Complex geometry and fundamental physical law

Abstract

We present here a product between vectors and scalars that mixes them within their own space, using imaginaries to describe geometric products between vectors as complex vectors, rather than introducing higher order/dimensional vector objects. This is done by means of a mixture tensor that lends itself naturally to tensor calculus. We use this to develop a notion of analyticity in higher dimensions based on the idea that a function can be made differentiable -- in a certain strong sense -- by permitting curvature of the underlying space, and we call this analytic curvature. To explore these ideas we use them to derive a few fundamental laws of physics which, while considered somewhat lightly, have nevertheless compelling features. The mixture, for instance, produces rich symmetries without adding dimensions beyond the familiar space-time, and its derivative produces familiar quantum field relations in which the field potentials are just derivatives of the coordinate basis.