Algebraic Quantum Mechanics and Pregeometry

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‐time 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.