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High Energy Physics - Theory, hep-th,High Energy Physics - Phenomenology, hep-ph,Mathematical Physics, math-ph,Mathematics, Mathematical Physics, math.MP
Abstract:
We introduce a new formulation of 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 ("spectral
triple") formulation of NCG are elegantly recovered from the simple requirement
that $B$ should be a differential graded $\ast$-algebra (or "$\ast$-DGA"). But
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. 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.
