The Standard Model as an extension of the noncommutative algebra of forms

Abstract

The Standard Model of particle physics can be deduced from a small number of axioms within Connes' noncommutative geometry (NCG). Boyle and Farnsworth [New J. Phys. 16 (2014) 123027] proposed to interpret Connes' approach as an algebra extension in the sense of Eilenberg. By doing so, they could deduce three axioms of the NCG Standard Model (i.e. order zero, order one and massless photon) from the single requirement that the extended algebra be associative. However, their approach was only applied to the finite algebra and fails the full model.By taking into account the differential graded structure of the algebra of noncommutative differential forms, we obtain a formulation where the same three axioms are deduced from the associativity of the extended differential graded algebra, but which is now also compatible with the full Standard Model.Finally, we present a Lorentzian version of the noncommutative geometry of the Standard Model and we show that the three axioms still hold if the four-dimensional manifold has a Lorentzian metric.

A remark on the mathematics of the seesaw mechanism

“…The requirement of renormalizability gives the further contraint that m L = 0. (Note that the Noncommutative Geometry approach to the Standard Model naturally predicts a matrix of this type with m L = 0 without any consideration of renormalizability, see [1], [2]).…”

The graded product of real spectral triples

“…NCG is a generalization of Riemannian geometry which (amongst other applications) provides an elegant framework for describing gauge theories coupled to gravity. In this capacity, it's main physical interest is in constraining the allowed extensions of the standard model of particle physics [10,[16][17][18][19][20][21][22][23][24][25][26][27]. The basic idea of NCG is to replace the familiar manifold and metric data {M, g} of Riemannian geometry with a 'spectral triple' of data {A, H, D}, which consist of a 'coordinate' algebra A that provides topological information, a Dirac operator D which provides metric information, and a Hilbert space H that provides a place for A and D to interact.…”

Wick rotation and fermion doubling in noncommutative geometry

“…It is based on results valid for (compact) Riemannian manifolds, and by its nature not immediately suited to accommodate a Lorentzian signature of the space. Although there are attempts in this direction -using either Krein spaces [4][5][6][7], covariant approaches [8], Wick rotations on pseudo-Riemanninan structures [9],or algebraic characterizations of causal structures [10][11][12] -it is fair to say that we are still far away from a full understanding of the theory. This becomes a problem when the tools of noncommutative geometry are applied to physics, and in particular to the Standard Model via the spectral action [13][14][15].…”

“…This explains the terminology "Dirac operator" for arbitrary spectral triple 7. By finite a dimensional spectral triple we mean that the algebra and Hilbert spaces are finite dimensional vector spaces.…”