## Algebraic Quantum Mechanics and Pregeometry

**Authors: **Bohm, D., Davies and Hiley, B.

**Editors: **Adenier, G., Krennikov, A. and Nieuwenhuizen, T.M.

**Journal:** Quantum Theory: Reconsiderations of Foundations

**Volume:** 3

**Pages:** 314-324

**Publisher:** American Institute of Physics

**Abstract:**

We discuss the relation between the q-number approach to quantum mechanics suggested by Dirac and the notion of "pregeometry" introduced by Wheeler. By associating the q-numbers with the elements of an algebra and regarding the primitive idempotents as "generalized points" we suggest an approach that may make it possible to dispense with an a priori given space manifold. In this approach the algebra itself would carry the symmetries of translation, rotation, etc. Our suggestion is illustrated in a preliminary way by using a particular generalized Clifford algebra proposed originally by Weyl, which approaches the ordinary Heisenberg algebra a suitable limit. We thus obtain a certain insight into how quantum mechanics may be regarded as a purely algebraic theory, provided that we further introduce a new set of "neighbourhood operators", which remove an important kind of arbitrariness that has thus far been present in the attempt to treat quantum mechanics solely in terms of a Heisenberg algebra.

http://eprints.bournemouth.ac.uk/21620/

**Source:** Manual

## Algebraic Quantum Mechanics and Pregeometry

**Authors: **Bohm, D.J., Davies, P.G. and Hiley, B.J.

**DOI:** 10.1063/1.2158735

**Abstract:**

We discuss the relation between the q-number approach to quantum mechanics suggested by Dirac and the notion of "pregeometry" introduced by Wheeler. By associating the q-numbers with the elements of an algebra and regarding the primitive idempotents as "generalized points" we suggest an approach that may make it possible to dispense with an a priori given space manifold. In this approach the algebra itself would carry the symmetries of translation, rotation, etc. Our suggestion is illustrated in a preliminary way by using a particular generalized Clifford Algebra proposed originally by Weyl, which approaches the ordinary Heisenberg algebra in a suitable limit. We thus obtain a certain insight into how quantum mechanics may be regarded as a purely algebraic theory, provided that we further introduce a new set of "neighbourhood operators", which remove an important kind of arbitrariness that has thus far been present in the attempt to treat quantum mechanics solely in terms of a Heisenberg algebra.

http://eprints.bournemouth.ac.uk/21620/

http://dx.doi.org/10.1063/1.2158735

**Source:** arXiv