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Oxford Quantum Talks Archive

Towards a quantum geometry: groupoids, Clifford algebras and shadow manifolds

Basil Hiley, Birkbeck College London
Categories, Logic and Foundations of Physics II, May 2008, Imperial College London

I will present some ideas, which although not yet formulated in terms of categories, uses the spirit of category theory especially as articulated by Lawvere. Starting from first principles, I follow Kauffman in making distinctions between two aspects of each indivisible sub-process and then order these processes to form a groupoid, which is then generalised into what I call an algebra of process. The resulting algebra is shown to be isomorphic to a hierarchy of orthogonal Clifford algebras, which, of course, include the Pauli and Dirac algebras. I then exploit the minimal left and right ideals to map from the algebra of process to a vector space inducing a light cone structure thus inverting the usual approach since here the vector space inherits its structure from the fundamental processes. Although I am working with the algebra of process, all of this is still within the conceptual framework used by Clifford himself, namely, classical physics. I then present an argument to generalise this structure to include symplectic Clifford algebras enabling us to introduce the Heisenberg group. Using the techniques applied to orthogonal CAs, I am able to show that the idempotents of this algebra map onto the points of an underlying manifold. Since the symplectic group acts in (x, p) phase space algebra, quantum mechanics demands that this structure is non-commutative and it is this feature that produces shadow manifolds. I will discuss the significance of these results.

[video] [streaming video] [slides]