arXiv subjects

Horace W. Crater

Publications and source records attributed to Horace W. Crater.

At least 19 recordsLinked to original sources

arXiv2014

On the Question of the Point-Particle Nature of the Electron

The electron and the positron treated as point particles in the Two Body Dirac equations of constraint dynamics for QED possess a new and as yet undiscovered peculiar ${}^1S_0$ bound-state which has a very large binding energy of about 300 keV, in addition to the usual ${}^1S_0$ positronium state with a binding energy of 6.8 eV. The production and detection of the peculiar ${}^1S_0$ state provide a test of the electron point-charge property. As the peculiar ${}^1S_0$ state lies lower than the usual ${}^1S_0$ state, the peculiar ${}^1S_0$ state can be produced by a two-photon decay of the usual $^{1}S_{0}$ state. We estimate the rate of the two-photon decay and show how it depends on the probability $P_{up}$ of the admixture of the peculiar component in the predominantly usual ${}^1S_0$ positronium. The produced peculiar ${}^1S_0$ state in turn annihilates into two photons with a total c.m. energy of about 723 keV. Thus the signature for this new peculiar ${}^1S_0$ positronium bound state would be the decay of the usual ${}^1S_0$ state into four photons, with two energies bunching around 150 and two around 360 keV. Such a four-photon decay of the usual ${}^1S_0$ state will not be present if the electron and positron are not point particles, or if the mixing probability $P_{up}$ is very small.

hep-ph↗

arXiv2014

Non-Inertial Frames in Minkowski Space-Time, Accelerated either Mathematical or Dynamical Observers and Comments on Non-Inertial Relativistic Quantum Mechanics

After a review of the existing theory of non-inertial frames and mathematical observers in Minkowski space-time we give the explicit expression of a family of such frames obtained from the inertial ones by means of point-dependent Lorentz transformations as suggested by the locality principle. These non-inertial frames have non-Euclidean 3-spaces and contain the differentially rotating ones in Euclidean 3-spaces as a subcase. Then we discuss how to replace mathematical accelerated observers with dynamical ones (their world-lines belong to interacting particles in an isolated system) and of how to define Unruh-DeWitt detectors without using mathematical Rindler uniformly accelerated observers. Also some comments are done on the transition from relativistic classical mechanics to relativistic quantum mechanics in non-inertial frames.

gr-qc↗

arXiv2014

Applications of Two Body Dirac Equations to Hadron and Positronium Spectroscopy

We review recent applications of the Two Body Dirac equations of constraint dynamics to meson spectroscopy and describe new extensions to three-body problems in their use in the study of baryon spectroscopy. We outline unique aspects of these equations for QED bound states that distinguish them among the various other approaches to the relativistic two body problem. Finally we discuss recent theorectial solutions of new peculiar bound states for positronium arising from the Two Body Dirac equations of constraint dynamics, assuming point particles for the electron and the positron.

hep-ph↗

arXiv2014

On Relativistic Entanglement and Localization of Particles and on their Comparison with the Non-Relativistic Theory

We make a critical comparison of relativistic and non-relativistic classical and quantum mechanics of particles in inertial frames and of the open problems in particle localization at the two levels. The solution of the problems of the relativistic center of mass, of the clock synchronization convention needed to define relativistic 3-spaces and of the elimination of the relative times in the relativistic bound states leads to a description with a decoupled non-local (non-measurable) relativistic center of mass and with only relative variables for the particles (single particle subsystems do not exist). We analyze the implications for entanglement of this relativistic spatial non-separability not existing in non-relativistic entanglement. Then we try to reconcile the two visions showing that also at the non-relativistic level in real experiments only relative variables are measured with their directions determined by the effective mean classical trajectories of particle beams present in the experiment. We discuss some of the existing problems with the observers and their reference frames, leaving to the future the discussion in non-inertial frames.

quant-ph↗

arXiv2013

Baryon Spectrum Analysis using Dirac's Covariant Constraint Dynamics

The energy spectrum of the baryons is determined by treating each of them as a three-body system with the interacting forces coming from a set of two-body potentials that depend on both the distance between the quarks and the spin and orbital angular momentum coupling terms. Constraint dynamics is first reviewed for a relativistic two-body system in order to assemble the necessary two body framework for the three-body problem and then we review the different types of covariant two-body interactions involved in constraint dynamics, including vector and scalar, and solve the problem of energy eigenstates using constraint dynamics. The Two Body Dirac equations of constraint dynamics derived by Crater and Van Alstine, matched with the quasipotential formalism of Todorov as the underlying two-body formalism are used, as well as the three-body constraint formalism of Sazdjian to integrate the three two-body equations into a single relativistically covariant three body equation for the bound state energies. The results are analyzed and compared to experiment using a best fit method and several different algorithms, including a gradient approach, and Monte Carlo method.

hep-ph↗

arXiv2013

On the Relativistic Micro-Canonical Ensemble and Relativistic Kinetic Theory for N Relativistic Particles in Inertial and Non-Inertial Rest Frames

A new formulation of relativistic classical mechanics allows a revisiting of old unsolved problems in relativistic kinetic theory and in relativistic statistical mechanics. In particular a definition of the relativistic micro-canonical partition function is given strictly in terms of the Poincaré generators of an interacting N-particle system both in the inertial and non-inertial rest frames. The non-relativistic limit allows a definition of both the inertial and non-inertial micro-canonical ensemble strictly in terms of the Galilei generators. Also the one-particle relativistic distribution function is defined and a new approach to the relativistic Boltzmann equation is delineated. Finally there are some comments on relativistic dissipative fluids.

math-ph↗

arXiv2012

Magnetic States at Short Distances

The magnetic interactions between a fermion and an antifermion of opposite electric or color charges in the 1S0 and 3P0 states are very attractive and singular near the origin and may allow the formation of new bound and resonance states at short distances. In the two body Dirac equations formulated in constraint dynamics, the short-distance attraction for these states for point particles leads to a quasipotential that behaves near the origin as -α^2/r^2. Both 1S0 and 3P0 states admit two types of eigenstates with drastically different behaviors for the radial wave function u=rψ. One type of states, with u growing as r^{1+\sqrt(1-4*α^2)/2} at small r, will be called usual states. The other type of states with u growing as r^{(1-\sqrt(1-4*α^2))/2} will be called peculiar states. Both of the usual and peculiar eigenstates have admissible behaviors at short distances. The usual bound 1S0 states possess attributes the same as those one usually encounters in QED and QCD. In contrast, the peculiar bound 1S0 states, yet to be observed, have distinctly different bound state properties and scattering phase shifts. For the 3P0 states, the usual solutions lead to the standard bound state energies and no resonance, but resonances have been found for the peculiar states whose energies depend on the description of the internal structure of the charges, the mass of the constituent, and the coupling constant. The existence of both usual and peculiar eigenstates in the same system leads to the non-self-adjoint property of the mass operator and two non-orthogonal complete sets. The mass operator can be made self-adjoint with a single complete set of admissible states by introducing a new peculiarity quantum number and an enlarged Hilbert space.

hep-ph↗

arXiv2011

On the Mass Difference between pion and rho meson using a Relativistic Two-Body Model

The big mass difference between the pion and rho meson possibly originates from the spin-dependent nature of the interactions in the two states since these two states are similar except for spin. Both pi and rho are quark-antiquark systems which can be treated using the two-body Dirac equations (TBDE) of constraint dynamics. This relativistic approach for two-body system has the advantage over the non-relativistic treatment in the sense that the spin-dependent nature is automatically coming out from the formalism. We employed Dirac's relativistic constraint dynamics to describe quark-antiquark systems. Within this formalism, the 16-component Dirac equation is reduced to the 4-component 2nd-order differential equation and the radial part of this equation is simply a Schroedinger-type equation with various terms calculated from the basic radial potential. We used a modified Richardson potential for quark-antiquark systems which satisfies the conditions of confinement and asymptotic freedom. We obtained the wave functions for these two mesons which are not singular at short distances. We also found that the cancellation between the Darwin and spin-spin interaction terms occurs in the pi mass but not in the rho mass and this is the main source of the big difference in the two meson masses.

hep-ph↗

arXiv2011

A Relativistic Version of the Two-Level Atom in the Rest-Frame Instant Form of Dynamics

We define a relativistic version of the two-level atom, in which an extended atom is replaced by a point particle carrying suitable Grassmann variables for the description of the two-level structure and of the electric dipole. After studying the isolated system "atom plus the electro-magnetic field" in the electric-dipole representation as a parametrized Minkowski theory, we give its restriction to the inertial rest frame and the explicit form of the Poincaré generators. After quantization we get a two-level atom with a spin 1/2 electric dipole and the relativistic generalization of the Hamiltonians of the Rabi and Jaynes-Cummings models.

math-ph↗

arXiv2010

Schwarzschild Solution of the Generally Covariant Quaternionic Field Equations of Sachs

Sachs has derived quaternion field equations that fully exploit the underlying symmetry of the principle of general relativity, one in which the fundamental 10 component metric field is replaced by a 16 component four-vector quaternion. Instead of the 10 field equations of Einstein's tensor formulation, these equations are 16 in number corresponding to the 16 analytic parametric functions {\partial}x^{μ'}/{\partial}x^{ν} of the Einstein Lie Group. The difference from the Einstein equations is that these equations are not covariant with respect to reflections in space-time, as a consequence of their underlying quaternionic structure. These equations can be combined into a part that is even and a part that is odd with respect to spatial or temporal reflections. This paper constructs a four-vector quaternion solution of the quaternionic field equation of Sachs that corresponds to a spherically symmetric static metric. We show that the equations for this four-vector quaternion corresponding to a vacuum solution lead to differential equations that are identical to the corresponding Schwarzschild equations for the metric tensor components.

gr-qc↗

arXiv2010

Applications of Two-Body Dirac Equations to the Meson Spectrum with Three versus Two Covariant Interactions, SU(3) Mixing, and Comparison to a Quasipotential Approach

In a previous paper Crater and Van Alstine applied the Two Body Dirac equations of constraint dynamics to the meson quark-antiquark bound states using a relativistic extention of the Adler-Piran potential and compared their spectral results to those from other approaches, ones which also considered meson spectroscopy as a whole and not in parts. In this paper we explore in more detail the differences and similarities in an important subset of those approaches, the quasipotential approach. In the earlier paper, the transformation properties of the quark-antiquark potentials were limited to a scalar and an electromagnetic-like four vector, with the former accounting for the confining aspects of the overall potential, and the latter the short range portion. A part of that work consisted of developing a way in which the static Adler-Piran potential was apportioned between those two different types of potentials in addition to covariantization. Here we make a change in this apportionment that leads to a substantial improvement in the resultant spectroscopy by including a time-like confining vector potential over and above the scalar confining one and the electromagnetic-like vector potential. Our fit includes 19 more mesons than the earlier results and we modify the scalar portion of the potential in such a way that allows this formalism to account for the isoscalar mesons η and η' not included in the previous work. Continuing the comparisons made in the previous paper with other approaches to meson spectroscopy we examine in this paper the quasipotential approach of Ebert, Faustov, and Galkin for a comparison with our formalism and spectral results.

hep-ph↗

arXiv2010

Massless Particles plus Matter in the Rest-Frame Instant Form of Dynamics

After introducing the parametrized Minkowski theory describing a positive-energy scalar massless particle, we study the rest-frame instant form of dynamics of such a particle in presence of another massive one (to avoid the front form of dynamics). Then we describe massless particles with Grassmann-valued helicity and their quantization.

hep-th↗

arXiv2009

Singularity Structures in Coulomb-Type Potentials in Two Body Dirac Equations of Constraint Dynamics

Two Body Dirac Equations (TBDE) of Dirac's relativistic constraint dynamics have been successfully applied to obtain a covariant nonperturbative description of QED and QCD bound states. Coulomb-type potentials in these applications lead naively in other approaches to singular relativistic corrections at short distances that require the introduction of either perturbative treatments or smoothing parameters. We examine the corresponding singular structures in the effective potentials of the relativistic Schroedinger equation obtained from the Pauli reduction of the TBDE. We find that the relativistic Schroedinger equation lead in fact to well-behaved wave function solutions when the full potential and couplings of the system are taken into account. The most unusual case is the coupled triplet system with S=1 and L={(J-1),(J+1)}. Without the inclusion of the tensor coupling, the effective S-state potential would become attractively singular. We show how including the tensor coupling is essential in order that the wave functions be well-behaved at short distances. For example, the S-state wave function becomes simply proportional to the D-state wave function and dips sharply to zero at the origin, unlike the usual S-state wave functions. Furthermore, this behavior is similar in both QED and QCD, independent of the asymptotic freedom behavior of the assumed QCD vector potential. Light- and heavy-quark meson states can be described well by using a simplified linear-plus-Coulomb-type QCD potential apportioned appropriately between world scalar and vector potentials. We use this potential to exhibit explicitly the origin of the large pi-rho splitting and effective chiral symmetry breaking. The TBDE formalism developed here may be used to study quarkonia in quark-gluon plasma environments.

hep-ph↗

arXiv2008

Towards Relativistic Atomic Physics. II. Collective and Relative Relativistic Variables for a System of Charged Particles plus the Electro-Magnetic Field

In this second paper we complete the classical description of an isolated system of "charged positive-energy particles, with Grassmann-valued electric charges and mutual Coulomb interaction, plus a transverse electro-magnetic field" in the rest-frame instant form of dynamics. In particular we show how to determine a collective variable associated with the internal 3-center of mass on the instantaneous 3-spaces, to be eliminated with the constraints ${\vec {\cal K}}_{(int)} \approx 0$. Here ${\vec {\cal K}}_{(int)}$ is the Lorentz boost generator in the unfaithful internal realization of the Poincare' group and its vanishing is the gauge fixing to the rest-frame conditions ${\vec {\cal P}}_{(int)} \approx 0$. We show how to find this collective variable for the following isolated systems: a) charged particles with a Coulomb plus Darwin mutual interaction; b) transverse radiation field; c) charged particles with a mutual Coulomb interaction plus a transverse electro-magnetic field. Then we define the Dixon multipolar expansion for the open particle subsystem. We also define the relativistic electric dipole approximation of atomic physics in the rest-frame instant form and we find the a possible relativistic generalization of the electric dipole representation.

hep-th↗

arXiv2007

Potential Models and Lattice Gauge Current-Current Correlators

We compare current-current correlators in lattice gauge calculations with correlators in different potential models, for a pseudoscalar charmonium in the quark-gluon plasma. An important ingredient in the evaluation of the current-current correlator in the potential model is the basic principle that out of the set of continuum states, only resonance states and Gamow states with lifetimes of sufficient magnitudes can propagate as composite objects and can contribute to the current-current correlator. When the contributions from the bound states and continuum states are properly treated, the potential model current-current correlators obtained with the potential proposed in Ref. [11] are consistent with the lattice gauge correlators. The proposed potential model thus gains support to be a useful tool to complement lattice gauge calculations for the study of $Q\bar Q$ states at high temperatures.

hep-ph↗

arXiv2007

Two gamma quarkonium and positronium decays with Two-Body Dirac equations of constraint dynamics

Two-Body Dirac equations of constraint dynamics provide a covariant framework to investigate the problem of highly relativistic quarks in meson bound states. This formalism eliminates automatically the problems of relative time and energy, leading to a covariant three dimensional formalism with the same number of degrees of freedom as appears in the corresponding nonrelativistic problem. It provides bound state wave equations with the simplicity of the nonrelativistic Schroedinger equation. Unlike other three-dimensional truncations of the Bethe-Salpeter equation, this covariant formalism has been thoroughly tested in nonperturbatives contexts in QED, QCD, and nucleon-nucleon scattering. Here we continue the important studies of this formalism by extending a method developed earlier for positronium decay into two photons to tests on the sixteen component quarkonium wave function solutions obtained in meson spectroscopy. We examine positronium decay and then the two-gamma quarkonium decays of eta_c, eta'_c, chi_0c, chi_2c, and pi-zero The results for the pi-zero, although off the experimental rate by 13%, is much closer than the usual expectations from a potential model.

hep-ph↗

arXiv2006

Hamiltonian Relativistic Two-Body Problem: Center of Mass and Orbit Reconstruction

After a short review of the history and problems of relativistic Hamiltonian mechanics with action-at-a-distance inter-particle potentials, we study isolated two-body systems in the rest-frame instant form of dynamics. We give explicit expressions of the relevant relativistic notions of center of mass, we determine the generators of the Poincare' group in presence of interactions and we show how to do the reconstruction of particles' orbits from the relative motion and the canonical non-covariant center of mass. In the case of a simple Coulomb-like potential model, it is possible to integrate explicitly the relative motion and show the two dynamical trajectories.

hep-th↗

arXiv2006

Tests of Two-Body Dirac Equation Wave Functions in the Decays of Quarkonium and Positronium into Two Photons

Two-Body Dirac equations of constraint dynamics provide a covariant framework to investigate the problem of highly relativistic quarks in meson bound states. This formalism eliminates automatically the problems of relative time and energy, leading to a covariant three dimensional formalism with the same number of degrees of freedom as appears in the corresponding nonrelativistic problem. It provides bound state wave equations with the simplicity of the nonrelativistic Schroedinger equation. Here we begin important tests of the relativistic sixteen component wave function solutions obtained in a recent work on meson spectroscopy, extending a method developed previously for positronium decay into two photons. Preliminary to this we examine the positronium decay in the 3P_{0,2} states as well as the 1S_0. The two-gamma quarkonium decays that we investigate are for the η_{c}, η_{c}^{\prime}, χ_{c0}, χ_{c2}, π^{0}, π_{2}, a_{2}, and f_{2}^{\prime} mesons. Our results for the four charmonium states compare well with those from other quark models and show the particular importance of including all components of the wave function as well as strong and CM energy dependent potential effects on the norm and amplitude. The results for the π^{0}, although off the experimental rate by 15%, is much closer than the usual expectations from a potential model. We conclude that the Two-Body Dirac equations lead to wave functions which provide good descriptions of the two-gamma decay amplitude and can be used with some confidence for other purposes.

hep-ph↗