Covariant Classical Mechanics Research Articles

Simulation of the Radiation Reaction Orbits of a Classical Relativistic Charged Particle withGeneralized Off‐Shell Lorentz Force

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 equations which result from the expansion of Green′s function in the Lorentz force equation for the self‐interaction. It appears that there is an attractor in this system which stabilizes the motion of the relativistic charged electron. The attractor may acquire fractal characteristics in the presence of an external field and thus become a strange attractor.

On a manifestly covariant classical mechanics or ruminations on “The Computable Universe” and the role of mathematical physics in solving the natural resource problems of the future

AbstractWe informally discuss a number of puzzling fundamental issues that illustrate one might view the laws of physics as a loosely connected patchwork of different theories, rather than a grand unified scheme, even within the realm of chemistry. One long‐lasting difficulty is the merging of special relativity and quantum mechanics: the many‐body Dirac equation, which is the most satisfactory description of the effects of relativity in chemistry, is not Lorentz invariant. In this article, we address the formulation of a manifestly Lorentz invariant classical mechanics in which each particle is described by individual space and time coordinates, while the system as a whole evolves according to a Hamiltonian dynamics based upon a universal evolution parameter. The physical interpretation of the theoretical framework, or its agreement with experiment, in particular when the transition would be made to a quantum theory in four dimensions, is not clear at present. Our search for an understanding of the physical concepts underlying the covariant mechanics has led to a broader view on the use of mathematical and theoretical physics: it could possibly be at the basis for a “computable physics” in which the laws of physics are redesigned such that they are optimally suited for computer simulations, rather than aim for the accurate description of physical reality. In the final section of the article, it is argued that this may be a fertile way to address some of the natural resource problems the world is likely to face in the future. This article essentially covers much of the material presented by one of the authors (MN) at the Odyssey meeting in Edmonton, June 2008, and we hope reflects some of the stimulating discourse that can ensue when people from different disciplines are brought together to share their thoughts. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Classification of infinitesimal symmetries in covariant classical mechanics

In the framework of general relativistic classical mechanics on a spacetime with absolute time, we classify the infinitesimal symmetries of the classical structure by means of distinguished Lie subalgebras of the Lie algebra of “special phase function.” These subalgebras are crucial also for the classification of infinitesimal quantum symmetries, which will be analyzed in a forthcoming paper.

Could the classical relativistic electron be a strange attractor?

We review the formulation of the problem of the electromagnetic self‐interaction of a relativistic charged particle in the framework of the manifestly covariant classical mechanics of Stueckelberg, Horwitz, and Piron. The gauge fields of this theory, in general, cause the mass of the particle to change. We show that the nonlinear Lorentz force equation for the self‐interaction resulting from the expansion of Green′s function has chaotic solutions. We study the autonomous equation for the off‐shell particle mass here, for which the effective charged particle mass achieves a macroscopic average value determined by what appears to be a strange attractor.

A differential equations approach to Hamiltonian systems

We present an intrinsic definition of a (possibly time-dependent) Hamiltonian differential equation as a submanifold of the first-order jet bundle over a fibred cosymplectic manifold. The equivalence of the standard constraint algorithm in mechanics to the completion procedure in differential equations theory is explicitly demonstrated. As an application, we study covariant classical mechanics. Finally, some problems with constraint algorithms in field theories are indicated.

Symmetries in covariant classical mechanics

In the framework of covariant classical mechanics (i.e., general relativistic classical mechanics on a space–time with absolute time), developed by Jadczyk and Modugno, we analyze systematically the relationship between symmetries of geometric objects. We show that the (holonomic) infinitesimal symmetries of the cosymplectic structure on space–time and of its horizontal potentials are also symmetries of spacelike metric, gravitational and electromagnetic fields, Euler–Lagrange morphism and Lagrangians. Then, we provide a definition for a covariant momentum map associated with a group of cosymplectic symmetries by means of a covariant lift of functions of phase space. In the case of holonomic symmetries, we see that any covariant momentum map takes values in the quantizable functions in the sense of Jadczyk and Modugno, i.e., functions quadratic in velocities with leading coefficient proportional to the spacelike metric. Finally, we illustrate the results by some examples.

The Classical Coulomb Problem in Pre-Maxwell Electrodynamics

We explore certain difficulties in the covariant classical mechanics associated with off-shell electrodynamics, through an examination of the classical Coulomb problem. We present a straightforward solution of the classical equations of motion for a test event traversing the field induced by a “fixed” event (an event moving uniformly along the time axis at a fixed point in space). This solution reveals the essential difficulties in the formalism at the classical level. We then offer a new model of the particle, as a certain distribution of events on the worldline, which eliminates these difficulties and permits comparison of classical off-shell electrodynamics with the standard Maxwell theory In this model, the “fixed” event induces a Yukawa-type potential, permitting a semiclassical identification of the pre-Maxwell time scale λ with the inverse mass of the intervening photon. Numerical solutions to the equations of motion are compared with the standard Maxwell solutions—they are seen to coincide when λ > 10–6 sec, providing an initial estimate of this parameter.

Particles and events in classical off-shell electrodynamics

Despite the many successes of the relativistic quantum theory developed by Horwitz, et. al., certain difficulties persist in the associated covariant classical mechanics. In this paper, we explore these difficulties through an examination of the classical Coulomb problem in the framework of off-shell electrodynamics. As the local gauge theory of a covariant quantum mechanics with evolution parameter $\tau$, off-shell electrodynamics constitutes a dynamical theory of spacetime events, interacting through five $\tau$-dependent pre-Maxwell potentials. We present a straightforward solution of the classical equations of motion, which is seen to be unsatisfactory, and reveals the essential difficulties in the formalism at the classical level. We then offer a new model of the particle current -- as a certain distribution of the event currents on the worldline -- which eliminates these difficulties and permits comparison of classical off-shell electrodynamics with the standard Maxwell theory. In this model, the ``fixed'' event induces a Yukawa-type potential, permitting a semi-classical identification of the pre-Maxwell time scale $\lambda$ with the inverse mass of the intervening photon. Numerical solutions to the equations of motion are compared with the standard Maxwell solutions, and are seen to coincide when $\lambda \gtrsim 10^{-6}$ seconds, providing an initial estimate of this parameter. It is also demonstrated that the proposed model provides a natural interpretation for the photon mass cut-off required for the renormalizability of the off-shell quantum electrodynamics.