Quantum Field Theory and Gravity: Conceptual and Mathematical ...

On the Motion of Point Defects in Relativistic Fields 307
electrodynamics, the self-interaction of a point charge with its own Liénard–
Wiechert field, simply does not exist!
In the Fokker–Schwarzschild–Tetrode theory, the law of motion of an
electron is given by equations (2.17), (2.18), though with E and B now standing
for the arithmetic means 1 ret
2
(E lw + E adv
lw )and 1 2 (Bret lw + Badv lw )oftheretarded
and advanced Liénard–Wiechert fields summed over all the other point
electrons. Nontrivial motions can occur only in the many-particle Fokker–
Schwarzschild–Tetrode theory.
While the Fokker–Schwarzschild–Tetrode electrodynamics does not suffer
from the infinities of formal Lorentz electrodynamics, it raises another
formidable problem: its second-order equations of motion for the system of
point charges do not pose a Cauchy problem for the traditional classical state
variables Q(t) and ˙Q(t) of these point charges. Instead, given the classical
state variables Q(t 0 )and ˙Q(t 0 )ofeachelectronattimet 0 , the computation
of their accelerations at time t 0 requires the knowledge of the states of
motion of all point electrons at infinitely many instances in the past and in
the future. While it would be conceivable in principle, though certainly not
possible in practice, to find some historical records of all those past events,
how could we anticipate the future without computing it from knowing the
present and the past?
Interestingly enough, though, Wheeler and Feynman showed that the
Fokker–Schwarzschild–Tetrode equations of motion can be recast as Abraham–Lorentz–Dirac
equations of motion for each point charge, though with
the external Maxwell–Maxwell fields replaced by the retarded Liénard–Wiechert
fields of all the other point charges, provided the Wheeler–Feynman
absorber identity is valid. While this is still not a Cauchy problem for the
classical state variables Q(t) and ˙Q(t) of each electron, at least one does not
need to anticipate the future anymore.
The Fokker–Schwarzschild–Tetrode and Wheeler–Feynman theories are
mathematically fascinating in their own right, but pose very difficult problems.
Rigorous studies [Bau1998], [BDD2010], [Dec2010] have only recently
begun.
4. Nonlinear electromagnetic field equations
Gustav Mie [Mie1912/13] worked out the special-relativistic framework for
fundamentally nonlinear electromagnetic field equations without point charges
(for a modern treatment, see [Chri2000]). Twenty years later Mie’s work
became the basis for Max Born’s assault on the infinite self-energy problem
of a point charge.
4.1. Nonlinear self-regularization
Born [Bor1933] argued that the dilemma of the infinite electromagnetic selfenergy
of a point charge in formal Lorentz electrodynamics is caused by