Our aim in this review article is to present the applications of Connes' noncommutative geometry to elementary particle physics. Whereas the existing literature is mostly focused on a mathematical audience, in this article we introduce the ideas and concepts from noncommutative geometry using physicists' terminology, gearing towards the predictions that can be derived from the noncommutative description. Focusing on a light package of noncommutative geometry (so-called 'almost-commutative manifolds'), we shall introduce in steps: electrodynamics, the electroweak model, culminating in the full Standard Model. We hope that our approach helps in understanding the role noncommutative geometry could play in describing particle physics models, eventually unifying them with Einstein's (geometrical) theory of gravity.One of the outstanding quests in modern theoretical physics is the unification of the four fundamental forces. There have been several theories around that partly fulfill this goal, all succeeding in some (different) aspects of such a theory. We will give an introduction to one of them, namely noncommutative geometry. It is a bottom-up approach in that it unifies the well-established Standard Model of highenergy physics with Einstein's general theory of relativity, thus not starting with extra dimensions, loops or strings. All this fits nicely in a mathematical framework, which was established by Connes in the 1980s [27].Of course, there is a price that one has to pay for having such a rigid mathematical basis: at present the unification has been obtained only at the classical level. The main reason for this can actually be found in any gauge theory (such as Yang-Mills theory) as well: its quantization is still waiting for a sound mathematical description. The noncommutative geometrical description of (classical) Yang-Mills theories -or the Standard Model, for that matter -minimally coupled to gravity encounters the same trouble in formulating the corresponding quantum theory, in addition troubled by the quantization of gravity. It needs no stressing that this situation needs to be improved (though some progress has been made recently, see the Outlook), and it is our hope that this review article strengthens the dialogue with for instance string theory or quantum gravity. Intriguingly, noncommutative spacetimes naturally appear in the context of both of these theories. In string theory, this started with the work of Seiberg and Witten [87], in loop quantum gravity the quantized area operator is a manifestation of an underlying quantum geometry (cf.[85] and references therein). This has lead to a fruitful acre where ideas from the fields involved are combined: noncommutative geometry and string theory already in [31], see the recent account in [61] and references therein; noncommutative geometry and loop quantum gravity in [1-3] and more recently in [72].

We define pseudo-Riemannian spectral triples, an analytic context broad enough to encompass a spectral description of a wide class of pseudo-Riemannian manifolds, as well as their noncommutative generalisations. Our main theorem shows that to each pseudo-Riemannian spectral triple we can associate a genuine spectral triple, and so a K-homology class. With some additional assumptions we can then apply the local index theorem. We give a range of examples and some applications. The example of the harmonic oscillator in particular shows that our main theorem applies to much more than just classical pseudo-Riemannian manifolds.

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.

Motivated by the space of spinors on a Lorentzian manifold, we define Krein spectral triples, which generalise spectral triples from Hilbert spaces to Krein spaces. This Krein space approach allows for an improved formulation of the fermionic action for almost-commutative manifolds. We show by explicit calculation that this action functional recovers the correct Lagrangians for the cases of electrodynamics, the electro-weak theory, and the Standard Model. The description of these examples does not require a real structure, unless one includes Majorana masses, in which case the internal spaces also exhibit a Krein space structure.