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2014

Quanta of Geometry: Noncommutative Aspects

Chamseddine, Ali H., Connes, Alain, Mukhanov, Viatcheslav

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In the construction of spectral manifolds in noncommutative geometry, a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of real scalar fields naturally appears and implies, by equality with the index formula, the quantization of the volume.

We first show that this condition implies that the manifold decomposes into disconnected spheres which will represent quanta of geometry.
We then refine the condition by involving the real structure and two types of geometric quanta, and show that connected spin-manifolds with large quantized volume are then obtained as solutions.
The two algebras M_2(H) and M_4(C) are obtained which are the exact constituents of the Standard Model.

Quanta of Geometry: Noncommutative Aspects · Around

Built on

H. Whitney, The self-intersection of a smooth n n -manifold in 2 n 2n -space
1944
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W. Greub, S. Halperin and R. Vanstone, Connections, Curvature and Cohomology,
1973
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J. D. Bekenstein, The quantum mass spectrum of the Kerr black hole
1986
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M. Henneaux and C. Teitelboim, The Cosmological Constant and General Covariance
1989
Earlier work this paper cites.

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J. D. Bekenstein and V. Mukhanov, Spectroscopy of the quantum black hole
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A. H. Chamseddine and A. Connes, The uncanny precision of the spectral action
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A. Connes, A short Survey of Noncommutative Geometry
Cited in the paper.
A. H. Chamseddine, A. Connes and V. Mukhanov, In preparation
Cited in the paper.

Then

A. H. Chamseddine and V. Mukhanov, Mimetic dark matter,
2013
Later among the works it cites.
Ali H. Chamseddine, Alain Connes, Viatcheslav Mukhanov, Geometry and the Quantum:Basics, arXiv:1411.0977, JHEP
Original
2014
Closest in time.
A. H. Chamseddine, V. Mukhanov and A. Vikman, Cosmology with Mimetic Matter
2014
Closest in time.
C. Gomez and G. Dvali, Quantum Compositness of Gravity: Black Holes, AdS and Inflation,
2014
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