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

SIGMA 8 (2012), 018, 25 pages      arXiv:1203.6164      https://doi.org/10.3842/SIGMA.2012.018
Contribution to the Special Issue “Loop Quantum Gravity and Cosmology”

Intersecting Quantum Gravity with Noncommutative Geometry - a Review

Johannes Aastrup a and Jesper Møller Grimstrup b
a) Institut für Analysis, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany
b) Wildersgade 49b, 1408 Copenhagen, Denmark

Received October 06, 2011, in final form March 16, 2012; Published online March 28, 2012

Abstract
We review applications of noncommutative geometry in canonical quantum gravity. First, we show that the framework of loop quantum gravity includes natural noncommutative structures which have, hitherto, not been explored. Next, we present the construction of a spectral triple over an algebra of holonomy loops. The spectral triple, which encodes the kinematics of quantum gravity, gives rise to a natural class of semiclassical states which entail emerging fermionic degrees of freedom. In the particular semiclassical approximation where all gravitational degrees of freedom are turned off, a free fermionic quantum field theory emerges. We end the paper with an extended outlook section.

Key words: quantum gravity; noncommutative geometry; semiclassical analysis.

pdf (498 kb)   tex (57 kb)

References

  1. Aastrup J., Grimstrup J.M., From quantum gravity to quantum field theory via noncommutative geometry, arXiv:1105.0194.
  2. Aastrup J., Grimstrup J.M., Intersecting Connes noncommutative geometry with quantum gravity, Internat. J. Modern Phys. A 22 (2007), 1589-1603, hep-th/0601127.
  3. Aastrup J., Grimstrup J.M., Spectral triples of holonomy loops, Comm. Math. Phys. 264 (2006), 657-681, hep-th/0503246.
  4. Aastrup J., Grimstrup J.M., Nest R., A new spectral triple over a space of connections, Comm. Math. Phys. 290 (2009), 389-398, arXiv:0807.3664.
  5. Aastrup J., Grimstrup J.M., Nest R., Holonomy loops, spectral triples and quantum gravity, Classical Quantum Gravity 26 (2009), 165001, 17 pages, arXiv:0902.4191.
  6. Aastrup J., Grimstrup J.M., Nest R., On spectral triples in quantum gravity. I, Classical Quantum Gravity 26 (2009), 065011, 53, arXiv:0802.1783.
  7. Aastrup J., Grimstrup J.M., Nest R., On spectral triples in quantum gravity. II, J. Noncommut. Geom. 3 (2009), 47-81, arXiv:0802.1784.
  8. Aastrup J., Grimstrup J.M., Paschke M., Emergent Dirac Hamiltonians in quantum gravity, arXiv:0911.2404.
  9. Aastrup J., Grimstrup J.M., Paschke M., On a derivation of the Dirac Hamiltonian from a construction of quantum gravity, arXiv:1003.3802.
  10. Aastrup J., Grimstrup J.M., Paschke M., Quantum gravity coupled to matter via noncommutative geometry, Classical Quantum Gravity 28 (2011), 075014, 10 pages, arXiv:1012.0713.
  11. Aastrup J., Grimstrup J.M., Paschke M., Nest R., On semi-classical states of quantum gravity and noncommutative geometry, Comm. Math. Phys. 302 (2011), 675-696, arXiv:0907.5510.
  12. Ashtekar A., New Hamiltonian formulation of general relativity, Phys. Rev. D 36 (1987), 1587-1602.
  13. Ashtekar A., New variables for classical and quantum gravity, Phys. Rev. Lett. 57 (1986), 2244-2247.
  14. Ashtekar A., Lewandowski J., Background independent quantum gravity: a status report, Classical Quantum Gravity 21 (2004), R53-R152, gr-qc/0404018.
  15. Ashtekar A., Lewandowski J., Representation theory of analytic holonomy C*-algebras, in Knots and Quantum Gravity (Riverside, CA, 1993), Oxford Lecture Ser. Math. Appl., Vol. 1, Oxford Univ. Press, New York, 1994, 21-61, gr-qc/9311010.
  16. Bahr B., Thiemann T., Gauge-invariant coherent states for loop quantum gravity. I. Abelian gauge groups, Classical Quantum Gravity 26 (2009), 045011, 22 pages, arXiv:0709.4619.
  17. Bahr B., Thiemann T., Gauge-invariant coherent states for loop quantum gravity. II. Non-Abelian gauge groups, Classical Quantum Gravity 26 (2009), 045012, 45 pages, arXiv:0709.4636.
  18. Bellissard J., K-theory of C*-algebras in solid state physics, in Statistical Mechanics and Field Theory: Mathematical Aspects (Groningen, 1985), Lecture Notes in Phys., Vol. 257, Springer, Berlin, 1986, 99-156.
  19. Bellissard J., Ordinary quantum Hall effect and noncommutative cohomology, in Localization in Disordered Systems (Bad Schandau, 1986), Teubner-Texte Phys., Vol. 16, Teubner, Leipzig, 1988, 61-74.
  20. Borchers H.J., On revolutionizing quantum field theory with Tomita's modular theory, J. Math. Phys. 41 (2000), 3604-3673.
  21. Bratteli O., Robinson D.W., Operator algebras and quantum statistical mechanics. I. C*- and W*-algebras, symmetry groups, decomposition of states, 2nd ed., Texts and Monographs in Physics, Springer-Verlag, New York, 1987.
  22. Bratteli O., Robinson D.W., Operator algebras and quantum statistical mechanics. II. Models in quantum statistical mechanics, 2nd ed., Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997.
  23. Cacic B., A reconstruction theorem for almost-commutative spectral triples, arXiv:1101.5908.
  24. Carey A.L., Phillips J., Sukochev F.A., On unbounded p-summable Fredholm modules, Adv. Math. 151 (2000), 140-163, math.OA/9908091.
  25. 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.
  26. Christensen E., Ivan C., Sums of two-dimensional spectral triples, Math. Scand. 100 (2007), 35-60, math.OA/0601024.
  27. Connes A., Gravity coupled with matter and the foundation of non-commutative geometry, Comm. Math. Phys. 182 (1996), 155-176, hep-th/9603053.
  28. Connes A., Noncommutative geometry, Academic Press Inc., San Diego, CA, 1994.
  29. Connes A., On the spectral characterization of manifolds, arXiv:0810.2088.
  30. Connes A., Une classification des facteurs de type III, Ann. Sci. École Norm. Sup. (4) 6 (1973), 133-252.
  31. Connes A., Marcolli M., Noncommutative geometry, quantum fields and motives, American Mathematical Society Colloquium Publications, Vol. 55, American Mathematical Society, Providence, RI, 2008, available at http://www.alainconnes.org/en/downloads.php.
  32. Connes A., Rovelli C., von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories, Classical Quantum Gravity 11 (1994), 2899-2917, gr-qc/9406019.
  33. Denicola D., Marcolli M., Zainy al Yasry A., Spin foams and noncommutative geometry, Classical Quantum Gravity 27 (2010), 205025, 53 pages, arXiv:1005.1057.
  34. Doná P., Speziale S., Introductory lectures to loop quantum gravity, arXiv:1007.0402.
  35. Haag R., Local quantum physics, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992.
  36. Hall B.C., Phase space bounds for quantum mechanics on a compact Lie group, Comm. Math. Phys. 184 (1997), 233-250.
  37. Hall B.C., The Segal-Bargmann "coherent state" transform for compact Lie groups, J. Funct. Anal. 122 (1994), 103-151.
  38. Kadison R.V., Ringrose J.R., Fundamentals of the theory of operator algebras, Vol. I, Graduate Studies in Mathematics, Vol. 15, American Mathematical Society, Providence, RI, 1997.
  39. Kadison R.V., Ringrose J.R., Fundamentals of the theory of operator algebras, Vol. II, Graduate Studies in Mathematics, Vol. 16, American Mathematical Society, Providence, RI, 1997.
  40. Kaminski D., Algebras of quantum variables for loop quantum gravity. I. Overview, arXiv:1108.4577.
  41. Lai A., The JLO character for the noncommutative space of connections of Aastrup-Grimstrup-Nest, arXiv:1010.5226.
  42. Lewandowski J., Okolów A., Quantum group connections, J. Math. Phys. 50 (2009), 123522, 31 pages, arXiv:0810.2992.
  43. Lord S., Rennie A., Varilly J.C., Riemannian manifolds in noncommutative geometry, arXiv:1109.2196.
  44. Martins R.D., An outlook on quantum gravity from an algebraic perspective, arXiv:1003.4434.
  45. Mislin G., Valette A., Proper group actions and the Baum-Connes conjecture, Advanced Courses in Mathematics, Birkhäuser Verlag, Basel, 2003.
  46. Nicolai H., Matschull H.J., Aspects of canonical gravity and supergravity, J. Geom. Phys. 11 (1993), 15-62.
  47. Rovelli C., Quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004.
  48. Sahlmann H., Loop quantum gravity - a short review, arXiv:1001.4188.
  49. Takesaki M., Tomita's theory of modular Hilbert algebras and its applications, Lecture Notes in Mathematics, Vol. 128, Springer-Verlag, Berlin, 1970.
  50. Thiemann T., Modern canonical quantum general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2007, gr-qc/0110034.
  51. Thiemann T., Winkler O., Gauge field theory coherent states (GCS). IV. Infinite tensor product and thermodynamical limit, Classical Quantum Gravity 18 (2001), 4997-5053, hep-th/0005235.

Previous article  Next article   Contents of Volume 8 (2012)