Algebraic Quantum Mechanics and Pregeometry
Abstract
We discuss the relation between the qnumber approach to quantum mechanics suggested by Dirac and the notion of "pregeometry" introduced by Wheeler. By associating the qnumbers 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 spacetime 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.
 Publication:

Quantum Theory: Reconsideration of Foundations
 Pub Date:
 January 2006
 DOI:
 10.1063/1.2158735
 arXiv:
 arXiv:quantph/0612002
 Bibcode:
 2006AIPC..810..314B
 Keywords:

 03.65.Fd;
 03.65.Vf;
 Algebraic methods;
 Phases: geometric;
 dynamic or topological;
 Quantum Physics;
 Mathematical Physics;
 Mathematics  Quantum Algebra
 EPrint:
 This paper was originally written in 1981 and published as a supplement to my Ph.D. thesis. (Davies, P., (1981) The Weyl Algebra and an Algebraic Mechanics. Ph.D thesis, Birkbeck College, University of London.) It is believed to be one of the "lost papers" of David Bohm as it was is not listed among his completed works and is set forth here for historical completeness