The Supergeometric Algebra

TL;DR

This paper proposes the supergeometric algebra as a fundamental framework where spinors are primary, offering a new perspective on their role in physics and their algebraic properties.

Contribution

It introduces the supergeometric algebra, emphasizing spinors as more fundamental than geometric algebra, and explores their algebraic interactions and implications for physical principles.

Findings

Supergeometric algebra unifies scalars, spinors, and multivectors.

Inner and outer products of spinors reproduce key physical principles.

Spinor indices can be viewed as bitcodes, linking algebra to information theory.

Abstract

Spinors are central to physics: all matter (fermions) is made of spinors, and all forces arise from symmetries of spinors. It is common to consider the geometric (Clifford) algebra as the fundamental edifice from which spinors emerge. This paper advocates the alternative view that spinors are more fundamental than the geometric algebra. The algebra consisting of linear combinations of scalars, column spinors, row spinors, multivectors, and their various products, can be termed the supergeometric algebra. The inner product of a row spinor with a column spinor yields a scalar, while the outer product of a column spinor with a row spinor yields a multivector, in accordance with the Brauer-Weyl (1935) theorem. Prohibiting the product of a row spinor with a row spinor, or a column spinor with a column spinor, reproduces the exclusion principle. The fact that the index of a spinor is a bitcode is highlighted.