An Extra Structure of Spacetime: A Space of Points, Areas and Volumes

TL;DR

This paper proposes a unified geometric framework called C-space, where points, areas, and volumes are treated equally, extending spacetime to higher dimensions to unify fundamental interactions and incorporate gravity and gauge fields.

Contribution

It introduces a 16-dimensional C-space that generalizes spacetime, unifies interactions via higher-dimensional geometry, and relates conserved charges to C-space isometries.

Findings

C-space is a 16-dimensional manifold unifying geometric objects.

Gravity and gauge fields are incorporated into the C-space metric.

Conserved charges relate to isometries in C-space, generalizing angular momentum and spin.

Abstract

A theory in which points, lines, areas and volumes are on on the same footing is investigated. All those geometric objects form a 16-dimensional manifold, called C-space, which generalizes spacetime. In such higher dimensional space fundamental interactions can be unified \` a la Kaluza-Klein. The ordinary, 4-dimensional, gravity and gauge fields are incorporated in the metric and spin connection, whilst the conserved gauge charges are related to the isometries of curved C-space. It is shown that a conserved generator of an isometry in C-space contains a part with derivatives, which generalizes orbital angular momentum, and a part with the generators of Clifford algebra, which generalizes spin.