Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 010, 23 pages      arXiv:1310.8225      https://doi.org/10.3842/SIGMA.2014.010
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

Exploring the Causal Structures of Almost Commutative Geometries

Nicolas Franco a and Michał Eckstein b
a) Copernicus Center for Interdisciplinary Studies, ul. Sławkowska 17, 31-016 Kraków, Poland
b) Faculty of Mathematics and Computer Science, Jagellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland

Received October 31, 2013, in final form January 20, 2014; Published online January 28, 2014

Abstract
We investigate the causal relations in the space of states of almost commutative Lorentzian geometries. We fully describe the causal structure of a simple model based on the algebra $\mathcal{S}(\mathbb{R}^{1,1}) \otimes M_2(\mathbb{C})$, which has a non-trivial space of internal degrees of freedom. It turns out that the causality condition imposes restrictions on the motion in the internal space. Moreover, we show that the requirement of causality favours a unitary evolution in the internal space.

Key words: noncommutative geometry; causal structures; Lorentzian spectral triples.

pdf (483 kb)   tex (45 kb)

References

  1. Barrett J.W., Lorentzian version of the noncommutative geometry of the standard model of particle physics, J. Math. Phys. 48 (2007), 012303, 7 pages, hep-th/0608221.
  2. Bengtsson I., Życzkowski K., Geometry of quantum states. An introduction to quantum entanglement, Cambridge University Press, Cambridge, 2006.
  3. Besnard F., A noncommutative view on topology and order, J. Geom. Phys. 59 (2009), 861-875, arXiv:0804.3551.
  4. Bieliavsky P., Claessens L., Sternheimer D., Voglaire Y., Quantized anti de Sitter spaces and non-formal deformation quantizations of symplectic symmetric spaces, in Poisson Geometry in Mathematics and Physics, Contemp. Math., Vol. 450, Amer. Math. Soc., Providence, RI, 2008, 1-24, arXiv:0705.4179.
  5. Bieliavsky P., Detournay S., Rooman M., Spindel Ph., BTZ black holes, WZW models and noncommutative geometry, hep-th/0511080.
  6. Bieliavsky P., Detournay S., Spindel Ph., Rooman M., Star products on extended massive non-rotating BTZ black holes, J. High Energy Phys. 2004 (2004), no. 6, 031, 25 pages, hep-th/0403257.
  7. Bognár J., Indefinite inner product spaces, Springer-Verlag, New York, 1974.
  8. Cagnache E., D'Andrea F., Martinetti P., Wallet J.-C., The spectral distance in the Moyal plane, J. Geom. Phys. 61 (2011), 1881-1897, arXiv:0912.0906.
  9. Chamseddine A.H., Connes A., Marcolli M., Gravity and the standard model with neutrino mixing, Adv. Theor. Math. Phys. 11 (2007), 991-1089, hep-th/0610241.
  10. Connes A., Noncommutative geometry, Academic Press Inc., San Diego, CA, 1994.
  11. Connes A., Marcolli M., Noncommutative geometry, quantum fields and motives, American Mathematical Society Colloquium Publications, Vol. 55, Amer. Math. Soc., Providence, RI, 2008.
  12. D'Andrea F., Martinetti P., On Pythagoras theorem for products of spectral triples, Lett. Math. Phys. 103 (2013), 469-492, arXiv:1203.3184.
  13. Franco N., Lorentzian approach to noncommutative geometry, Ph.D. Thesis, University of Namur, Belgium, 2011, arXiv:1108.0592.
  14. Franco N., Temporal Lorentzian spectral triple, arXiv:1210.6575.
  15. Franco N., Eckstein M., An algebraic formulation of causality for noncommutative geometry, Classical Quantum Gravity 30 (2013), 135007, 18 pages, arXiv:1212.5171.
  16. Gayral V., Gracia-Bondía J.M., Iochum B., Schücker T., Várilly J.C., Moyal planes are spectral triples, Comm. Math. Phys. 246 (2004), 569-623, hep-th/0307241.
  17. Iochum B., Krajewski T., Martinetti P., Distances in finite spaces from noncommutative geometry, J. Geom. Phys. 37 (2001), 100-125, hep-th/9912217.
  18. Kadison R.V., Ringrose J.R., Fundamentals of the theory of operator algebras. Vol. II. Advanced theory, Pure and Applied Mathematics, Vol. 100, Academic Press Inc., Orlando, FL, 1986.
  19. Marcolli M., Pierpaoli E., Teh K., The spectral action and cosmic topology, Comm. Math. Phys. 304 (2011), 125-174, arXiv:1005.2256.
  20. Moretti V., Aspects of noncommutative Lorentzian geometry for globally hyperbolic spacetimes, Rev. Math. Phys. 15 (2003), 1171-1217, gr-qc/0203095.
  21. Nakahara M., Geometry, topology and physics, 2nd ed., Graduate Student Series in Physics, Institute of Physics, Bristol, 2003.
  22. Nelson W., Sakellariadou M., Cosmology and the noncommutative approach to the standard model, Phys. Rev. D 81 (2010), 085038, 7 pages, arXiv:0812.1657.
  23. Paschke M., Sitarz A., Equivariant Lorentzian spectral triples, math-ph/0611029.
  24. Strohmaier A., On noncommutative and pseudo-Riemannian geometry, J. Geom. Phys. 56 (2006), 175-195, math-ph/0110001.
  25. van den Dungen K., Paschke M., Rennie A., Pseudo-Riemannian spectral triples and the harmonic oscillator, J. Geom. Phys. 73 (2013), 37-55, arXiv:1207.2112.
  26. van den Dungen K., van Suijlekom W.D., Particle physics from almost-commutative spacetimes, Rev. Math. Phys. 24 (2012), 1230004, 105 pages, arXiv:1204.0328.
  27. van Suijlekom W.D., The noncommutative Lorentzian cylinder as an isospectral deformation, J. Math. Phys. 45 (2004), 537-556, math-ph/0310009.
  28. Verch R., Quantum Dirac field on Moyal-Minkowski spacetime - illustrating quantum field theory over Lorentzian spectral geometry, Acta Phys. Polon. B Proc. Suppl. 4 (2011), 507-527, arXiv:1106.1138.

Previous article  Next article   Contents of Volume 10 (2014)