|
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
- 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.
- Bengtsson I., Życzkowski K., Geometry of quantum states. An introduction to
quantum entanglement, Cambridge University Press, Cambridge, 2006.
- Besnard F., A noncommutative view on topology and order, J. Geom.
Phys. 59 (2009), 861-875, arXiv:0804.3551.
- 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.
- Bieliavsky P., Detournay S., Rooman M., Spindel Ph., BTZ black holes, WZW models
and noncommutative geometry, hep-th/0511080.
- 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.
- Bognár J., Indefinite inner product spaces, Springer-Verlag, New York,
1974.
- 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.
- 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.
- Connes A., Noncommutative geometry, Academic Press Inc., San Diego, CA, 1994.
- Connes A., Marcolli M., Noncommutative geometry, quantum fields and motives,
American Mathematical Society Colloquium Publications, Vol. 55,
Amer. Math. Soc., Providence, RI, 2008.
- D'Andrea F., Martinetti P., On Pythagoras theorem for products of spectral
triples, Lett. Math. Phys. 103 (2013), 469-492,
arXiv:1203.3184.
- Franco N., Lorentzian approach to noncommutative geometry, Ph.D. Thesis,
University of Namur, Belgium, 2011, arXiv:1108.0592.
- Franco N., Temporal Lorentzian spectral triple, arXiv:1210.6575.
- Franco N., Eckstein M., An algebraic formulation of causality for
noncommutative geometry, Classical Quantum Gravity 30
(2013), 135007, 18 pages, arXiv:1212.5171.
- 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.
- Iochum B., Krajewski T., Martinetti P., Distances in finite spaces from
noncommutative geometry, J. Geom. Phys. 37 (2001),
100-125, hep-th/9912217.
- 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.
- Marcolli M., Pierpaoli E., Teh K., The spectral action and cosmic topology,
Comm. Math. Phys. 304 (2011), 125-174, arXiv:1005.2256.
- Moretti V., Aspects of noncommutative Lorentzian geometry for globally
hyperbolic spacetimes, Rev. Math. Phys. 15 (2003),
1171-1217, gr-qc/0203095.
- Nakahara M., Geometry, topology and physics, 2nd ed., Graduate Student Series
in Physics, Institute of Physics, Bristol, 2003.
- Nelson W., Sakellariadou M., Cosmology and the noncommutative approach to the
standard model, Phys. Rev. D 81 (2010), 085038, 7 pages,
arXiv:0812.1657.
- Paschke M., Sitarz A., Equivariant Lorentzian spectral triples,
math-ph/0611029.
- Strohmaier A., On noncommutative and pseudo-Riemannian geometry,
J. Geom. Phys. 56 (2006), 175-195,
math-ph/0110001.
- 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.
- 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.
- van Suijlekom W.D., The noncommutative Lorentzian cylinder as an isospectral
deformation, J. Math. Phys. 45 (2004), 537-556,
math-ph/0310009.
- 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.
|
|