arXiv subjects

Martin Land

Publications and source records attributed to Martin Land.

15 recordsLinked to original sources

arXiv2023

An Evolving Spacetime Metric Induced by a `Static' Source

In a series of recent papers we developed a formulation of general relativity in which spacetime and the dynamics of matter evolve with a Poincar\'e invariant parameter $\tau$. In this paper, we apply the formalism to derive the metric induced by a `static' event evolving uniformly along its $t$-axis at the spatial origin ${\mathbf x} = 0$. The metric is shown to vary with $t$ and $\tau$, as well as spatial distance $r$, taking its maximum value for a test particle at the retarded time $\tau = t - r/c$. In the resulting picture, an event localized in space and time produces a metric field similarly localized, where both evolve in $\tau$. We first derive this metric as a solution to the wave equation in linearized field theory, and discuss its limitations by studying the geodesic motion it produces for an evolving event. By then examining this solution in the 4+1 formalism, which poses an initial value problem for the metric under $\tau$-evolution, we clarify these limitations and indicate how they may be overcome in a solution to the full nonlinear field equations.

physics.gen-ph↗

arXiv2022

Local Metric with Parameterized Evolution

We present a canonical Hamiltonian formulation of GR in which $\tau$, the parameter of system evolution, is external to spacetime, playing a role similar to what we call time in nonrelativistic mechanics. This approach, known as Stueckelberg-Horwitz-Piron (SHP) theory, inherits the full computational power of classical analytical mechanics while maintaining manifest covariance throughout and eliminating possible conflict with general diffeomorphism invariance. In particular, SHP simplifies the initial value problem with potential applications in highly dynamical interactions, such as black hole collisions. By allowing the energy-momentum tensor and metric to depend explicitly on $\tau$, we may describe particle motion in geodesic form with respect to a dynamically evolving background metric. As a toy model, we consider a $\tau$-dependent mass $M(\tau)$, first as a perturbation in the Newtonian approximation and then for a Schwarzschild-like metric. As expected, the extended Einstein equations imply a non-zero energy-momentum tensor, proportional to $dM / d\tau$, representing a flow of mass and energy into spacetime that corresponds to the changing source mass. In $\tau$-equilibrium, this system becomes a generalized Schwarzschild solution for which the extended Ricci tensor and energy-momentum tensor vanish.

gr-qc↗

arXiv2022

The Particle as a Statistical Ensemble of Events in Stueckelberg-Horwitz-Piron Electrodynamics

In classical Maxwell electrodynamics, charged particles following deterministic trajectories are described by currents that induce fields, mediating interactions with other particles. Statistical methods are used when needed to treat complex particle and/or field configurations. In Stueckelberg-Horwitz-Piron (SHP) electrodynamics, the classical trajectories are traced out dynamically, through the evolution of a 4D spacetime event $x^\mu (\tau)$ as $\tau$ grows monotonically. Stueckelberg proposed to formalize the distinction between coordinate time $x^0 = ct$ (measured by laboratory clocks) and chronology $\tau$ (the temporal ordering of event occurrence) in order to describe antiparticles and resolve problems of irreversibility such as grandfather paradoxes. Consequently, in SHP theory, the elementary object is not a particle (a 4D curve in spacetime) but rather an event (a single point along the dynamically evolving curve). Following standard deterministic methods in classical relativistic field theory, one is led to Maxwell-like field equations that are \hbox{$\tau$-dependent} and sourced by a current that represents a statistical ensemble of instantaneous events distributed along the trajectory. The width $\lambda$ of this distribution defines a correlation time for the interactions and a mass spectrum for the photons emitted by particles. As $\lambda$ becomes very large, the photon mass goes to zero and the field equations become $\tau$-independent Maxwell's equations. Maxwell theory thus emerges as an equilibrium limit of SHP, in which $\lambda$ is larger than any other relevant time scale. Thus, statistical mechanics is a fundamental ingredient in SHP electrodynamics, and its insights are required to give meaning to the concept of a particle.

physics.gen-ph↗

arXiv2022

A 4+1 Formalism for the Evolving Stueckelberg-Horwitz-Piron Metric

We propose a field theory for the local metric in Stueckelberg--Horwitz--Piron (SHP) general relativity, a framework in which the evolution of classical four-dimensional (4D) worldlines $x^\mu \left( \tau \right)$ ($\mu = 0,1,2,3 $) is parameterized by an external time $\tau$. Combining insights from SHP electrodynamics and the ADM formalism in general relativity, we generalize the notion of a 4D spacetime $\mathcal{M}$ to a formal manifold $\mathcal{M}_5 = \mathcal{M} \times R$, representing an admixture of geometry (the diffeomorphism invariance of $\mathcal{M}$) and dynamics (the system evolution of $\mathcal{M} \left( \tau \right)$ with the monotonic advance of $\tau \in R$). Strategically breaking the formal 5D symmetry of a metric $g_{\alpha\beta}(x,\tau)$ ($\alpha,\beta = 0,1,2,3,5 $) posed on $\mathcal{M}_5$, we obtain ten unconstrained Einstein equations for the $\tau$-evolution of the 4D metric $\gamma_{\mu\nu}(x,\tau)$ and five constraints that are to be satisfied by the initial conditions. The resulting theory differs from five-dimensional (5D) gravitation, much as SHP U(1) gauge theory differs from 5D electrodynamics.

gr-qc↗

arXiv2016

Abraham-Lorentz-Dirac Equation in 5D Stuekelberg Electrodynamics

We derive the Abraham-Lorentz-Dirac (ALD) equation in the framework of the electrodynamic theory associated with Stueckelberg manifestly covariant canonical mechanics. In this framework, a particle worldline is traced out through the evolution of an event $x^\mu(\tau)$. By admitting unconstrained commutation relations between the positions and velocities, the associated electromagnetic gauge fields are in general dependent on the parameter $\tau$, which plays the role of time in Newtonian mechanics. Standard Maxwell theory emerges from this system as a $\tau$-independent equilibrium limit. In this paper, we calculate the $\tau$-dependent field induced by an arbitrarily evolving event, and study the long-range radiation part, which is seen to be an on-shell plane wave of the Maxwell type. Following Dirac's method, we obtain an expression for the finite part of the self-interaction, which leads to the ALD equation that generalizes the Lorentz force. This third-order differential equation is then converted to an integro-differential equation, identical to the standard Maxwell expression, except for the $\tau$-dependence of the field. By studying this $\tau$-dependence in detail, we show that field can be removed from the integration, so that the Lorentz force depends only on the instantaneous external field and an integral over dynamical variables of the event evolution. In this form, pre-acceleration of the event by future values of the field is not present.

physics.gen-ph↗

arXiv2016

Electrostatics in Stueckelberg-Horwitz-Piron Electrodynamics

In this paper, we study fundamental aspects of electrostatics as a special case in Stueckelberg-Horwitz electromagnetic theory. In this theory, spacetime events $x^\mu(\tau)$ evolve in an unconstrained 8-dimensional phase space, interacting through five $\tau$-dependent gauge fields induced by the current densities associated with their evolutions. The chronological time $\tau$ was introduced as an independent evolution parameter in order to free the laboratory clock $x^0$ to evolve alternately 'forward' and 'backward' in time according to the sign of the energy, thus providing a classical implementation of the Feynman-Stueckelberg interpretation of pair creation/annihilation. The resulting theory differs in its underlying mechanics from conventional electromagnetism, but coincides with Maxwell theory in an equilibrium limit. After a brief review of Stueckelberg-Horwitz electrodynamics, we obtain the field produced by an event in uniform motion and verify that it satisfies the field equations. We study this field in the rest frame of the event, where it depends explicitly on coordinate time $x^0$ and the parameter $\tau$, as well as spatial distance $R$. Calculating with this generalized Coulomb field, we demonstrate how Gauss's theorem and Stoke's theorem apply in 4D spacetime, and obtain the fields associated with a charged line and a charged sheet. Finally, we use the field of the charged sheet to study a static event in the vicinity of a potential barrier. In all of these cases, we observe a small transfer of mass from the field to the particle. It is seen that for an event in the field of an oppositely charged sheet of sufficient density, the event can reverse time direction, providing a specific model for pair phenomena.

physics.gen-ph↗

arXiv2016

Field signature for apparently superluminal particle motion

In the context of Stueckelberg's covariant symplectic mechanics, Horwitz and Aharonovich have proposed a simple mechanism by which a particle traveling below light speed almost everywhere may exhibit a transit time that suggests superluminal motion. This mechanism, which requires precise measurement of the particle velocity, involves a subtle perturbation affecting the particle's recorded time coordinate caused by virtual pair processes. The Stueckelberg framework is particularly well suited to such problems, because it permits pair creation/annihilation at the classical level. In this paper, we study a trajectory of the type proposed by Horwitz and Aharonovich, and derive the Maxwell 4-vector potential associated with the motion. We show that the resulting fields carry a signature associated with the apparent superluminal motion, providing an independent test for the mechanism that does not require direct observation of the trajectory, except at the detector.

physics.gen-ph↗

arXiv2016

Speeds of light and mass stability in Stueckelberg-Horwitz-Piron electrodynamics

It is well-known that the 5D gauge structure of Stueckelberg-Horwitz-Piron (SHP) electrodynamics permits the exchange of mass between particles and the fields induced by their motion, even at the classical level. This phenomenon presents two closely related problems: (1) What accounts for the stability of the measured masses of the known particles? (2) Under what circumstances can real particles evolve sufficiently off-shell to account for mass changing phenomena such as flavor-changing neutrino interactions and low energy nuclear reactions? To approach these questions, we introduce a constant $c_5$ associated with the invariant time $\tau$, in analogy with the constant $c$ that associates a unit of length with intervals of time $t$ in standard relativity. It follows that electromagnetic mass exchange can be a small effect, in proportion to $c_5 / c$. We show that this structure permits a classical self-interaction that tends to restore on-shell propagation. Finally we propose a model in which a particle evolving through a complex charged environment can acquire a significant mass shift for a short time.

physics.gen-ph↗

arXiv2016

Pair production in classical Stueckelberg-Horwitz-Piron electrodynamics

In this paper we calculate pair production from bremsstrahlung as a classical effect in Stueckelberg-Horwitz-Piron electrodynamics. In this framework, worldlines are traced out dynamically through the evolution of events $x^\mu(\tau)$ parameterized by a chronological time $\tau$ that is independent of the spacetime coordinates. These events, defined in an unconstrained 8D phase space, interact through five $\tau$-dependent gauge fields induced by the event evolution. The resulting theory differs in its underlying mechanics from conventional electromagnetism, but coincides with Maxwell theory in an equilibrium limit. In particular, the total mass-energy-momentum of particles and fields is conserved, but the mass-shell constraint is lifted from individual interacting events, so that the standard Feynman-Stueckelberg interpretation of pair creation/annihilation is implemented in classical mechanics.

physics.gen-ph↗

arXiv2009

Harmonic Oscillator States with Non-Integer Orbital Angular Momentum

We study the quantum mechanical harmonic oscillator in two and three dimensions, with particular attention to the solutions as represents of their respective symmetry groups: O(2), O(3), and O(2,1). Solving the Schrodinger equation by separating variables in polar coordinates, we obtain wavefunctions characterized by a principal quantum number, the group Casimir eigenvalue, and one observable component of orbital angular momentum, with eigenvalue $m+s$, for integer $m$ and real constant parameter $s$. In each symmetry group, $s$ splits the solutions into two inequivalent representations, one associated with $s=0$, which recovers the familiar description of the oscillator as a product of one-dimensional solutions, and the other with $s>0$ (in three dimensions, $s=0, 1/2$) whose solutions are non-separable in Cartesian coordinates, and are hence overlooked by the standard Fock space approach. In two dimensions, a single set of creation and annihilation operators forms a ladder representation for the allowed oscillator states for any $s$, and the degeneracy of energy states is always finite. However, in three dimensions, the integer and half-integer eigenstates are qualitatively different: the former can be expressed as finite dimensional irreducible tensors under O(3) or O(2,1), and a ladder representation can be constructed via irreducible tensor products of the vector creation operator multiplet, while the latter exhibit infinite degeneracy and the finite-dimensional ladder representation fails for these states. These results are closely connected to the breaking of a unitary symmetry of the harmonic oscillator Hamiltonian recently discussed by Bars.

math-ph↗

arXiv2006

Dirac Monopole from Lorentz Symmetry in N-Dimensions: II. The Generalized Monopole

In a previous paper, we found an extension of the N-dimensional Lorentz generators that partially restores the closed operator algebra in the presence of a Maxwell field, and is conserved under system evolution. Generalizing the construction found by Berard, Grandati, Lages and Mohrbach for the angular momentum operators in the O(3)-invariant nonrelativistic case, we showed that the construction can be maximally satisfied in a three dimensional subspace of the full Minkowski space; this subspace can be chosen to describe either the O(3)-invariant space sector, or an O(2,1)-invariant restriction of spacetime. When the O(3)-invariant subspace is selected, the field solution reduces to the Dirac monopole field found in the nonrelativistic case. For the O(2,1)-invariant subspace, the Maxwell field can be associated with a Coulomb-like potential on spacetime, similar to that used by Horwitz and Arshansky to obtain a covariant generalization of the hydrogen-like bound state. In this paper we elaborate on the generalization of the Dirac monopole to N-dimensions.

hep-th↗

arXiv2006

Discrete Symmetries of Off-Shell Electromagnetism

We discuss the discrete symmetries of the Stueckelberg-Schrodinger relativistic quantum theory and its associated 5D local gauge theory, a dynamical description of particle/antiparticle interactions, with monotonically increasing Poincare-invariant parameter. In this framework, worldlines are traced out through the parameterized evolution of spacetime events, advancing or retreating with respect to the laboratory clock, with negative energy trajectories appearing as antiparticles when the observer describes the evolution using the laboratory clock. The associated gauge theory describes local interactions between events (correlated by the invariant parameter) mediated by five off-shell gauge fields. These gauge fields are shown to transform tensorially under under space and time reflections, unlike the standard Maxwell fields, and the interacting quantum theory therefore remains manifestly Lorentz covariant. Charge conjugation symmetry in the quantum theory is achieved by simultaneous reflection of the sense of evolution and the fifth scalar field. Applying this procedure to the classical gauge theory leads to a purely classical manifestation of charge conjugation, placing the CPT symmetries on the same footing in the classical and quantum domains. In the resulting picture, interactions do not distinguish between particle and antiparticle trajectories -- charge conjugation merely describes the interpretation of observed negative energy trajectories according to the laboratory clock.

hep-th↗

arXiv2006

Duality in Off-Shell Electromagnetism

In this paper, we examine the Dirac monopole in the framework of Off-Shell Electromagnetism, the five dimensional U(1) gauge theory associated with Stueckelberg-Schrodinger relativistic quantum theory. After reviewing the Dirac model in four dimensions, we show that the structure of the five dimensional theory prevents a natural generalization of the Dirac monopole, since the theory is not symmetric under duality transformations. It is shown that the duality symmetry can be restored by generalizing the electromagnetic field strength to an element of a Clifford algebra. Nevertheless, the generalized framework does not permit us to recover the phenomenological (or conventional) absence of magnetic monopoles.

hep-th↗

arXiv2006

Dirac Monopole from Lorentz Symmetry in N-Dimensions: I. The Generator Extension

It is by now well-known that a Lorentz force law and the homogeneous Maxwell equations can be derived from commutation relations among Euclidean coordinates and velocities, without explicit reference to momentum, action or variational principle. This result was extended to the relativistic case and shown to correspond to a Stueckelberg-type quantum theory, in which gauge transformations may depend on the invariant evolution parameter, such that the associated the five-dimensional electromagnetism becomes standard Maxwell theory in the equilibrium limit. Building on the work of Berard, Grandati, Lages and Mohrbach, we construct an extension of the Lorentz generators in N-dimensions that restores the closed commutation relations in the presence of a Maxwell field, and renders the extended generators constants of the classical motion. The algebra imposes conditions on the Maxwell field, leading to a Dirac monopole solution. The construction can be maximally satisfied in a three dimensional subspace of the full Minkowski space; this subspace can be chosen to describe either the O(3)-invariant space sector, generalizing the nonrelativistic result, or an O(2,1)-invariant restriction of spacetime, and leading to a relativistic Coulomb-like potential of the type used by Horwitz and Arshansky to obtain a covariant generalization of the hydrogen-like bound state.

hep-th↗

arXiv2006

Higher-Order Kinetic Term for Controlling Photon Mass in Off-Shell Electrodynamics

In a relativistic classical and quantum mechanics with Poincare-invariant parameter, particle worldlines are traced out by the evolution of spacetime events. In pre-Maxwell electrodynamics -- the local gauge theory associated with this framework -- events induce five local off-shell fields, which mediate interactions between instantaneous events, not between the worldlines which represent entire particle histories. The fifth field, required to compensate for dependence of gauge transformations on the evolution parameter, enables the exchange of mass between particles and fields. In the equilibrium limit, these pre-Maxwell fields are pushed onto the zero-mass shell, but during interactions there is no mechanism regulating the mass that photons may acquire, even when event trajectories evolve far into the spacelike region. This feature of the off-shell formalism requires the application of some ad hoc mechanism for controlling the photon mass in low energy classical Coulomb scattering of charged events, and in the renormalization of off-shell quantum electrodynamics. We discuss a nonlocal, higher derivative correction to the photon kinetic term, which provides regulation of the photon mass in a manner that preserves the gauge invariance and Poincare covariance of the original theory. We demonstrate that the inclusion of this term is equivalent to an earlier solution to the classical Coulomb problem, and that the resulting quantum field theory is renormalized.

hep-th↗