L. P. Horwitz

Tel Aviv University | TAU · Department of Physics and Astronomy

In this paper we review the fundamental concepts of entropy bounds put forward by Bousso and its relation to the holographic principle. We relate covariant entropy with logarithmic distance of separation of nearby geodesics. We also give sufficient arguments to show that the origin of entropy bounds is not indeed thermodynamic, but statistical.

The Raychaudhuri equation is derived by assuming geometric flow in space–time M of n+1 dimensions. The equation turns into a harmonic oscillator form under suitable transformations. Thereby, a relation between geometrical entropy and mean geodesic deviation is established. This has a connection to chaos theory where the trajectories diverge exponen...

Tunneling as a Source for Quantum Chaos - Lecture Presentation

Raychaudhuri equation is derived by assuming geometric flow in spacetime M of n+1 dimensions. The equation turns into a harmonic oscillator form under suitable transformations.Thereby a relation between geometrical entropy and mean geodesic deviation is established. This has a connection to chaos theory where the trajectories diverge exponentially....

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...

In this chapter, we discuss the scattering theory of Lax and Phillips (Lax and Phillips 1967), originally developed for the description of resonances in the scattering of classical waves, such as electromagentic or acoustic waves, off compactly supported obstacles. In Appendix B, we give an extensive treatment of the mathematical background enablin...

In this chapter we discuss the construction of self-adjoint operators indicating the direction of time within the framework of standard quantum mechanics (Strauss 2011). Such operators will be referred to as Lyapunov operators. The particular construction of Lyapunov operators in this section will enable us to develop a formalism providing a good a...

After some years of observed radioactivity, Gamow (Gamow (1928)) wrote down a formula in quantum mechanics that was supposed to govern the decay of the nucleus of an atom into a lighter nucleus with the emission of an alpha particle (helium atoms without their electron cloud, i.e. ionized helium). This formula, a Schrödinger equation for a stable p...

As we have discussed in Chap. 2, several aspects of the Lax–Phillips scattering theory (Lax and Phillips 1967) distinguish it as an appealing abstract formalism for implementation even in situations outside of the strict range of problems for which it has been originally devised. The description of resonances in the framework of the Lax–Phillips th...

In this chapter we show that the characterization of unstable Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor in terms of the structure of a geometric type Hamiltonian can be applied to a wide class of potential models of standard form through definition of a conformal metric.

In this chapter, 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 one to one corresponding motions are along geodesic curves.

The Hamilton equations for Hamiltonian systems of the type discussed, for example, by Gutzwiller (Gutzwiller 1990) and Curtiss and Miller (Curtiss and Miller 1985), with Hamiltonian of the form.

This book focuses on unstable systems both from the classical and the quantum mechanical points of view and studies the relations between them. The first part deals with quantum systems. Here the main methods are critically described, such as the Gamow approach, the Wigner-Weisskopf formulation, the Lax-Phillips theory, and a method developed by th...

A consistent (off-shell) 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 br...

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, using local coordinate transformations, following the original procedure of Einstein, this theory is embedded into the framework o...

A consistent (off-shell) 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 br...

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...

Abstract --------------- 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 Hamilto...

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, using local coordinate transformations, following the original procedure of Einstein, this theory is embedded into the framework o...

Classical chaos is often characterized as exponential divergence of nearby trajectories. In many interesting cases these trajectories can be identified with geodesic curves. We define here the entropy by $S = \ln \chi (x)$ with $\chi(x)$ being the distance between two nearby geodesics. We derive an equation for the entropy which by transformation t...

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...

The relativistic quantum theory of Stueckelberg, Horwitz and Piron (SHP) describes in a simple way the experiment on interference in time of an electron emitted by femtosecond laser pulses carried out by Lindner {\it et al}. In this paper, we show that, in a way similar to our study of the Lindner {\it et al} experiment (with some additional discus...

This work concerns a study of the quantum mechanical extension of the work of Horwitz et al. [1] on the stability of classical Hamiltonian systems by geometrical methods. Simulations are carried out for several important examples, these show that the quantum mechanical extension of the classical method, for which trajectories are plotted as expecta...

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...

We review our previous work on the existence of Lorentz invariant Berry phases generated in the Stueckleberg-Horwitz-Piron manifestly covariant quantum theory (SHP), also reviewed here, by a perturbed four dimensional harmonic oscillator. These phases are associated with a fractional perturbation of the azimuthal symmetry of the oscillator. They ar...

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...

A relativistic 4D string is described in the framework of the covariant quantum theory first introduced by Stueckelberg (1941) [1], and further developed by Horwitz and Piron (1973) [2], and discussed at length in the book of Horwitz (2015) [3]. We describe the space-time string using the solutions of relativistic harmonic oscillator [4]. We first...

We construct a model for a particle in the framework of the theory of Stueckelberg, Horwitz and Piron (SHP) as an ensemble of events subject to the laws of covariant classical equilibrium statistical mechanics. The canonical and grand canonical emsembles are constructed without an a priori constraint on the total mass of the system. We show that th...

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 gen...

Introduction and some problems encountered in the construction of a relativistic quantum theory.- Relativistic Classical and Quantum Mechanics.- Spin, Statistics and Correlations.- Gauge Fields and Flavor Oscillations.- The Relativistic Action at a Distance Two Body Problem.- Experimental Consequences of Coherence in Time.- Scattering Theory and Re...

One of the deepest and most difficult problems of theoretical physics in the past century has been the construction of a simple, well-defined one-particle theory which unites the ideas of quantum mechanics and relativity. Early attempts, such as the construction of the Klein-Gordon equation and the Dirac equation were inadequate to provide such a t...

To develop the foundations of a manifestly covariant mechanics, we must first examine the Einstein notion of time and its physical meaning. We will then be in a position to introduce the relativistic quantum theory developed by Stueckelberg (1941) and Horwitz and Piron (1973). We describe in this chapter a simple and conceptual understanding of the...

We shall discuss in this chapter the basic idea of a relativistic particle with spin , based on Wigner’s seminal work (Wigner 1939).The theory is adapted here to be applicable to relativistic quantum theory; in this form, Wigner’s theory, together with the requirements imposed by the observed correlation between spin and statistics in nature for id...

Relativistic scattering theory has been generally based on quantum field theory, providing methods of computing an S matrix (transition amplitude operator) by the semi-axiomatic approach of Lehmann et al. (1955) or through the use of interaction picture expansion of the perturbed field equations (Schweber 1964; Jauch and Rohrlich 1955; Schwinger-To...

In this chapter we describe three important applications of the theory. In the first section, we discuss the application of the Stueckelberg theory to the calculation of the anomalous moment of the electron (Bennett 2012). The original work of Schwinger (1951), and many later treatments (Itzykson 1980) use the standard formalism of quantum field th...

In this chapter we discuss the general formulation of gauge fields in the quantum theory, both abelian and nonabelian. A generalization of the elementary Stueckelberg diagram (Fig. 2. 1), demonstrating a “classical” picture of pair annihilation and creation, provides a similar picture of a process involving two or more vertices (diagrams of this ty...

Models with action at a distance potentials, such as the Coulomb potential, have been very useful in nonrelativistic mechanics. They provide a simpler framework than the perhaps more fundamental field mediated models for interaction, and are also straightforwardly amenable to rigorous mathematical analysis. In this Newtonian-Galilean view, all even...

In this chapter we shall discuss the notion of coherence in time, and, in particular, describe in some detail the experiment of Lindner et al. (2005) in which it was demonstrated that an electron wave packet undergoing a sequential ionizing perturbation in time (from Argon gas) undergoes interference phenomena, and the careful analysis and design o...

In this chapter, we shall discuss the statistical mechanics of a many event system, for which the points in space time constitute the fundamental entities for which distribution functions must be constructed to achieve a manifestly covariant theory. Assuming that each event is part of an evolving world line, as in our construction of Chap. 4. the c...

In this chapter, we discuss the cosmological problem of accounting for the radiation curves of galaxies. It has commonly been assumed that the disagreement of simulations using the Newtonian form for gravitational attraction (with forces proportional to \(1/r^2\) between stellar bodies) with the Tulley-Fisher radiation curves (Tulley 1977) is due t...

We study a class of dynamical systems for which the motions can be described in terms of geodesics on a manifold (ordinary potential models can be cast into this form by means of a conformal map). It is rigorously proven that the geodesic deviation equation of Jacobi, constructed with a second covariant derivative, is unitarily equivalent to that o...

The most recent meeting took place at the University of Connecticut, Storrs, on June 9-13, 2014. This meeting forms the basis for the Proceedings that are recorded in this issue of the Journal of Physics: Conference Series. Along with the work of some of the founding members of the Association, we were fortunate to have lecturers from application a...

We show that a modification of Wigner's induced representation for the description of a relativistic particle with spin can be used to construct spinors and tensors of arbitrary rank, with invariant decomposition over angular momentum. In particular, scalar and vector fields, as well as the representations of their transformations, are constructed....

A relativistic 4D string is described in the framework of the covariant quantum theory first introduced by Stueckelberg (1941), and further developed by Horwitz and Piron (1973). We describe the spacetime string using the solutions of relativistic harmonic oscillator. The mass and energy spectrum are derived. We first study the problem of the discr...

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 indep...

An investigation of dynamical properties of solutions of a 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 a given model. It is shown that, at large rapidity, the “fan” amplitude is also a leading...

We show here that a recently developed criterion for the stability of conservative Hamiltonian systems can be extended to Hamiltonians with weak time dependence. In this method, the geodesic equations contain the Hamilton equations of the original potential model through an inverse map in the tangent space in terms of a geometric embedding. The sec...

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 Lor...

An experiment is proposed which can distinguish between two approaches to the reality of the electric field, and whether it has mechanical properties such as mass and stress. A charged pendulum swings within the field of a much larger charge. The two fields manifest the familiar apparent curvature of their field-lines, "bent" so as not to cross eac...

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...

We show that there is nontrivial Berry relativistically covariant phase generated by a perturbed relativistic oscillator. This phase is associated with a fractional perturbation of the azimuthal symmetry of the oscillator.

Under a proper assignment of a metric and a connection, the (classical) dynamical trajectories can be identified as geodesics of the underlying manifold. We show how these geometric structures can be derived; specifically, we construct them explicitly for configuration and phase spaces of Hamiltonian systems. We demonstrate how the correspondence b...

We consider the two-level system approximation of a single emitter driven by a continuous laser pump and simultaneously coupled to the electromagnetic vacuum and to a thermal reservoir beyond the Markovian approximation. We discuss the connection between a rigorous microscopic theory and the phenomenological spectral diffusion approach, used to mod...

The adaptation of Wigner's induced representation for a relativistic quantum theory making possible the construction of wavepackets and admitting covariant expectation values for the coordinate operator x^\mu introduces a foliation on the Hilbert space of states. The spin-statistics relation for fermions and bosons implies the universality of the p...

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...

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.

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...

In this essay we discuss the geometrical embedding method (GEM) for the analysis of the stability of Hamiltonian systems using geometrical techniques familiar from general relativity. This method has proven to be very effective. In particular, we show that although the application of standard Lyapunov analysis predicts that completely integrable Ke...

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 produ...

Although the subject of relativistic dynamics has been explored from both classical and quantum mechanical points of view since the work of Einstein and Dirac, its most striking development has been in the framework of quantum field theory. The very accurate calculations of spectral and scattering properties, for example, of the anamolous magnetic...

The role of time has changed conceptually moving from classical Newtonian physics to general relativity and is one of the main obstacles avoiding a clear unification between a covariant quantum mechanics theory and a theory of gravity. In quantum mechanics as in Newtonian physics, time is an evolutional causal parameter, while in general relativity...

In previous papers 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 τ). In this paper, we reconcile these derivations resulting in different explicit forms, and relate our results to the convention...

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 Lore...

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. [5] 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 g...

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 t...

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 infinit...

In [Y. Strauss, “Self-adjoint Lyapunov variables, temporal ordering and irreversible representations of Schrödinger evolution”, J. Math. Phys. 51, No. 2, Article ID 022104 (2010)] 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...

The International Association for Relativistic Dynamics was organized in February 1998 in Houston, Texas, with John R. Fanchi as president. Although the subject of relativistic dynamics has been explored, from both classical and quantum mechanical points of view, since the work of Einstein and Dirac, its most striking development has been in the fr...

We review the formulation of the problem of electromagnetic self-interaction of a relativistic charged particle in the framework of the manifestly covariant classical mechanics of Stueckeleberg, Horwitz, and Piron. The gauge fields of this theory, in general, cause the mass of the particle to change. We study the four dynamical off-mass-shell orbit...

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 t...

An effective characterization of chaotic conservative Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor derived from the structure of the Hamiltonian has been extended to a wide class of potential models of standard form through definition of a conformal metric. The geodesic equations reproduce the Hamilton eq...

In this paper we explore the problem of fields generated by a source undergoing hyperbolic motion in the framework of Stueckelberg manifestly covariant relativistic dynamics. The resulting gauge fields are computed numerically using Green-Functions which are retarded in the Stueckelberg absolute time τ, and qualitatively compared with Maxwell field...

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

It has been shown that the orbits of motion for a wide class of nonrelativistic Hamiltonian systems can be described as geodesic flows on a manifold and an associated dual. This method can be applied to a four dimensional manifold of orbits in spacetime associated with a relativistic system. We show that a relativistic Hamiltonian which generates E...

It has been shown that the orbits of motion for a wide class of nonrelativistic Hamiltonian systems can be described as geodesic flow on a manifold and an associated dual. This method can be applied to a four dimensional manifold of orbits in space-time associated with a relativistic system. One can study the consequences on the geometry of the int...

1. Foundations of quantum statistical mechanics; 2. Elementary examples; 3. Quantum statistical master equation; 4. Quantum kinetic equations; 5. Quantum irreversibility; 6. Entropy and dissipation: the microscopic theory; 7. Global equilibrium: thermostatics and the microcanonical ensemble; 8. Bose-Einstein ideal gas condensation; 9. Scaling, reno...

This book concentrates on quantum statistical mechanics. It contains chapters on foundations, including entanglement and irreversibility, quantum optics and damping, unstable systems, relativistically covariant statistical mechanics of many particle systems and many other topics.

Using a recently developed geometrical method, we study the transition from order to chaos in an important class of Hamiltonian systems. We show agreement between this geometrical method and the surface of section technique applied to detect chaotic behavior. We give, as a particular illustration, detailed results for an important class of potentia...

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 conti...

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 stud...

We use a geometrical method to distinguish between ordered and chaotic motion in three-dimensional Hamiltonian systems. We show that this method gives results in agreement with the computation of Lyapunov characteristic exponents. We discuss some examples of unstable Hamiltonian systems in three dimensions, giving, as a particular illustration, det...

It is shown that the lifetime of the hyperfine level giving rise to the 21 cm hydrogen line might be affected by the so-called Zeno effect, and hence that the amount of neutral hydrogen in our galaxy may be larger than previously deduced.

It is shown that a Zeno-type effect due to disturbances caused by collisions associated with correlated pairs inside the nucleus may significantly modify double-beta decay rates. Several nuclei for which double-beta decay data is available are studied, and it is shown that the trend of the effect is consistent with this interpretation. The resultin...

The characterization of chaotic Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor in the structure of the Hamiltonian is extended to a wide class of potential models of standard form through definition of a conformal metric. The geodesic equations reproduce the Hamilton equations of the original potential mode...

The scattering theory of Lax and Phillips, designed primarily for hyperbolic systems, such as electromagnetic or acoustic waves, is described. The embedding of the quantum theory into this structure, carried out by Flesia and Piron, is reviewed. We show how the density matrix for an effectively pure state can evolve to an effectively mixed state (d...

The recently developed quantum theory utilizing the ideas and results of Lax and Phillips for the description of scattering and resonances, or unstable systems, is reviewed. The framework for the construction of the Lax-Phillips theory is given by a functional space which is the direct integral over time of the usual quantum mechanical Hilbert spac...

I discuss the interpretation of a recent experiment showing quantum interference in time. It is pointed out that the standard nonrelativistic quantum theory does not have the property of coherence in time, and hence cannot account for the results found. Therefore, this experiment has fundamental importance beyond the technical advances it represent...

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...

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 a...

We show that the method of stochastic reduction of linear superpositions can be applied to the process of disentanglement for the spin-0 state of two spin-1/2 particles. We describe the geometry of this process in the framework of the complex projective space

The requirement of gauge invariance for the Stckelberg equation for the relativistically covariant wave function of a system evolving according a universal world (or historical) time T implies the existence of a five dimensional pre-Maxwell field on the manifold of spacetime and r. The Maxwell theory is contained in this theory; integration of the...

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I comment on the interpretation of a recent experiment showing quantum interference in time. It is pointed out that the standard nonrelativistic quantum theory, used by the authors in their analysis, cannot account for the results found, and therefore that this experiment has fundamental importance beyond the technical advances it represents. Some...