Reviews of Modern Physics, 67, Jan, 1995 113-155
Commentary by Jack Sarfatti
Classical radiation damping requires precognition
If we use only retarded electromagnetic waves that propagate on the future light cone of their source events, conservation of energy for an accelerating classical point charge implies HN's Lorentz-covariant tensor eq. 2.2 on p.116
The first term on the RHS is the external Lorentz force for the point charge. The second term involving the third derivative of the particle's postion relative to the proper time along its world line is the self-force or radiation reaction. This equation cannot be deduced from the Lagrangian of traditional classical electrodynamics for point charges in purely retarded causal electromagnetic fields. It is put in adhoc in order to obey conservation of energy. If you choose a time symmetric sum of advanced and retarded waves there is no radiaton and no radiation reaction. Anti-causal advanced waves propagate on the past light cone of their source events. Dirac used HN's eq. 2.3 which is half the difference of the retarded and advanced waves.
Dirac [1] showed that the individual self-fields diverge for the point charge, but their difference is finite and it is exactly equal to the adhoc radiation reaction term in eq. 2.2 above. The classical self-force when quantized is responsible for the spontaneous emission of bound atomic electrons in excited energy levels. This is amazing and highly suggestive that we are close to the secrets of Einstein's "Old One".
Dirac then considered an external impulse force on the charge. What happens after the force disappears? There are only two possible solutions. The first is precognitive in which the charge anticipates that the force will hit it in the future. The radiation reaction self-force accelerates the charge before it is hit giving it just the right velocity it needs at the moment it is hit. The second solution is not precognitive, but the self-force rapidly accelerates the charge to infinite energy. The time scale for both solutions is 1/137 of the time it takes light to cross the Compton wavelength of the charge. The precognitive solution is, therefore, preferred for the tiny electron since it would be hard to measure the precognitive effect. Remember this is all classical electrodynamics with no quantum mechanics. Roger Penrose has a nice discussion of all this in The Emperor's New Mind.
The Wheeler-Feynman Trick
Wheeler and Feynman (WF) [2] postulated a Lorentz-invariant delayed action at a distance between charges along null world lines. Tetrode and Fokker independently had similar ideas in the 1920s. The electromagnetic fields are there formally but they have no independent degrees of freedom as in the traditional theory. An example is HN eq. 1.9 on p. 115 for a "direct particle potential". WF further postulate that there is no bare self-force and that each charge feels the sum of one half of the advanced and retarded "direct particle potentials" of all the other charges. The field acting on a particular charge a is assumed to be
The total absorber boundary condition is that
Therefore, we can add this "0" to the first term without changing anything and the result for the field on charge a is obviously the Dirac formula from above
Cosmology
HN develop the classical WF theory in the curved spacetime of general relativity. They use the Robertson-Walker line element for homogeneous isotropic solutions to Einstein's gravitational field equations for the spacetime metric tensor. In particular, their "Table II" on page 125 shows that the standard Big Bang model for the open universe violates the WF boundary condition. The case for the closed universe is in their words "ambiguous". One might say that traditional causality in which the future cannot influence the past requires a closed universe if the standard Big Bang is correct. All current observational evidence seems to favor an open universe.
HN then develop the quantum version of their classical theory using the Feynman path method. The quantum "influence functional" of the future of our Universe replaces the classical absorber boundary condition. They show that there are no renormalization infinities in the delayed action-at-a-distance theory at the quantum level because of damping by the cosmological influence functional of the entire future Universe on every charged particle using their new improved steady state model.
... the apparently local behavior of a quantum system actually involves the response of the Universe via an influence functional which arises when we take into account how the absorber reacts back (via advanced potentials) on the local system. The influence functional enters into any probability calculation in the path integral approach whenever the effects of external variables on the local system are integrated out. It is a double integral over paths and conjugate paths. ... the conjugate paths arise in the calculation of probability for spontaneous transition of the atomic electron, involving the response of the Universe, when the effects of the individual absorber particles are integrated out. p. 147The congugate paths in this case carry negative energy em waves propagating backwards in time. They are the final cause of past spontaneous emissions in the action at a distance theoy and they replace the virtual photon vacuum fluctuations of the traditional quantum electrodynamics.
The finite mass renormalization of the electron in the HN theory is 0.15 for the mass shift divided by the bare mass (i.e., eq. 5.9, p. 151). The corresponding charge renormalization is much smaller 0.04 (i.e., eq. 5.15, p. 151).
HN discuss the prospects for the experimental measurement of advanced electromagnetic waves if our Universe obeys the standard Big Bang and is open. One obvious prediction (not in their paper) is that there should be blue-shifted atomic spectra from stars whose past light cones intersect our telescopes.
[1] Dirac, P.A.M. Proc. R. Soc. A, 165, 199 and 167, 148 (1938)
[2] Wheeler, J.A. & Feynman, R.P. , Rev. Mod. Phys. 17, 157 (1945)