## The Profound Significance of the B(3) Field

The profound significance of the B(3) field is that it introduces into electrodynamics a higher topology. The B(3) field is reviewed for example by Donald Reed, in “Advances in Chemical Physics”, volume 119(3), pp. 532 ff., published by Wiley Interscience in 2001 as a special topical issue called “Modern Non-Linear Optics”. Reed describes how Cartan’s calculus of differential forms introduces a higher order topology than that used by Heaviside in his vector development of the equations of electrodynamics. The index (3) of the B(3) field is part of the complex circular representation ((1), (2), (3)) of space. In ECE theory, this is the a index. There are various decompositions of the vector field, the first was by Helmholtz, who developed the vector field in terms of lamellar (curl free) and solenoidal (divergenceless) components. In 1971, H. E. Moses showed that any smooth vector field may be separated into circularly polarized vectors and into three modes. This method simplifies the analysis of three dimensional classical flow fields. The lamellar mode of Moses implies the existence of a scalar potential and has eigenvalue zero. The B(3) field comes from this mode because the B(3) field is phaseless as is well known. The solenoidal vector field component of Moses divides into two chiral components of different handedness. In O(3) electrodynamics these are denoted B(1) and B(2), which in free space are divergenceless. In ECE theory these modes are a = (1), (2) and (3). In the arena of incompressible and viscous flow dynamics, vorticity can be described as a superposition of circularly polarized modes only, and the normally nonlinear convection term of the Navier Stokes equation drops out, allowing for exact solutions. In the area of non-linear optics these complex circular modes allow the fundamental B(3) field of electrodynamics to be defined within a factor -ig as the conjugate product of non-linear optics, observed whenever the inverse Faraday effect is observed.