The Lorentz force and energy-momentum for off-shell electromagnetism

Journal of Mathematical Physics

Shnerb

Horwitz

1995

Self Cite

In 1948, Feynman showed Dyson how the Lorentz force law and homogeneous Maxwell equations could be derived from commutation relations among Euclidean coordinates and velocities, without reference to an action or variational principle. When Dyson published the work in 1990, several authors noted that the derived equations have only Galilean symmetry and so are not actually the Maxwell theory. In particular, Hojman and Shepley proved that the existence of commutation relations is a strong assumption, sufficient to determine the corresponding action, which for Feynman's derivation is of Newtonian form. In a recent paper, Tanimura generalized Feynman's derivation to a Lorentz covariant form with scalar evolution parameter, and obtained an expression for the Lorentz force which appears to be consistent with relativistic kinematics and relates the force to the Maxwell field in the usual manner. However, Tanimura's derivation does not lead to the usual Maxwell theory either, because the force equation depends on a fifth (scalar) electromagnetic potential, and the invariant evolution parameter cannot be consistently identified with the proper time of the particle motion. Moreover, the derivation cannot be made reparameterization invariant; the scalar potential causes violations of the mass-shell constraint which this invariance should guarantee.In this paper, we examine Tanimura's derivation in the framework of the proper time method in relativistic mechanics, and use the technique of Hojman and Shepley to study the unconstrained commutation relations. We show that Tanimura's result then corresponds to the five-dimensional electromagnetic theory previously derived from a 1 Stueckelberg-type quantum theory in which one gauges the invariant parameter in the proper time method. This theory provides the final step in Feynman's program of deriving the Maxwell theory from commutation relations; the Maxwell theory emerges as the "correlation limit" of a more general gauge theory, in which it is properly contained.

“…We may find the Lorentz force [36] by applying the Euler-Lagrange equations to (4.17), which in the notation of (2.33), i.e., α, β = 0, 1, 2, 3, 4, is…”

Section: Implications For Gauge Theory In Covariant Quantum Mechanicsmentioning

confidence: 99%

On Feynman’s approach to the foundations of gauge theory

Journal of Mathematical Physics

Shnerb

Horwitz

1995

Self Cite

“…However, one is immediately confronted by conceptual difficulties in attempting to describe even the simple case of low energy Coulomb scattering. First, although the current j α (x, τ) and the potential a α (x, τ) are individually constructed to be vector + scalar representations of O(3,1) on physical grounds, the 5D scalar structure of the electromagnetic action (22) places all five components on the same algebraic footing. This suggests an underlying formal symmetry larger than O(3,1) but containing it as a subgroup: O(4,1) for the choice g 55 = 1 or O(3,2) symmetry for g 55 = −1.…”

Section: Classical Shp Electrodynamicsmentioning

confidence: 99%

“…Thus, the factor λ that characterizes the width of the ensemble and represents the average event inter-occurrence time, also determines the mass spectrum of the photons mediating the interaction between events. As proposed by Stueckelberg and seen in the second of (19), the masses of the particles and fields are not separately conserved, although Noether's theorem for affine τ-displacement symmetry guarantees that the total mass is a constant of the motion [22]. If λ is small (so that ϕ/λ approaches a delta function and the current narrows to a small neighborhood around the event), the mass spectrum becomes wide and the interaction range and cross-section decreases.…”

Section: Non-local Field Kinetics ↔ Ensemble Of Eventsmentioning

confidence: 99%

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

2017

Entropy

Abstract: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 µ (τ) as τ grows monotonically. Stueckelberg proposed to formalize the distinction between coordinate time x 0 = ct (measured by laboratory clocks) and chronology τ (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 τ-dependent and sourced by a current that represents a statistical ensemble of instantaneous events distributed along the trajectory. The width λ of this distribution defines a correlation time for the interactions and a mass spectrum for the photons emitted by particles. As λ becomes very large, the photon mass goes to zero and the field equations become τ-independent Maxwell's equations. Maxwell theory thus emerges as an equilibrium limit of SHP, in which λ 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.

“…This clasical theory can be derived [4] from the quantum theory proposed by Sa'ad, Horwitz and Arshansky [2] …”

Section: Off-shell Electromagnetismmentioning

confidence: 99%

Duality in Off-Shell Electromagnetism

2005

Found Phys

In this paper, we examine the Dirac monopole in the framework of Off-Shell Electromagnetism, the five dimensional U(1) gauge theory associated with StueckelbergSchrodinger 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.