Pointless Spaces and Quantum Gravity

Abstract

We sketch first a survey of some papers on the themes in the title (with embellishments and connections). Thus in Section 2 we consider ideas from [81, 283, 418, 486, 558, 577, 578, 579, 580, 621, 679, 680, 681, 682]. There are connections here to quantum mechanics (QM) on a discrete background as in [38, 39, 40, 70, 109, 221, 429, 430, 669, 670], to differential calculi on finite sets as in [29, 84, 109, 191, 191, 193, 194, 196, 197, 198, 199, 200, 201, 202, 203, 204], fuzzy physics (cf. [429, 447] and references there), and to noncommutative geometry (NCG) (cf. [109, 155, 156, 280, 429, 447]). Some of the latter material will also be sketched. A basic driving impulse in all this is to understand QM and provide a properly (discrete) geometrical setting for QM, relativity, and classical mechanics (CM). In this direction there are many approaches to relating QM and CM behavior (see e.g. [109] and references there for a survey of some of this). To amplify this a little we will sketch some papers and results which are either more recent or have recently come to our attention; for example we cite [575, 390, 391, 392].