Symmetries from mixtures: an algebra for gauge theories

Abstract

We present here a product between vectors and scalars that {\it 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 {\it mixture tensor} that captures the rich geometries of geometric algebras, and simultaneously 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 {\it analytic curvature}. To explore these ideas we use them to derive a few fundamental laws and operators of physics which, while considered somewhat lightly, have 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. The symmetries of physical laws are seen to arise not directly from the bases, but from their mixtures.