The Uncanny Precision of the Spectral Action

@article{Chamseddine2008TheUP,
  title={The Uncanny Precision of the Spectral Action},
  author={Ali H. Chamseddine and Alain Connes},
  journal={Communications in Mathematical Physics},
  year={2008},
  volume={293},
  pages={867-897}
}
Noncommutative geometry has been slowly emerging as a new paradigm of geometry which starts from quantum mechanics. One of its key features is that the new geometry is spectral in agreement with the physical way of measuring distances. In this paper we present a detailed introduction with an overview on the study of the quantum nature of space-time using the tools of noncommutative geometry. In particular we examine the suitability of using the spectral action as an action functional for the… 
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References

SHOWING 1-10 OF 37 REFERENCES
Noncommutative geometry and the standard model with neutrino mixing
We show that allowing the metric dimension of a space to be independent of its KO-dimension and turning the finite noncommutative geometry F– whose product with classical 4-dimensional space-time
Conceptual explanation for the algebra in the noncommutative approach to the standard model.
The purpose of this Letter is to remove the arbitrariness of the ad hoc choice of the algebra and its representation in the noncommutative approach to the standard model, which was begging for a
Action Integrals and Partition Functions in Quantum Gravity
One can evaluate the action for a gravitational field on a section of the complexified spacetime which avoids the singularities. In this manner we obtain finite, purely imaginary values for the
Gravity coupled with matter and the foundation of non-commutative geometry
We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds. Its unitary representations correspond
Scale invariance in the spectral action
TLDR
All desirable features with correct signs for the relevant terms are obtained uniquely and without any fine tuning in the spectral action of the noncommutative space defined by the standard model.
Why the Standard Model
The Spectral Action Principle
Abstract:We propose a new action principle to be associated with a noncommutative space . The universal formula for the spectral action is where is a spinor on the Hilbert space, is a scale and a
Quantum gravity boundary terms from the spectral action of noncommutative space.
TLDR
It is proved that the spectral action predicts uniquely the gravitational boundary term required for consistency of quantum gravity with the correct sign and coefficient.
Lorentzian version of the noncommutative geometry of the standard model of particle physics
A formulation of the noncommutative geometry for the standard model of particle physics with a Lorentzian signature metric is presented. The elimination of the fermion doubling in the Lorentzian case
Asymptotics and Hamiltonians in a First order formalism
We consider four-dimensional spacetimes which are asymptotically flat at spatial infinity and show that, in the first-order framework, the action principle for general relativity is well defined
...
...