Process, Distinction, Groupoids and Clifford Algebras: an Alternative View of the Quantum Formalism

TL;DR

This paper presents a novel mathematical framework using groupoids and Clifford algebras to unify classical and quantum physics, connecting space-time structures with quantum phenomena and Bohmian mechanics.

Contribution

It introduces a new approach that structures processes into groupoids and Clifford algebras, linking classical space-time and quantum mechanics through geometric and algebraic methods.

Findings

Orthogonal Clifford algebras describe space-time and quantum formalisms.

Bohmian mechanics emerges naturally from the Clifford connection.

Non-commutative geometry relates to phase space and implicate-explicate order.

Abstract

In this paper we start from a basic notion of process, which we structure into two groupoids, one orthogonal and one symplectic. By introducing additional structure, we convert these groupoids into orthogonal and symplectic Clifford algebras respectively. We show how the orthogonal Clifford algebra, which include the Schr\"oodinger, Pauli and Dirac formalisms, describe the classical light-cone structure of space-time, as well as providing a basis for the description of quantum phenomena. By constructing an orthogonal Clifford bundle with a Dirac connection, we make contact with quantum mechanics through the Bohm formalism which emerges quite naturally from the connection, showing that it is a structural feature of the mathematics. We then generalise the approach to include the symplectic Clifford algebra, which leads us to a non-commutative geometry with projections onto shadow manifolds. These shadow manifolds are none other than examples of the phase space constructed by Bohm. We also argue that this provides us with a mathematical structure that fits the implicate-explicate order proposed by Bohm.