paper we explore the old problem of fields generated by a source undergoing hyperbolic motion in this framework. The 5D fields are computed numerically using absolute time $\tau$-retarded Green-functions, and qualitatively compared with Maxwell fields generated by the same motion. We find that although the zero mode of all fields coincides with the corresponding Maxwell problem, the non-zero mode should affect, through the Lorentz force, the observed motion of test particles., Comment: 36 pages, 8 figures

This paper presents an analysis of the band structure of a spacetime potential lattice created by a standing electromagnetic wave. We show that there are energy band gaps. We estimate the effect, and propose a measurement that could confirm the existence of such phenomena., Comment: 8 pages. 2 figures

In this letter we show that although the application of standard Lyapunov analysis predicts that completely integrable Kepler motion is unstable, the geometrical analysis of Horwitz et al [1] predicts the observed stability. This seems to us to provide evidence for both the incompleteness of the standard Lyapunov analysis and the strength of the geometrical analysis. Moreover, we apply this approach to the three body problem in which the third body is restricted to move on a circle of large radius which induces an adiabatic time dependent potential on the second body. This causes the second body to move in a very interesting and intricate but periodic trajectory; however, the standard Lyapunov analysis, as well as methods based on the parametric variation of curvature associated with the Jacobi metric, incorrectly predict chaotic behavior. The geometric approach predicts the correct stable motion in this case as well., Comment: 9 pages, 14 figures

We introduce a self-adjoint operator that indicates the direction of time within the framework of standard quantum mechanics. That is, as a function of time its expectation value decreases monotonically for any initial state. This operator can be defined for any system governed by a Hamiltonian with a uniformly finitely degenerate, absolutely continuous and semibounded spectrum. We study some of the operator's properties and illustrate them for a large equivalence class of scattering problems. We also discuss some previous attempts to construct such an operator, and show that the no-go theorems developed in this context are not applicable to our construction., Comment: 5 pages, 2 figures

We consider the EPR experiment in the energy-based stochastic reduction framework. A gedanken set up is constructed to model the interaction of the particles with the measurement devices. The evolution of particles' density matrix is analytically derived. We compute the dependence of the disentanglement rate on the parameters of the model, and study the dependence of the outcome probabilities on the noise trajectories. Finally, we argue that these trajectories can be regarded as non-local hidden variables., Comment: 11 pages, 5 figures

The semigroup decomposition formalism makes use of the functional model for $C_{.0}$ class contractive semigroups for the description of the time evolution of resonances. For a given scattering problem the formalism allows for the association of a definite Hilbert space state with a scattering resonance. This state defines a decomposition of matrix elements of the evolution into a term evolving according to a semigroup law and a background term. We discuss the case of multiple resonances and give a bound on the size of the background term. As an example we treat a simple problem of scattering from a square barrier potential on the half-line., Comment: LaTex 22 pages 3 figures

Gauge fields associated with the manifestly covariant dynamics of particles in (3,1) spacetime are five-dimensional. We provide solutions of the classical 5D gauge field equations in both (4,1) and (3,2) flat spacetime metrics for the simple example of a uniformly moving point source. Green functions for the 5D field equations are obtained, which are consistent with the solutions for uniform motion obtained directly from the field equations with free asymptotic conditions., Comment: 32 pages, latex

A non-linear relativistic 4D field model of a quantum particle which emerges from the internal dynamics in the quantum phase space $CP(N-1)$ is proposed. In this model there is no distinction between `particle' and its `surrounding field', and the space-time manifold emerges from the description of the quantum state. The quantum observables of the `quantum particle field' are described in terms of the affine parallel transport of the local dynamical variables in $CP(N-1)$., Comment: LATEX, 20 pages, 6 figures

The scattering theory of Lax and Phillips, originally developed to describe resonances associated with classical wave equations, has been recently extended to apply as well to the case of the Schroedinger equation in the case that the wave operators for the corresponding Lax-Phillips theory exist. It is known that the bound state levels of an atom become resonances (spectral enhancements) in the continuum in the presence of an electric field (in all space) in the quantum mechanical Hilbert space. Such resonances appear as states in the extended Lax-Phillips Hilbert space. We show that for a simple version of the Stark effect, these states can be explicitly computed, and exhibit the (necessarily) semigroup property of decay in time. The widths and location of the resonances are those given by the poles of the resolvent of the standard quantum mechanical form., Comment: Plain TeX, 19 pages

We review the formulation of the problem of the electromagnetic self-interaction of a relativistic charged particle in the framework of the manifestly covariant classical mechanics of Stueckelberg, Horwitz and Piron. The gauge fields of this theory, in general, cause the mass of the particle to change. We show that the non-linear Lorentz force equation for the self-interaction resulting from the expansion of the Green's function has chaotic solutions. We study the autonomous equation for the off-shell particle mass here, for which the effective charged particle mass achieves a macroscopic average value determined by what appears to be a strange attractor., Comment: 19 pages PLain TeX, 1 page Captions, 18 figure (.eps files)

We solve the problem of formulating Brownian motion in a relativistically covariant framework in 3+1 dimensions. We obtain covariant Fokker-Planck equations with (for the isotropic case) a differential operator of d'Alembert form. Treating the spacelike and timelike fluctuations separately in order to maintain the covariance property, we show that it is essential to take into account the analytic continuation of "unphysical" fluctuations. We discuss the notion of locality in this framework and possible implications for entangled states., Comment: 19 pages Latex, one figure ps

We analyze in this work the energy transfer process of accelerated charges, the mass fluctuations accompanying this process, and their inertial properties. Based on a previous work, we use here the dipole antenna, which is a very convenient framework for such analysis, for analyzing those characteristics. We show that the radiation process can be viewed by two energy transfer processes: one from the energy source to the charges and the second from the charges into the surrounding space. Those processes, not being in phase, result in mass fluctuations. The same principle is true during absorption. We show that in a transient period between absorption and radiation the dipole antenna gains mass according to the amount of absorbed energy and loses this mass as radiated energy. We rigorously prove that the gain of mass, resulting from electrical interaction has inertial properties in the sense of Newton's third low. We arrive to this result by modeling the reacting spacetime region by an electric dipole., Comment: 19 pages, 5 figures

We review the classical and quantum mechanics of Stueckelberg, and introduce the compensation fields necessary for the gauge covariance of the Stueckelberg- Schr\"odinger equation. To achieve this, one must introduce a fifth, Lorentz scalar, compensation field, in addition to the four vector fields which compensate the action of the space-time derivatives. A generalized Lorentz force can be derived from the classical Hamilton equations associated with this evolution function. We show that the fifth (scalar) field can be eliminated through the introduction of a conformal metric on the spacetime manifold. The geodesic equation associated with this metric coincides with the Lorentz force, and is therefore dynamically equivalent. Since the generalized Maxwell equations for the five dimensional fields provide an equation relating the fifth field with the spacetime density of events, one can derive the spacetime event density associated with the Friedmann-Robertson-Walker solution of the Einstein equations. The resulting density, in the conformal coordinate space, is isotropic and homogeneous, decreasing as the square of the Robertson-Walker scale factor. Using the Einstein equations, one sees that both for the static and matter dominated models, the conformal time slice in which the events which generate the world lines are contained becomes progressively thinner as the inverse square of the scale factor, establishing a simple correspondence between the configurations predicted by the underlying Friedmann-Robertson-Walker dynamical model and the configurations in the conformal coordinates., Comment: Plain TeX, 9 pages

We solve the problem of formulating Brownian motion in a relativistically covariant framework in 1+1 and 3+1 dimensions. We obtain covariant Fokker-Planck equations with (for the isotropic case) a differential operator of invariant d'Alembert form. Treating the spacelike and timelike fluctuations separately, we show that it is essential to take into account the analytic continuation of "unphysical" fluctuations in order to achieve these results., Comment: Plain TeX, 10 pages

Using the Gell-Mann-Hartle-Griffiths formalism in the framework of the Flesia-Piron form of the Lax-Phillips theory we show that the Schr\"oedinger equation may be derived as a condition of stability of histories. This mechanism is realized in a mathematical structure closely related to the Zeno effect., Comment: 14 pages

The analysis of the EM radiation from a single charge shows that the radiated power depends on the retarded acceleration of the charge. Therefore, for consistency, an accelerated charge, free from the influence of external forces, should gradually lose its acceleration, until its total energy is radiated. Calculations show that the self force of a charge, which compensates for its radiation, is proportional to the derivative of the acceleration. However, when using this self-force in the equation of motion of the charge, one gets a diverging solution, for which the acceleration runs away to infinity. This means that there is an inconsistency in the solution of the single charge problem. However, in the construction of the conserved Maxwell charge density, there is implicitly an integral over the corresponding world line which corresponds to a collection of charged spacetime events. One may therefore consistently think of the ``self force'' as the force on a charge due to another charge at the retarded position. From this point of view, the energy is evidently conserved and the radiation process appears as an absorbing resistance to the feeding source. The purpose of this work is to learn about the behavior of single charges from the behavior of a real current, corresponding to the set of chargesa moving on a world line, and to study the analog of the self force of a charge associated with the radiation resistance of a continuum of charges., Comment: 13 pages, 2 figures

We first derive the relation between the eikonal approximation to the Maxwell wave equations in an inhomogeneous anisotropic medium and geodesic motion in a three dimensional Riemannian manifold using a method which identifies the symplectic structure of the corresponding mechanics. We then apply an analogous method to the five dimensional generalization of Maxwell theory required by the gauge invariance of Stueckelberg's covariant classical and quantum dynamics to demonstrate, in the eikonal approximation, the existence of geodesic motion for the flow of mass in a four dimensional pseudo-Riemannian manifold. These results provide a foundation for the geometrical optics of the five dimensional radiation theory and establish a model in which there is mass flow along geodesics. Finally we discuss the case of relativistic quantum theory in an anisotropic medium as well. In this case the eikonal approximation to the relativistic quantum mechanical current coincides with the geodesic flow governed by the pseudo-Riemannian metric obtained from the eikonal approximation to solutions of the Stueckelberg-Schr\"odinger equation. This construction provides a model for an underlying quantum mechanical structure for classical dynamical motion along geodesics on a pseudo-Riemannian manifold. The locally symplectic structure which emerges is that of Stueckelberg's covariant mechanics on this manifold., Comment: TeX file. 17 pages. Rewritten for clarity

We have previously studied properties of a one-dimensional potential with $N$ equally spaced identical barriers in a (fixed) finite interval for both finite and infinite $N$. It was observed that scattering and spectral properties depend sensitively on the ratio $c$ of spacing to width of the barriers (even in the limit $N \to \infty$). We compute here the specific heat of an ensemble of such systems and show that there is critical dependence on this parameter, as well as on the temperature, strongly suggestive of phase transitions., Comment: 27 pages, 7 figures, 3 new figures have been added with the accompanying text

We discuss the properties of a large number N of one-dimensional (bounded) locally periodic potential barriers in a finite interval. We show that the transmission coefficient, the scattering cross section $\sigma$, and the resonances of $\sigma$ depend sensitively upon the ratio of the total spacing to the total barrier width. We also show that a time dependent wave packet passing through the system of potential barriers rapidly spreads and deforms, a criterion suggested by Zaslavsky for chaotic behaviour. Computing the spectrum by imposing (large) periodic boundary conditions we find a Wigner type distribution. We investigate also the S-matrix poles; many resonances occur for certain values of the relative spacing between the barriers in the potential.

In this work we study the relativistic mechanics of continuous media on a fundamental level using a manifestly covariant proper time procedure. We formulate equations of motion and continuity (and constitutive equations) that are the starting point for any calculations regarding continuous media. In the force free limit, the standard relativistic equations are regained, so that these equations can be regarded as a generalization of the standard procedure. In the case of an inviscid fluid we derive an analogue of the Bernoulli equations. For irrotational flow we prove that the velocity field can be derived from a potential. If, in addition, the fluid is incompressible, the potential must obey the d'Alembert equation, and thus the problem is reduced to solving the d'Alembert equation with specific boundary conditions (in both space and time). The solutions indicate the existence of light velocity sound waves in an incompressible fluid (a result known from previous literature [19]). Relaxing the constraints and allowing the fluid to become linearly compressible, one can derive a wave equation from which the sound velosity can again be computed. For a stationary background flow, it has been demonstrated that the sound velocity attains its corrrect values for the incompressible and non-relatvistic limits. Finally, viscosity is introduced, bulk and shear viscosity constants are defined, and we formulate equations for the motion of a viscous fluid., Comment: Latex, 44 pages, 5 figures

The one-channel Wigner-Weisskopf survival amplitude may be dominated by exponential type decay in pole approximation at times not too short or too long, but, in the two channel case, for example, the pole residues are not orthogonal, and the pole approximation evolution does not correspond to a semigroup (experiments on the decay of the neutral K-meson system support the semigroup evolution postulated by Lee, Oehme and Yang, and Yang and Wu, to very high accuracy). The scattering theory of Lax and Phillips, originally developed for classical wave equations, has been recently extended to the description of the evolution of resonant states in the framework of quantum theory. The resulting evolution law of the unstable system is that of a semigroup, and the resonant state is a well-defined funtion in the Lax-Phillips Hilbert space. In this paper we apply this theory to relativistically covarant quantum field theoretical form of the (soluble) Lee model. We show that this theory provides a rigorous underlying basis for the Lee-Oehme-Yang-Wu construction., Comment: Plain TeX, 34 pages

It has been shown by Gupta and Padmanabhan that the radiation reaction force of the Abraham-Lorentz-Dirac equation can be obtained by a coordinate transformation from the inertial frame of an accelerating charged particle to that of the laboratory. We show that the problem may be formulated in a flat space of five dimensions, with five corresponding gauge fields in the framework of the classical version of a fully gauge covariant form of the Stueckelberg- Feynman-Schwinger covariant mechanics (the zero mode fields of the 0,1,2,3 components correspond to the Maxwell fields). Without additional constraints, the particles and fields are not confined to their mass shells. We show that in the mass-shell limit, the generalized Lorentz force obtained by means of the retarded Green's functions for the five-dimensional field equations provides the classical Abraham-Lorentz-Dirac radiation reaction terms (with renormalized mass and charge). We also obtain general coupled equations for the orbit and the off-shell dynamical mass during the evolution. The theory does not admit radiation if the particle remains identically on-shell. The structure of the equations implies that the mass-shell deviaiton is bounded when the external field is removed., Comment: Plain TeX, 12 pages

We apply the quantum Lax-Phillips scattering theory to a relativistically covariant quantum field theoretical form of the (soluble) Lee model. We construct the translation representations with the help of the wave operators, and show that the resulting Lax-Phillips $S$-matrix is an inner function (the Lax-Phillips theory is essentially a theory of translation invariant subspaces). We then discuss the non-relativistic limit of this theory, and show that the resulting kinematic relations coincide with the conditions required for the Galilean description of a decaying system., Comment: Tex, 15 pages, report TAUP 2588-99. Proceedings of Solvay Conference Les Treilles 3-8 July, 1999. To be published in Chaos, Solitons and Fractals

A relativistic charged particle moving in a uniform magnetic field and kicked by an electric field is considered. Under the assumption of small magnetic field, an iterative map is developed. We consider both the case in which no radiation is assumed and the radiative case, using the Lorentz-Dirac equation to describe the motion. Comparison between the non-radiative case and the radiative case shows that in both cases one can observe a stochastic web structure for weak magnetic fields, and, although there are global differences in the result of the map, that both cases are qualitatively similar in their small scale behavior. We also develop an iterative map for strong magnetic fields. In that case the web structure no longer exists; it is replaced by a rich chaotic behavior. It is shown that the particle does not diffuse to infinite energy; it is limited by the boundaries of an attractor (the boundaries are generally much smaller than light velocity). Bifurcation occurs, converging rapidly to Feigenbaum's universal constant. The chaotic behavior appears to be robust. For intermediate magnetic fields, it is more difficult to observe the web structure, and the influence of the unstable fixed point is weaker., Comment: 13 revtex pages including 11 figures

The quantum mechanical description of the evolution of an unstable system defined initially as a state in a Hilbert space at a given time does not provide a semigroup (exponential) decay law. The Wigner-Weisskopf survival amplitude, describing reversible quantum transitions, may be dominated by exponential type decay in pole approximation at times not too short or too long, but, in the two channel case, for example, the pole residues are not orthogonal, and the evolution does not correspond to a semigroup (experiments on the decay of the neutral $K$-meson system strongly support the semigroup evolution postulated by Lee, Oehme and Yang, and Yang and Wu). The scattering theory of Lax and Phillips, originally developed for classical wave equations, has been recently extended to the description of the evolution of resonant states in the framework of quantum theory. The resulting evolution law of the unstable system is that of a semigroup, and the resonant state is a well-defined function in the Lax-Phillips Hilbert space. In this paper we apply this theory to a relativistically covariant quantum field theoretical form of the (soluble) Lee model. We construct the translation representations with the help of the wave operators, and show that the resulting Lax-Phillips $S$-matrix is an inner function (the Lax-Phillips theory is essentially a theory of translation invariant subspaces). In the special case that the $S$-matrix is a rational inner function, we obtain the resonant state explicitly and analyze its particle ($V, N, \theta$) content. If there is an exponential bound, the general case differs only by a so-called trivial inner factor, which does not change the complex spectrum, but may affect the wave function of the resonant state., Comment: Plain TeX, 33 pages

A charged particle circling in a uniform magnetic field and kicked by an electric field is considered. Under the assumption of small magnetic field, an iterative map is developed. Comparison between the (relativistic) non-radiative case and the (relativistic) radiative case shows that in both cases one can observe a stochastic web structure, and that both cases are qualitatively similar., Comment: 8 Revtex pages including 4 figures

The basic work of Zaslavskii et al showed that the classical non-relativistic electromagnetically kicked oscillator can be cast into the form of an iterative map on the phase space; the resulting evolution contains a stochastic flow to unbounded energy. Subsequent studies have formulated the problem in terms of a relativistic charged particle in interaction with the electromagnetic field. We review the structure of the covariant Lorentz force used to study this problem. We show that the Lorentz force equation can be derived as well from the manifestly covariant mechanics of Stueckelberg in the presence of a standard Maxwell field, establishing a connection between these equations and mass shell constraints. We argue that these relativistic generalizations of the problem are intrinsically inaccurate due to an inconsistency in the structure of the relativistic Lorentz force, and show that a reformulation of the relativistic problem, permitting variations (classically) in both the particle mass and the effective ``mass'' of the interacting electromagnetic field, provides a consistent system of classical equations for describing such processes., Comment: 14 tex pages + 1 eps figure

A brief summary of the mathematical structure of the Lee-Friedrichs model is given, including the natural extension of its Hamiltonian to a Gel'fand triple, containing a generalized eigenstate for the complex pole. Contains references to some physical applications and relativistic generalizations., Comment: To be published in Kluwer Encyclopedia of Mathematics;review

We review the application of the Wigner-Weisskopf model to the two-channel decay problem for the neutral $K$ meson system in the resolvent formalism. The residues in the pole approximation are not orthogonal, leading to additional interference terms in the $K_S-K_L 2\pi$ channel. We show that these terms lead to non-trivial changes in the exit beam in comparison to the result calculated with the assumptions of Lee, Oehme and Yang, and Wu and Yang, corresponding to semigroup evolution for which the pole residues are orthogonal, and hence appear to rule out the applicability of the Wigner-Weisskopf model for the computation of regeneration in the neutral $K$ meson system., Comment: 7 pages, Plain TeX