Particles vs. events: The concatenated structure of world lines in relativistic quantum mechanics

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.

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

“…Under the conditions j 5 → 0 and f 5µ → 0, pointwise in x as τ → ±∞, integration of (24) over τ , called concatenation of events into a worldlines [9], recovers the relations…”

Section: Concatenationmentioning

confidence: 84%

Discrete Symmetries of Off-Shell Electromagnetism

2005

Found Phys

This paper discusses the discrete symmetries of off-shell electromagnetism, the Stueckelberg-Schrodinger relativistic quantum theory and its associated 5D local gauge theory. Seeking a dynamical description of particle/antiparticle interactions, Stueckelberg developed a covariant mechanics with a monotonically increasing Poincaré-invariant parameter. In Stueckelberg's framework, worldlines are traced out through the parameterized evolution of spacetime events, which may advance or retreat with respect to the laboratory clock, depending on the sign of the energy, so that negative energy trajectories appear 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 CP T 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.

“…The connection with Maxwell theory is made by extending an observation by Stueckelberg: if (9) holds asymptotically, pointwise in x as τ → ±∞, then integration of (3) over τ , called concatenation of events into a worldlines [7], recovers standard…”

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.