Advertisement

Quantum Gravity via Manifold Positivity

  • Michael H. Freedman
Chapter

Abstract

The macroscopic dimensions of space-time should not be input but rather output of a general model for physics. Here, dimensionality arises from a recently discovered mathematical bifurcation: “positive versus indefinite manifold pairings.” It is used to build actions on a “formal chain” of combinatorial space-times of arbitrary dimension. The context for such actions is 2-field theory where Feynman integrals are not over classical, but previously quantized configurations. A topologically enforced singularity of the action can terminate the dimension at four and, in fact, the final fourth dimension is Lorentzian due to light-like vectors in the four dimensional manifold pairing. Our starting point is the action of “causal dynamical triangulations” but in a dimension-agnostic setting. Curiously, some hint of extra compact dimensions emerges from our action.

Keywords

Hilbert Space Quantum Gravity Kinetic Term Formal Chain Wick Rotation 
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.

References

  1. 1.
    J. Ambjørn, R. Loll, Non-perturbative Lorentzian quantum gravity, causality and topology change. Nucl. Phys. B 536(1–2), 407–434 (1998)Google Scholar
  2. 2.
    J. Ambjørn, J. Jurkiewicz, R. Loll, Reconstructing the universe. Phys. Rev. D 72(6), 064014 (2005)Google Scholar
  3. 3.
    J. Ambjørn, J. Jurkiewicz, R. Loll, Quantum gravity as sum over spacetimes. arXiv:0906.3947v2 (2009)Google Scholar
  4. 4.
    D. Calegari, M. Freedman, K. Walker, Positivity in the universal pairing in 3 dimensions. J. Am. Math. Soc. 23, 107–188 (2010)Google Scholar
  5. 5.
    S. DeMichelis, M. Freedman, Uncountably many exotic \({\mathbb{R}}^{4}\)’s in standard 4-space. J. Differ. Geom. 35, 219–254 (1992)Google Scholar
  6. 6.
    R. Dijkgraaf, E. Witten, Topological gauge theories and group cohomology. Commun. Math. Phys. 129(2), 393–429 (1990)Google Scholar
  7. 7.
    S. Donaldson, P. Kronheimer, The Geometry of Four-Manifolds. Oxford Mathematical Monographs (Oxford University Press, New York, 1997)Google Scholar
  8. 8.
    P. Ehrenfest, Welche Rolle spielt die Dreidimensionalität des Raumes in den Grundgesetzen der Physik? Ann. Phys. 61, 440 (1920)Google Scholar
  9. 9.
    M. Freedman, The topology of four dimensional manifolds. J. Differ. Geom. 17, 357–453 (1982)Google Scholar
  10. 10.
    M. Freedman, F. Quinn, Topology of 4-Manifolds (Princeton University Press, Princeton, 1990)Google Scholar
  11. 11.
    M. Freedman, A. Kitaev, C. Nayak, J.K. Slingerland, K. Walker, Z. Wang, Universal manifold pairings and positivity. Geom. Topol. 9, 2303–2317 (2005)Google Scholar
  12. 12.
    S. Giddings, A. Strominger, Baby universe, third quantization and the cosmological constant. Nucl. Phys. B 321(2), 481–508 (1989)Google Scholar
  13. 13.
    M. Gromov, Groups of polynomial growth and expanding maps. Sci. Publ. Math. 53, 53–73 (1981)Google Scholar
  14. 14.
    M. Kreck, P. Teichner, Positive topological field theories and manifolds of dimension > 4. J. Topol. 1, 663–670 (2008)Google Scholar
  15. 15.
    R. Penrose, Applications of negative dimensional tensors, in Combinatorial Mathematics and Its Applications, ed. by D. Welsh (Academic, London, 1972)Google Scholar
  16. 16.
    S. Smale, Generalized Poincare’s conjecture in dimensions greater than 4. Ann. Math. 74(2), 391–406 (1961)Google Scholar
  17. 17.
    M. Srednicki, Infinite quantization. UCSB 88–07. The paper is still in preprintGoogle Scholar
  18. 18.
    F. Tangherlini, Schwarzchild field in n dimensions and the dimensionality of space problem. Nuovo Cimento 27(10), 636–651 (1963)Google Scholar
  19. 19.
    C. Taubes, Self-dial Yang-Mills connections over non-self-dual 4-manifolds. J. Differ. Geom. 17, 139–170 (1982)Google Scholar
  20. 20.
    W. Thurston, Hyperbolic structures on 3-manifolds I: deformation of acylindrical manifolds. Ann. Math. 124(2), 203–246 (1986)Google Scholar
  21. 21.
    K. Uhlenbeck, Connections with \({L}_{p}\) bounds on curvature. Commun. Math. Phys. 83, 31–42 (1982)Google Scholar
  22. 22.
    H. Whitney, The self-intersections of smooth n-manifolds in 2n-space. Ann. Math. 45(2), 220–246 (1944)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Microsoft CorporationUniversity of CaliforniaOaklandUSA

Personalised recommendations