Journal of Noncommutative Geometry

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Volume 7, Issue 2, 2013, pp. 433–456
DOI: 10.4171/JNCG/122

Electrodynamics from noncommutative geometry

Koen van den Dungen[1] and Walter D. van Suijlekom[2]

(1) Mathematical Science Institute, The Australian National University, John Dedman Building 27, Union Lane, ACT 2601, Canberra, Australia
(2) IMAPP, Faculty of Science, Radboud Universiteit Nijmegen, Heyendaalseweg 135, 6525 AJ, Nijmegen, Netherlands

Within the framework of Connes’ noncommutative geometry, the notion of an almost commutative manifold can be used to describe field theories on compact Riemannian spin manifolds. The most notable example is the derivation of the Standard Model of high energy physics from a suitably chosen almost commutative manifold. In contrast to such a non-abelian gauge theory, it has long been thought impossible to describe an abelian gauge theory within this framework. The purpose of this paper is to improve on this point. We provide a simple example of a commutative spectral triple based on the two-point space and show that it yields a U(1)-gauge theory. Then we slightly modify the spectral triple such that we obtain the full classical theory of electrodynamics on a curved background manifold.

Keywords: Noncommutative geometry, abelian gauge theory, electrodynamics

van den Dungen Koen, van Suijlekom Walter: Electrodynamics from noncommutative geometry. J. Noncommut. Geom. 7 (2013), 433-456. doi: 10.4171/JNCG/122