# Beyond Hilbert space

1. Jun 13, 2006

### CarlB

Is anyone out there working on a theory of elementary particles that is basic quantum mechanics without the Hilbert space? The reason I'm asking is because I found this article by B. J. Hiley:

Algebraic Quantum Mechanics, Algebraic Spinors and Hilbert Space.
The orthogonal Clifford algebra and the generalised Clifford algebra, $$C^n$$, (discrete Weyl algebra) is re-examined and it is shown that the quantum mechanical wave function (element of left ideal), density operator (element of a two sided ideal) and mean values (algebraic trace) can be constructed from entirely within the algebra. No appeal to Hilbert space is necessary. We show how the GNS construction can be obtained from within both algebras. The limit of $$C^n$$ as $$n \to \infty$$ is shown to be the extended Heisenberg algebra. Finally the relationship to the usual Hilbert space approach is discussed.
http://www.bbk.ac.uk/tpru/BasilHiley/Algebraic Quantum Mechanic 5.pdf

Carl