Abstract
This is a report on our joint work with A. Chamseddine and M. Marcolli. This essay gives a short introduction to a potential application in physics of a new type of geometry based on spectral considerations which is convenient when dealing with non-commutative spaces, i.e., spaces in which the simplifying rule of commutativity is no longer applied to the coordinates. Starting from the phenomenological Lagrangian of gravity coupled with matter one infers, using the spectral action principle, that space-time admits a fine structure which is a subtle mixture of the usual 4-dimensional continuum with a finite discrete structure F. Under the (unrealistic) hypothesis that this structure remains valid (i.e., one does not have any “hyperfine” modification) until the unification scale, one obtains a number of predictions whose approximate validity is a basic test of the approach.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
J. Barrett, A Lorentzian version of the non-commutative geometry of the standard. model of particle physics hep-th/0608221.
C. Bordé, Base units of the SI, fundamental constants and modern quantum physics, Phil. Trans. R. Soc. A 363 (2005), 2177–2201.
A. Chamseddine, A. Connes, Universal Formula for Non-commutative Geometry. Actions: Unification of Gravity and the Standard Model, Phys. Rev. Lett. 77 (1996), 4868–4871.
A. Chamseddine, A. Connes, The Spectral Action Principle, Comm.Math. Phys. 186 (1997), 731–750.
A. Chamseddine, A. Connes, Scale Invariance in the Spectral Action, hep-th/0512169 to appear in Jour. Math. Phys.
A. Chamseddine, A. Connes, Inner fluctuations of the spectral action, hep-th/0605011.
A. Chamseddine, A. Connes, M. Marcolli, Gravity and the standard model with neutrino. mixing, hep-th/0610241.
S. Coleman, Aspects of symmetry, Selected Erice Lectures, Cambridge University Press, 1985.
A. Connes, Non-commutative geometry, Academic Press (1994).
A. Connes, Non-commutative geometry and reality, Journal of Math. Physics 36 no. 11 (1995).
A. Connes, Gravity coupled with matter and the foundation of non-commutative geometry, Comm. Math. Phys. (1995)
A. Connes, Non-commutative Geometry and the standard model with neutrino mixing, hep-th/0608226.
A. Connes, M. Marcolli Non-commutative Geometry, Quantum fields and Motives, Book in preparation.
A. Connes, H. Moscovici, The local index formula in non-commutative geometry, GAFA, Vol. 5 (1995), 174–243.
L. Dąbrowski, A. Sitarz, Dirac operator on the standard Podleśs quantum sphere, Non-commutative Geometry and Quantum Groups, Banach Centre Publications 61, Hajac, P.M. and Pusz, W. (eds.), Warszawa: IMPAN, 2003, pp. 49–58.
S. Giddings, D. Marolf, J. Hartle, Observables in effective gravity, hep-th/0512200.
J. Gracia-Bondia, B. Iochum, T. Schucker, The standard model in non-commutative. geometry and fermion doubling. Phys. Lett. B 416 no. 1–2 (1998), 123–128.
D. Kastler, Non-commutative geometry and fundamental physical interactions: The. Lagrangian level, Journal. Math. Phys. 41 (2000), 3867–3891.
M. Knecht, T. Schucker Spectral action and big desert hep-ph/065166
O. Lauscher, M. Reuter, Asymptotic Safety in Quantum Einstein Gravity: nonperturbative. renormalizability and fractal spacetime structure, hep-th/0511260.
F. Lizzi, G. Mangano, G. Miele, G. Sparano, Fermion Hilbert space and Fermion. Doubling in the Non-commutative Geometry Approach to Gauge Theories hepth/9610035.
J. Mather, Commutators of diffeomorphisms. II, Comment. Math. Helv. 50 (1975), 33–40.
R.N. Mohapatra, P.B. Pal, Massive neutrinos in physics and astrophysics, World Scientific, 2004.
M. Rieffel, Morita equivalence for C*-algebras and W*-algebras, J. Pure Appl. Algebra, 5 (1974), 51–96.
M. Sher, Electroweak Higgs potential and vacuum stability, Phys. Rep. Vol.179 (1989) N.5–6, 273–418.
M. Veltman, Diagrammatica: the path to Feynman diagrams, Cambridge Univ. Press, 1994.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Birkhäuser Verlag Basel
About this chapter
Cite this chapter
Connes, A. (2007). Non-commutative Geometry and the Spectral Model of Space-time. In: Duplantier, B. (eds) Quantum Spaces. Progress in Mathematical Physics, vol 53. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8522-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-7643-8522-4_5
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-8521-7
Online ISBN: 978-3-7643-8522-4
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)