  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
$B=A\oplus H$ 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 $\ast$-algebra (or "$\ast$-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.
