Global one-dimensionality conjecture within quantum general relativity

Lukasz Andrzej Glinka1


A simple quantum gravity model, based on a new conjecture within the canonically quantized 3 + 1 general relativity, is presented. The conjecture states that matter fields are functionals of an embedding volume form only and reduces to quantum geometrodynamics. By dimensional reduction the resulting theory is presented in the form of the Dirac equation, and application of the Fock quantization with the diagonalization procedure yields construction of the appropriate quantum field theory. The 1D wave function is derived, the corresponding 3-dimensional manifolds are discussed as well as physical scales associated with quantum correlations.


  1. I. L. Buchbinder, S. D. Odintsov, and I. L. Shapiro, Effective Action in Quantum Gravity (Institute of Physics Publishing, 1992); D. J. Gross, T. Piran, and S. Weinberg (eds.), Two-Dimensional Quantum Gravity and Random Surfaces (World Scientific, 1992); G. W. Gibbons and S. W. Hawking (eds.), Euclidean Quantum Gravity (World Scientific, 1993); G. Esposito, Quantum Gravity, Quantum Cosmology and Lorentzian Geometries (Springer, 1994); J. Ehlers and H. Friedrich (eds.), Canonical Gravity: From Classical to Quantum (Springer, 1994); E. Prugovecki, Principles of Quantum General Relativity (World Scientific, 1995); R. Gambini and J. Pullin, Loops, Knots, Gauge Theories and Quantum Gravity (Cambridge University Press, 1996); G. Esposito, A. Yu. Kamenshchik, and G. Pollifrone, Euclidean Quantum Gravity on Manifolds with Boundary (Springer, 1997); P. Fre, V. Gorini, G. Magli, and U. Moschella, Classical and Quantum Black Holes (Institute of Physics Publishing, 1999); I. G. Avramidi, Heat Kernel and Quantum Gravity (Springer, 2000); B. N. Kursunoglu, S. L. Mintz, and A. Perlmutter (eds.), Quantum Gravity, Generalized Theory of Gravitation and Superstring Theory-Based Unification (Kluwer, 2002); S. Carlip, Quantum Gravity in 2+1 Dimensions (Cambridge University Press, 2003); D. Giulini, C. Kiefer and C. Lammerzahl (eds.), Quantum Gravity. From Theory to Experimental Search (Springer, 2003); C. Rovelli, Quantum Gravity (Cambridge University Press, 2004); G. Amelino-Camelia and J. Kowalski-Glikman (eds.), Planck Scale Effects in Astrophysics and Cosmology (Springer, 2005); A. Gomberoff and D. Marolf (eds.), Lectures on Quantum Gravity (Springer, 2005); D. Rickles, S. French, and J. Saatsi (eds.), The Structural Foundations of Quantum Gravity (Clarendon Press, 2006); D. Gross, M. Henneaux, and A. Sevrin (eds.), The Quantum Structure of Space and Time (World Scientific, 2007); C. Kiefer, Quantum Gravity (2nd ed., Oxford University Press, 2007); T. Thiemann, Modern Canonical Quantum General Relativity (Cambridge University Press, 2007); D. Oriti, Approaches to Quantum Gravity. Toward a New Understanding of Space, Time, and Matter (Cambridge University Press, 2009).
  2. L. A. Glinka, AIP Conf. Proc. 1018, 94 (2008); arXiv:0801.4157 [gr-qc]; SIGMA 3, 087 (2007); arXiv:0707.3341 [gr-qc]; arXiv:gr-qc/0612079.
  3. J. B. Hartle and S. W. Hawking, Phys. Rev. D 28, 2960 (1983).
  4. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation. (Freeman, 1973); R. M. Wald, General Relativity (University of Chicago, 1984); S. Carroll, Spacetime and Geometry. An Introduction to General Relativity (Addison-Wesley, 2004).
  5. J. W. York, Phys. Rev. Lett. 28, 1082 (1972); G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977).
  6. J. F. Nash, Ann. Math. 56, 405 (1952); ibid., 63, 20 (1956); S. Masahiro, Nash Manifolds (Springer, 1987); M. Gunther, Ann. Global Anal. Geom. 7, 69 (1989); Math. Nachr. 144, 165 (1989).
  7. R. Arnowitt, S. Deser and C. W. Misner, in Gravitation: An Introduction to Current Research, ed. by L. Witten, p. 227, (Wiley, 1962); B. DeWitt, The Global Approach to Quantum Field Theory (Vol. 1 and 2, Clarendon Press, 2003).
  8. A. Hanson, T. Regge, and C. Teitelboim, Constrained Hamiltonian Systems (Accademia Nazionale dei Lincei, 1976).
  9. P. A. M. Dirac, Lectures on Quantum Mechanics (Belfer Graduate School of Science, Yeshiva University, 1964).
  10. B. S. DeWitt, Phys. Rev. 160, 1113 (1967).
  11. L. D. Faddeev, Usp. Fiz. Nauk 136, 435 (1982).
  12. J. A. Wheeler, Geometrodynamics (Academic Press, 1962); Einsteins Vision (Springer, 1968).
  13. A. E. Fischer, Gen. Rel. Grav. 15, 1191 (1983); J. Math. Phys. 27, 718 (1986).
  14. V. V. Fernandez, A. M. Moya, and W. A. Rodrigues Jr, Adv. Appl. Clifford Alg. 11, 1 (2001).
  15. N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (3rd ed., John Wiley and Sons, 1980).
For more information about this paper please visit Springer's Home Page of this paper.

Back to The Contents Page