Title: Second quantization of spectral geometry: emergence of the Standard Model and gravity

Abstract:

We start with a gentle introduction to the spectral approach to (noncommutative) geometry. We insist on the
presence of fermions, so that a central role is played by a Dirac operator, acting on a one-particle Hilbert space.
The combination with a coordinate algebra then allows for a full reconstruction of the spacetime geometry
from the spectral data. Moreover, when allowing for noncommutative spaces described by an algebra of
matrix-valued coordinates, the Dirac operator gets minimally coupled to a gauge field, as well as to the Higgs
field.
The dynamics of all of these fields will emerge from the second quantization of the system. After a careful
exposition of the latter procedure in the present context, we will analyze the information theoretic von Neu-
mann entropy of the unique equilibrium state. We show that this entropy is given by the spectral action for a
specific universal function. In the case of a curved spacetime manifold one finds that there is an asymptotic
expansion of this entropy yielding the Einstein-Hilbert action, plus higher-order gravitational terms. Finally,
in the presence of gauge and Higgs fields, the very same procedure yields additionally the Yang-Mills action
functional and the scalar terms describing a Higgs mechanism.