A new algebraic structure in the standard model of particle physics

Abstract

We introduce a new formulation of the real-spectral-triple formalism in non-commutative 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.