Abstract
We calculate pair production from bremsstrahlung as a classical effect in Stueckelberg-Horwitz electrodynamics. In this framework, worldlines are traced out dynamically through the evolution of events xμ(τ) parameterized by a chronological time τ that is independent of the spacetime coordinates. These events, defined in an unconstrained 8D phase space, interact through five τ-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 Feynman-Stueckelberg interpretation of pair creation/annihilation is implemented in classical mechanics.We consider a three-stage interaction which when parameterized by the laboratory clock x0 appears as (1) particle-1 scatters on a heavy nucleus to produce bremsstrahlung, (2) the radiation field produces a particle/antiparticle pair, (3) the antiparticle is annihilated with particle-2 in the presence of a second heavy nucleus. When parameterized in chronological time τ, the underlying process develops as (1) particle-2 scatters on the second nucleus and begins evolving backward in time with negative energy, (2) particle-1 scatters on the first nucleus and releases bremsstrahlung, (3) particle-2 absorbs radiation which returns it to forward time evolution with positive energy.
Highlights
In the historical introduction to his book on quantum field theory [1], Weinberg devotes a paragraph to deprecation of Dirac’s hole theory of antiparticles, observing that QFT had made the theory ”unnecessary, even though it lingers on in many textbooks.” It might have been mentioned that the essential idea of hole theory lingers on productively as the quasiparticle formalism in condensed matter physics and many-particle theory.1 Still, as an interpretation of particle/antiparticle processes, the Feynman-Stueckelberg time reversal formalism, expressed in QFT through the Feynman propagator, has many conceptual advantages
In this paper we have shown that a classical equivalent of the Bethe-Heitler mechanism is permitted in Stueckelberg-Horwitz electrodynamics
The process begins at τ1 with a pair annihilation event produced by the scattering of an incoming particle in the Coulomb field of a target nucleus
Summary
In the historical introduction to his book on quantum field theory [1], Weinberg devotes a paragraph to deprecation of Dirac’s hole theory of antiparticles, observing that QFT had made the theory ”unnecessary, even though it lingers on in many textbooks.” It might have been mentioned that the essential idea of hole theory lingers on productively as the quasiparticle formalism in condensed matter physics and many-particle theory. Still, as an interpretation of particle/antiparticle processes, 1More generally, Dirac’s fundamental insight that the absence of a physical object can behave like the presence of an inverse object has been influential in many fields, especially psychology, and can be compared to the remark attributed to Miles Davis that, “Music is the space between the notes. The description of an antiparticle as a particle propagating backward in time was first proposed by Stueckelberg [2] in the context of classical relativity, without resort to quantum ideas or phenomena In this model, a pair process is represented by a single worldline, generated dynamically by a classical event whose time coordinate advances or retreats with respect to the laboratory clock, as its instantaneous energy changes sign under interaction with a field. Stueckelberg was not able to provide a classical account of pair processes, because the mass-shell constraint p2 = (Mx)2 = −M2 prevents continuous evolution of the event trajectory from the timelike region into the spacelike region on its way to time-reversed timelike motion He considered adding a vector component to his Lorentz force that would overcome the constraint, but dropped the idea, finding no justification from first principles.
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