Connes' notion of non-commutative geometry (NCG) generalizes Riemannian geometry and yields a striking reinterepretation of the standard model of particle physics, coupled to Einstein gravity. We suggest a simple reformulation with two key mathematical advantages: (i) it unifies many of the traditional NCG axioms into a single one; and (ii) it immediately generalizes from non-commutative to non-associative geometry. Remarkably, it also resolves a longstanding problem plaguing the NCG construction of the standard model, by precisely eliminating from the action the collection of seven unwanted terms that previously had to be removed by an extra, non-geometric, assumption. With this problem solved, the NCG algorithm for constructing the standard model action is tighter and more explanatory than the traditional one based on effective field theory.

Connes' non-commutative geometry (NCG) is a generalization of Riemannian geometry that is particularly apt for expressing the standard model of particle physics coupled to Einstein gravity. In a previous paper, we suggested a reformulation of this framework that is: (i) simpler and more unified in its axioms, and (ii) allows the Lagrangian for the standard model of particle physics (coupled to Einstein gravity) to be specified in a way that is tighter and more explanatory than the traditional algorithm based on effective field theory. Here we explain how this same reformulation yields a new perspective on the symmetries of a given NCG. Applying this perspective to the NCG traditionally used to describe the standard model we find, instead, an extension of the standard model by an extra − U (1) B L gauge symmetry, and a single extra complex scalar field σ, which is a singlet underThis field has cosmological implications, and offers a new solution to the discrepancy between the observed Higgs mass and the NCG prediction.

We show how twisting the spectral triple of the Standard Model of elementary particles naturally yields the Krein space associated with the Lorentzian signature of spacetime. We discuss the associated spectral action, both for fermions and bosons. What emerges is a tight link between twists and Wick rotation.

We argue that the ordinary commutative and associative algebra of spacetime coordinates (familiar from general relativity) should perhaps be replaced, not by a noncommutative algebra (as in noncommutative geometry), but rather by a Jordan algebra (leading to a framework which we term ‘Jordan geometry’). We present the Jordan algebra (and representation) that most nearly describes the standard model of particle physics, and we explain that it actually describes a certain (phenomenologically viable) extension of the standard model: by three right-handed (sterile) neutrinos, a complex scalar field φ, and a U(1) B−L gauge boson which is Higgsed by φ. We then note a natural extension of this construction, which describes the SU(4) × SU(2)L × SU(2)R Pati–Salam model. Finally, we discuss a simple and natural Jordan generalization of the exterior algebra of differential forms.

We introduce a new formulation of the real-spectral-triple formalism in noncommutative geometry (NCG): we explain its mathematical advantages and its success in capturing the structure of the standard model of particle physics. The idea, in brief, is to represent A (the algebra of differential forms on some possibly-noncommutative space) on H (the Hilbert space of spinors on that space); and to reinterpret this representation as a simple super-algebra B = A ⊕ H with even part A and odd part H. B is the fundamental object in our approach: we show that (nearly) all of the basic axioms and assumptions of the traditional real-spectral-triple formalism of NCG are elegantly recovered from the simple requirement that B should be a differential graded * -algebra (or " * -DGA"). Moreover, this requirement also yields other, new, geometrical constraints. When we apply our formalism to the NCG traditionally used to describe the standard model of particle physics, we find that these new constraints are physically meaningful and phenomenologically correct. In particular, these new constraints provide a novel interpretation of electroweak symmetry breaking that is geometric rather than dynamical. This formalism is more restrictive than effective field theory, and so explains more about the observed structure of the standard model, and offers more guidance about physics beyond the standard model.