Metric Clifford Algebra

TL;DR

This paper introduces metric Clifford algebras for real vector spaces with a metric, defining a deformation of Euclidean Clifford products, and presents the golden formula relevant to geometry and physics.

Contribution

It formalizes the concept of metric Clifford algebra, introduces the gauge metric extensor, and proves the golden formula for applications in geometry and physics.

Findings

Defined metric Clifford algebra $ ext{Cl}(V,g)$ with signature $(p,q)$

Derived the gauge metric extensor $h$ encoding geometric info

Proved the golden formula connecting metric and Euclidean Clifford products

Abstract

In this paper we introduce the concept of metric Clifford algebra $\mathcal{C\ell}(V,g)$ for a $n$-dimensional real vector space $V$ endowed with a metric extensor $g$ whose signature is $(p,q)$, with $p+q=n$. The metric Clifford product on $\mathcal{C\ell}(V,g)$ appears as a well-defined \emph{deformation}(induced by $g$) of an euclidean Clifford product on $\mathcal{C\ell}(V)$. Associated with the metric extensor $g,$ there is a gauge metric extensor $h$ which codifies all the geometric information just contained in $g.$ The precise form of such $h$ is here determined. Moreover, we present and give a proof of the so-called \emph{golden formula,} which is important in many applications that naturally appear in ours studies of multivector functions, and differential geometry and theoretical physics.