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An affinity for affine quantum gravity

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To Andrei Alekseevich Slavnov, a scholar and gentleman of the first rank

Abstract

The main principle of affine quantum gravity is the strict positivity of the matrix \(\{ \hat g_{ab} (x)\} \) composed of the spatial components of the local metric operator. Canonical commutation relations are incompatible with this principle, and they can be replaced by noncanonical, affine commutation relations. Due to the partial second-class nature of the quantum gravitational constraints, it is advantageous to use the projection operator method, which treats all quantum constraints on an equal footing. Using this method, enforcement of regularized versions of the gravitational constraint operators is formulated quite naturally as a novel and relatively well-defined functional integral involving only the same set of variables that appears in the usual classical formulation. Although perturbatively nonrenormalizable, gravity may possibly be understood nonperturbatively from a hard-core perspective that has proved valuable for specialized models.

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References

  1. “String Theory,” Wikipedia: The Free Encyclopedia (Jan. 5, 2011), http://en.wikipedia.org/w/index.php?title=String_theory&oldid=406091132

  2. J. Polchinski, String Theory (Cambridge Univ. Press, Cambridge, 1998), Cambridge Monogr. Math. Phys.

    Book  Google Scholar 

  3. “Loop Quantum Gravity,” Wikipedia: The Free Encyclopedia (Jan. 9, 2011), http://en.wikipedia.org/w/index.php?title=Loop_quantum_gravity&oldid=406894581

  4. C. Rovelli, Quantum Gravity (Cambridge Univ. Press, Cambridge, 2004), Cambridge Monogr. Math. Phys.

    Book  MATH  Google Scholar 

  5. J. R. Klauder, “Noncanonical Quantization of Gravity. I: Foundations of Affine Quantum Gravity,” J. Math. Phys. 40, 5860–5882 (1999); arXiv: gr-qc/9906013.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. R. Klauder, “Noncanonical Quantization of Gravity. II: Constraints and the Physical Hilbert Space,” J. Math. Phys. 42, 4440–4465 (2001); arXiv: gr-qc/0102041.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. R. Klauder, “Ultralocal Fields and Their Relevance for Reparametrization-Invariant Quantum Field Theory,” J. Phys. A: Math. Gen. 34, 3277–3288 (2001); arXiv: quant-ph/0012076.

    Article  MathSciNet  MATH  Google Scholar 

  8. N. Aronszajn, “La théorie des noyaux reproduisants et ses applications. Première partie,” Proc. Cambridge Philos. Soc. 39, 133–153 (1943).

    Article  MathSciNet  Google Scholar 

  9. N. Aronszajn, “Theory of Reproducing Kernels,” Trans. Am. Math. Soc. 68, 337–404 (1950).

    Article  MathSciNet  MATH  Google Scholar 

  10. H. Meschkowski, Hilbertsche Räume mit Kernfunktion (Springer, Berlin, 1962).

    MATH  Google Scholar 

  11. J. R. Klauder, “Quantization = Geometry + Probability,” in Probabilistic Methods in Quantum Field Theory and Quantum Gravity, Ed. by P. H. Damgaard, H. Hüffel, and A. Rosenblum (Plenum Press, New York, 1990), pp. 73–85.

    Google Scholar 

  12. G. Watson and J. R. Klauder, “Metric and Curvature in Gravitational Phase Space,” Class. Quantum Grav. 19, 3617–3623 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  13. J. R. Klauder, “Universal Procedure for Enforcing Quantum Constraints,” Nucl. Phys. B 547, 397–412 (1999); arXiv: hep-th/9901010.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. R. Klauder, “Coherent State Quantization of Constraint Systems,” Ann. Phys. 254, 419–453 (1997); arXiv: quant-ph/9604033.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. R. Klauder, “Quantization of Constrained Systems,” in Methods of Quantization (Springer, Berlin, 2001), Lect. Notes Phys. 572, pp. 143–182; arXiv: hep-th/0003297.

    Chapter  Google Scholar 

  16. J. S. Little and J. R. Klauder, “Elementary Model of Constraint Quantization with an Anomaly,” Phys. Rev. D 71, 085014 (2005).

    Article  MathSciNet  Google Scholar 

  17. J. R. Klauder, “Field Structure through Model Studies: Aspects of Nonrenormalizable Theories,” Acta Phys. Austriaca, Suppl. 11, 341–387 (1973).

    Google Scholar 

  18. J. R. Klauder, “On the Meaning of a Non-renormalizable Theory of Gravitation,” Gen. Relativ. Gravit. 6, 13–19 (1975).

    Article  MathSciNet  Google Scholar 

  19. J. R. Klauder, “Continuous and Discontinuous Perturbations,” Science 199, 735–740 (1978).

    Article  Google Scholar 

  20. J. R. Klauder, Beyond Conventional Quantization (Cambridge Univ. Press, Cambridge, 2000, 2005).

    MATH  Google Scholar 

  21. O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural’ceva, Linear and Quasi-linear Equations of Parabolic Type (Am. Math. Soc., Providence, RI, 1968), Transl. Math. Monogr. 23.

    Google Scholar 

  22. J. R. Klauder, “Taming Nonrenormalizability,” J. Phys. A.: Math. Theor. 42, 335208 (2009); arXiv: 0811.3386.

    Article  MathSciNet  Google Scholar 

  23. J. R. Klauder, “Rethinking Renormalization,” in The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, Ed. by K. Alladi, J.R. Klauder, C.R. Rao (Springer, New York, 2010), pp. 503–528; arXiv: 0904.2869.

    Chapter  Google Scholar 

  24. R. Fernández, J. Fröhlich, and A. D. Sokal, Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory (Springer, Berlin, 1992).

    MATH  Google Scholar 

  25. J. R. Klauder, “Weak Correspondence Principle,” J. Math. Phys. 8, 2392–2399 (1967).

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Physics and Department of Mathematics, University of Florida, Gainesville, FL, 32611, USA

    John R. Klauder

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Klauder, J.R. An affinity for affine quantum gravity. Proc. Steklov Inst. Math. 272, 169–176 (2011). https://doi.org/10.1134/S0081543811010159

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