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

  • B.J. Hiley
Part of the Lecture Notes in Physics book series (LNP, volume 813)


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 Schrödinger, 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.


Dirac Operator Light Cone Lorentz Transformation Left Ideal Clifford Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



I should like to thank in particular Bob Callaghan for his patience during the many discussions we had on various aspects of this subject. I would also like to thank Ernst Binz, Ray Brummelhuis, Bob Coecke, Maurice de Gosson and Clive Kilmister for their continual encouragement. Without Ray Brummelhuis’ generous support this work would not have reached the light of day.


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