Born’s theory starts from describing the field by two vectors (or a “six-vector”), B, E, the magnetic induction and electric field-strength respectively. A second pair of vectors (or a second six-vector) H, D, is introduced, merely an abbreviation, if you please, for the partial derivatives of the Lagrange function with respect to the components of B and E respectively (though with the negative sign for E). H is called magnetic field and D dielectric displacement. It was pointed out by Born that it is possible to choose the independent vectors in different ways.
Four
different and, to a certain extent, equivalent and symmetrical representations of the theory can be given by combining each of the two “magnetic” vectors with each of the two “electric” vectors to form the set of six independent variables. Every one of these four representations can be derived from a variation principle, using, of course, entirely different Lagrange functions—physically different, that is, though their analytic expressions by the respective variables are either identical or very similar to each other. In studying Born’s theory I came across a further representation, which is so entirely different from all the aforementioned, and presents such curious analytical aspects, that I desired to have it communicated. The idea is to use two complex combinations of B, E, H, D as independent variables, but in such a way that their “conjugates,”
i. e.
, the partial derivatives of
L
, equal their
complex
conjugates.
Derivative expansion for the one-loop effective Lagrangian in QED
