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 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.
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- 1.
The term “movement” is being used, not to describe movements of objects but in a more general sense, implying more subtle orders of change, development and evolution of every kind [14].
- 2.
We assume the real field for convenience and leave open the possibility for a more fundamental structure.
- 3.
Note this immediately gives us an explanation as to why we identify \(\alpha_{0i}\) with the velocity in the Dirac theory.
- 4.
This suggests that these operators could be used to describe the creation and annihilations of extensions.
- 5.
This is essentially the same idea that led to the notion of the anti-particle “going backward in time”, but at this stage we are not considering anti-matter.
- 6.
While I was preparing this article, Aharonov reminded me of the similarity of these ideas to his on pre- and post-selection [36].
- 7.
\(|P_{1}\rangle\) and \(\langle P_{2}|\) are not to be taken as elements in a Hilbert space and its dual.
- 8.
Physicists should not be put off by the notation because the es are exactly the γs used in standard Dirac theory We have changed notation simply to bring out the fact that we do not need to go to a matrix representation, although that possibility is always open to us should we wish to take advantage of it.
- 9.
It should also be noted that these derivatives are related to the sum and differences of the boundary, δ *, and co-boundary, δ operators in the exterior calculus.
- 10.
The results discussed here are for a fixed number of particles. To discuss creation and annihilation of particles we must go to a field theory as discussed in Bohm et al. [54].
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Acknowledgment
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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Hiley, B. (2010). Process, Distinction, Groupoids and Clifford Algebras: an Alternative View of the Quantum Formalism. In: Coecke, B. (eds) New Structures for Physics. Lecture Notes in Physics, vol 813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12821-9_12
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