[Paper Review] Process, Distinction, Groupoids and Clifford Algebras: an Alternative View of the Quantum Formalism

B. J. Hiley|arXiv (Cornell University)|Nov 9, 2012

Quantum Mechanics and Applications53 references19 citations

TL;DR

This paper proposes a foundational framework for quantum mechanics based on a primitive notion of process, using orthogonal and symplectic groupoids to generate Clifford algebras. It shows that the Schrödinger, Pauli, and Dirac formalisms emerge naturally from orthogonal Clifford algebras, while symplectic structures yield non-commutative geometry and Bohmian phase space, unifying the implicate order and complementarity within a single algebraic framework.

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 Schröodinger, 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.

Motivation & Objective

• To develop a foundational quantum theory based not on particles or fields, but on a primitive notion of process as the fundamental ontological element.
• To show that space-time and quantum phenomena emerge from algebraic structures built on process, rather than being imposed a priori.
• To demonstrate that the Bohmian approach and quantum potential are inherent features of Clifford algebraic structures, not ad hoc additions.
• To unify the implicate and explicate orders via projections onto shadow manifolds in non-commutative geometry.
• To establish a mathematically coherent framework where quantum mechanics, relativity, and complementarity arise from a single algebraic foundation.

Proposed method

• Start from a basic process represented as a distinguishable pair [T₁T₂], formalized as a brace to preserve indivisibility while allowing distinction.
• Construct orthogonal and symplectic groupoids from the process, then promote them to orthogonal and symplectic Clifford algebras via additional algebraic structure.
• Use a discrete structure with shift and phase operators (U^s, V^t) to model spatial and momentum-like transformations, then take the continuum limit to recover the Heisenberg algebra.
• Introduce a Dirac connection on a Clifford bundle to derive dynamical equations equivalent to those in the Bohm formalism, without relying on Hilbert space representations.
• Generalize to symplectic Clifford algebras to describe non-commutative geometry, where position and momentum cannot be simultaneously defined, realizing Bohr's complementarity.
• Project the non-commutative structure onto 'shadow manifolds' to model the explicate order, with phase space as a specific example.

Experimental results

Research questions

• RQ1 Can quantum mechanics and space-time structure emerge from a fundamental notion of process rather than from pre-existing manifolds?
• RQ2 How do the Schrödinger, Pauli, and Dirac formalisms arise naturally from orthogonal Clifford algebras?
• RQ3 Is the Bohmian approach, including the quantum potential, a structural feature of Clifford algebras rather than a phenomenological addition?
• RQ4 Can the implicate-explicate order of Bohm be mathematically realized through non-commutative geometry derived from process-based groupoids?
• RQ5 How does the continuum limit of discrete process structures lead to the Heisenberg algebra and standard quantum mechanics?

Key findings

• The orthogonal Clifford algebra, including the Pauli and Dirac algebras, naturally describes the classical light-cone structure of space-time, showing that quantum algebraic structures have classical geometric roots.
• By introducing a Dirac connection on a Clifford bundle, the Bohmian dynamical equations emerge as a structural feature of the mathematics, not a separate postulate.
• The Schrödinger, Pauli, and Dirac theories form a natural hierarchy within Clifford algebras, each embedded in the next, reflecting physical hierarchy from non-relativistic to relativistic spinors.
• The symplectic Clifford algebra, in the continuum limit, yields the Heisenberg algebra and a non-commutative geometry where points in position and momentum space cannot be simultaneously defined, realizing Bohr's complementarity.
• Projections onto 'shadow manifolds' from the non-commutative structure provide a mathematical realization of Bohm's explicate order, with phase space as a concrete example.
• The quantum potential in the Dirac theory is shown to be the exact relativistic generalization of the de Broglie-Bohm quantum potential, confirming the consistency of the approach in the relativistic domain.