Clifford Space as the Arena for Physics

TL;DR

This paper proposes a new theoretical framework where extended objects are described using multivectors in Clifford space ($C$-space), suggesting that physics occurs in $C$-space rather than Minkowski space, leading to novel insights into relativistic dynamics.

Contribution

It introduces the concept of $C$-space with multivector coordinates for extended objects, generalizing the center of mass, and shows how this leads to Stueckelberg dynamics in Minkowski space.

Findings

Extended objects are represented by multivectors in Clifford algebra.

Physics in $C$-space implies unconstrained Stueckelberg dynamics in Minkowski space.

$C$-space provides a new arena for formulating physical theories.

Abstract

A new theory is considered according to which extended objects in $n$-dimensional space are described in terms of multivector coordinates which are interpreted as generalizing the concept of centre of mass coordinates. While the usual centre of mass is a point, by generalizing the latter concept, we associate with every extended object a set of $r$-loops, $r=0,1,..., n-1$, enclosing oriented $(r+1)$-dimensional surfaces represented by Clifford numbers called $(r+1)$-vectors or multivectors. Superpositions of multivectors are called polyvectors or Clifford aggregates and they are elements of Clifford algebra. The set of all possible polyvectors forms a manifold, called $C$-space. We assume that the arena in which physics takes place is in fact not Minkowski space, but $C$-space. This has many far reaching physical implications, some of which are discussed in this paper. The most notable is the finding that although we start from the constrained relativity in $C$-space we arrive at the unconstrained Stueckelberg relativistic dynamics in Minkowski space which is a subspace of $C$-space.