Huygens’ principle in classical electrodynamics: a distributional approach

By combining Pauli algebra with distribution theory, we give a very compact and conceptually simple formulation of Huygens’ principle in classical electrodynamics. The Stratton-Chu and Kottler-Franz representations of an electromagnetic field as surface integrals are derived with minimal effort and maximal clarity. They are then generalized by allowing the integration surfaces to move freely, so that charge distributions in arbitrary motion are represented. 1 Electrodynamics with the Pauli algebra The Pauli algebra is a generalization of vector analysis in R. One defines the products of two vectors A,A by AA ′ = A · A′ + iA × A′. (1) Thus AA consists of a scalar part and a (pseudo-) vector part, the latter represented by an imaginary axial vector, which is actually the oriented area spanned by A and A: 〈AA′ 〉s = A · A ′, 〈AA′ 〉v = iA × A ′. The two common bilinear expressions are thus united into a single complex entity. A general element of the algebra, here called a Pauli number, is represented by a complex scalar plus a complex vector A = A0 + A, A0 ∈ C, A ∈ C ,