Electrodynamics

This episode discusses the Born-Infeld model for
Electromagnetodynamics. Here, the standard model are the Maxwell
equations coupling the interaction of magnetic and electric field
with the help of a system of partial differential equations. This
is a well-understood classical system. But in this classical model,
one serious drawback is that the action of a point charge (which is
represented by a Dirac measure on the right-hand side) leads to an
infinite energy in the electric field which physically makes no
sense.

On the other hand, it should be possible to study the electric
field of point charges since this is how the electric field is
created. One solution for this challenge is to slightly change
the point of view in a way similar to special relativity theory
of Einstein. There, instead of taking the momentum () as
preserved quantity and Lagrange parameter the Lagrangian is
changed in a way that the bound for the velocity (in relativity
the speed of light) is incorporated in the model.


In the electromagnetic model, the Lagrangian would have to
restrict the intensity of the fields. This was the idea which
Borne and Infeld published already at the beginning of the last
century. For the resulting system it is straightforward to
calculate the fields for point charges. But unfortunately it is
impossible to add the fields for several point charges (no
superposition principle) since the resulting theory (and the PDE)
are nonlinear. Physically this expresses, that the point charges
do not act independently from each other but it accounts for
certain interaction between the charges. Probably this
interaction is really only important if charges are near enough
to each other and locally it should be only influenced by the
charge nearest. But it has not been possible to prove that up to
now.


The electrostatic case is elliptic but has a singularity at each
point charge. So no classical regularity results are directly
applicable. On the other hand, there is an interesting interplay
with geometry since the PDE occurs as the mean curvature equation
of hypersurfaces in the Minkowski space in relativity.


The evolution problem is completely open. In the static case we
have existence and uniqueness without really looking at the PDEs
from the way the system is built. The PDE should provide at least
qualitative information on the electric field. So if, e.g., there
is a positive charge there could be a maximum of the field (for
negative charges a minimum - respectively), and we would expect
the field to be smooth outside these singular points. So a
Lipschitz regular solution would seem probable. But it is open
how to prove this mathematically.


A special property is that the model has infinitely many inherent
scales, namely all even powers of the gradient of the field. So
to understand maybe asymptotic limits in theses scales could be a
first interesting step.


Denis Bonheure got his mathematical education at the Free
University of Brussels and is working there as Professor of
Mathematics at the moment.
Literature and additional material

M. Kiessling: Electromagnetic Field Theory Without Divergence
Problems 1, The Born Legacy, Journal of Statistical Physics,
Volume 116, Issue 1, pp 1057-1122, 2004.

M. Kiessling: Electromagnetic Field Theory Without Divergence
Problems 2, A Least Invasively Quantized Theory, Journal of
Statistical Physics, Volume 116, Issue 1, pp 1123-1159, 2004.

M. Kiessling: On the motion of point defects in relativistic
fields, in Quantum Field Theory and Gravity, Conceptual and
Mathematical Advances in the Search for a Unified Framework,
Finster e.a. (ed.), Springer, 2012.

Y. Brenier: Some Geometric PDEs Related to Hydrodynamics and
Electrodynamics, ICM Vol. III pp 761--772, 2002.