arXiv subjects

L. P. Horwitz

Publications and source records attributed to L. P. Horwitz.

At least 55 records · Page 3Linked to original sources

arXiv2001

Lax-Phillips Scattering Theory of a Relativistic Quantum Field Theoretical Lee-Friedrichs Model and Lee-Oehme-Yang-Wu Phenomenology

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.

hep-ph↗

arXiv2000

Classical Radiation Reaction Off-Shell Corrections to the Covariant Lorentz Force

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.

physics.class-ph↗

arXiv1999

Representation of the Resonances of a Relativistic Quantum Field Theoretical Model in Lax-Phillips Scattering Theory

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.

quant-ph↗

arXiv1999

The Radiative Kicked Oscillator: A Stochastic Web or Chaotic Attractor ?

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.

chao-dyn↗

arXiv1999

Representation of the Resonance of a Relativistic Quantum Field Theoretical Lee-Friedrichs Model in Lax-Phillips Scattering Theory

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 ($V, N, θ$) 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.

math-ph↗

arXiv1999

The Effect of Radiation on the Stochastic Web

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.

chao-dyn↗

arXiv1999

Chaos and Maps in Relativistic Dynamical Systems

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.

chao-dyn↗

arXiv1998

Lee-Friedrichs Model

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.

hep-ph↗

arXiv1998

Dynamical Relativistic Systems and the Generalized Gauge Fields of Manifestly Covariant Theories

The problem of the classical non-relativistic electromagnetically kicked oscillator can be cast into the form of an iterative map on phase space. The original work of Zaslovskii {\it et al} showed that the resulting evolution contains a stochastic flow in phase space to unbounded energy. Subsequent studies have formulated the problem in terms of a relativistically charged particle in interaction with the electromagnetic field. We review the standard derivation of the covariant Lorentz force, and review the structure of the relativistic equations 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. We show how this agreement is achieved, and criticize some of the fundamental assumptions underlying these derivations. We argue that a more complete theory, involving ``off-shell'' electromagnetic fields should be utilized. We then discuss the formulation of the off-shell electromagnetism implied by the full gauge invariance of the Stueckelberg mechanics (based on its quantized form), and show that a more general class of physical phenomena can occur.

hep-th↗

arXiv1998

Effect of Non-Orthogonality of Residues in the Wigner-Weisskopf model on Regeneration for the Neutral $K$ Meson system

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 $K_S-K_L 2π$ 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.

hep-ph↗

arXiv1998

Non-Orthogonality of Residues in the Wigner-Weisskopf Model for Neutral K Meson Decay

We review the application of the Wigner-Weisskopf model for the neutral K meson system in the resolvent formalism. The Wigner-Weisskopf model is not equivalent to the Lee-Oehme-Yang-Wu formulation (which provides an accurate representation of the data). The residues in the pole approximation in the Wigner-Weisskopf model are not orthogonal, leading to additional interference terms in the $K_S-K_L 2π$ channel. We show that these terms would be detectable experimentally in the decay pattern of the beam emitted from a regenerator if the Wigner-Weisskopf theory were correct. The consistency of the data with the Lee-Oehme-Yang-Wu formulation appears to rule out the applicability of the Wigner-Weisskopf theory to the problem of neutral K meson decay.

hep-th↗

arXiv1998

Second Quantization of the Stueckelberg Relativistic Quantum Theory and Associated Gauge Fields

The gauge compensation fields induced by the differential operators of the Stueckelberg-Schrödinger equation are discussed, as well as the relation between these fields and the standard Maxwell fields. An action is constructed and the second quantization of the fields carried out using a constraint procedure. Some remarks are made on the properties of the second quantized matter fields.

hep-th↗

arXiv1998

Description of Unstable Systems in Relativistic Quantum Mechanics in the Lax-Phillips Theory

We discuss some of the experimental motivation for the need for semigroup decay laws, and the quantum Lax-Phillips theory of scattering and unstable systems. In this framework, the decay of an unstable system is described by a semigroup. The spectrum of the generator of the semigroup corresponds to the singularities of the Lax-Phillips $S$-matrix. In the case of discrete (complex) spectrum of the generator of the semigroup, associated with resonances, the decay law is exactly exponential. The states corresponding to these resonances (eigenfunctions of the generator of the semigroup) lie in the Lax-Phillips Hilbert space, and therefore all physical properties of the resonant states can be computed. We show that the parametrized relativistic quantum theory is a natural setting for the realization of the Lax-Phillips theory.

hep-th↗

arXiv1998

Representation of Quantum Mechanical Resonances in the Lax-Phillips Hilbert Space

We discuss the quantum Lax-Phillips theory of scattering and unstable systems. In this framework, the decay of an unstable system is described by a semigroup. The spectrum of the generator of the semigroup corresponds to the singularities of the Lax-Phillips $S$-matrix. In the case of discrete (complex) spectrum of the generator of the semigroup, associated with resonances, the decay law is exactly exponential. The states corresponding to these resonances (eigenfunctions of the generator of the semigroup) lie in the Lax-Phillips Hilbert space, and therefore all physical properties of the resonant states can be computed. We show that the Lax-Phillips $S$-matrix is unitarily related to the $S$-matrix of standard scattering theory by a unitary transformation parametrized by the spectral variable $σ$ of the Lax-Phillips theory. Analytic continuation in $σ$ has some of the properties of a method developed some time ago for application to dilation analytic potentials. We work out an illustrative example using a Lee-Friedrichs model for the underlying dynamical system.

hep-th↗

arXiv1998

Chaos of the Relativistic Parametrically Forced van der Pol Oscillator

A manifestly relativistically covariant form of the van der Pol oscillator in 1+1 dimensions is studied. We show that the driven relativistic equations, for which $x$ and $t$ are coupled, relax very quickly to a pair of identical decoupled equations, due to a rapid vanishing of the ``angular momentum'' (the boost in 1+1 dimensions). A similar effect occurs in the damped driven covariant Duffing oscillator previously treated. This effect is an example of entrainment, or synchronization (phase locking), of coupled chaotic systems. The Lyapunov exponents are calculated using the very efficient method of Habib and Ryne. We show a Poincaré map that demonstrates this effect and maintains remarkable stability in spite of the inevitable accumulation of computer error in the chaotic region. For our choice of parameters, the positive Lyapunov exponent is about 0.242 almost independently of the integration method.

chao-dyn↗

arXiv1997

New Quadratic Mass Relations for Heavy Mesons

By assuming the existence of (quasi)-linear Regge trajectories for heavy mesons, we derive new quadratic mass relations of non-Gell-Mann-Okubo type, 6M^2(q-qbar)+3M^2(c-cbar)=8M^2(c-qbar), 20M^2(q-qbar)+5M^2(b-bbar)= 16M^2(b-qbar), q=n(=u,d),s, which show excellent agreement with experiment. We also establish the sum rule M^2(i-ibar)+M^2(j-jbar)-2M^2(j-ibar)\approx const for any pair of flavors, (i,j).

hep-ph↗

arXiv1997

New quadratic baryon mass relations

By assuming the existence of (quasi)-linear baryon Regge trajectories, we derive new quadratic Gell-Mann-Okubo type baryon mass relations. These relations are used to predict the masses of the charmed baryons absent from the Baryon Summary Table so far, in good agreement with the predictions of many other approaches.

hep-ph↗

arXiv1997

New Mass Relation for Meson 25-plet

By assuming the existence of (quasi)-linear Regge trajectories for 25-plet mesons in the low energy region, we derive a new, 14th power, meson mass relation. This relation may be reduced to a quadratic Gell-Mann-Okubo type formula by fitting the values of the Regge slopes of these (quasi)-linear trajectories. Such a formula holds with an accuracy of ~2% for vector mesons, and also suggests that the quantum number I(J^P)=1/2(2^{+}) should be assigned to the both B_J(5732) and B_{sJ}(5850) mesons discovered recently. We also discuss reasons for the failure of group theory to produce correct mass relations for heavy quarkonia.

hep-ph↗