In an extraordinary book published in 1914, and before he became a California
immigrant, H. Bateman examined a number of
different solution equivalence classes to the Electromagnetic Wave equations of
One class of solutions was generated by a complex vector field, M= E + icB,
which satisfied the auxiliary constraint, M.M = 0. Such wave function solutions
to Maxwell's equations were defined to be self conjugate fields, for both
Poincare invariants of such elctromagnetic fields are zero.
That is: D.E-B.H = 0 and E.B=0.
In the modern language of Cartan both d(A^G) = 0 and d(A^F) = 0.
It is not clear that Bateman recognized that the constraint M.M=0 is sufficient
to guarantee that the complex vector M generates a minimal surface [Osserman 1983]. This result had its
basis in the S. Lie theorem, every holomorphic function generates a minimal
Now suppose such minimal surface solutions
have envelopes with 1-dimensional edges of regression, or self-intersections.
Then on the propagating wave solutions there could be 1-dimensional defects or
strings, and the strings or defects could themselves propagate. What are the
propagation speeds of the strings? As these characteristic solution defects are
not linear invariants, but are projective invariants, then such solutions to
Maxwell's equations are NOT limited to the finite propagation speed imposed by
the linear Lorentz equivalence class