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 su...

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Bibliographic Details
Main Authors: Bohm, D J, Davies, P G, Hiley, B J
Format: Conference Proceeding
Language:English
Subjects:
ALGEBRA
CLIFFORD ALGEBRA
GEOMETRY
PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
QUANTUM FIELD THEORY
QUANTUM MECHANICS
ROTATION
SPACE-TIME
SYMMETRY
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Summary: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.
ISSN:0094-243X
1551-7616
DOI:10.1063/1.2158735