Quantum General Invariance and Loop Gravity

Abstract

A quantum physical projector is proposed for generally covariant theories which are derivable from a Lagrangian. The projector is the quantum analogue of the integral over the generators of finite one-parameter subgroups of the gauge symmetry transformations which are connected to the identity. Gauge variables are retained in this formalism, thus permitting the construction of spacetime area and volume operators in a tentative spacetime loop formulation of quantum general relativity.

This is a preview of subscription content, log in to check access.

REFERENCES

  1. 1.

    J. M. Pons, D. C. Salisbury, and L. C. Shepley, “Gauge transformations in the Lagrangian and Hamiltonian formalisms of generally covariant systems,” Phys. Rev. D 55, 658–668 (1997), gr-qc/9612037.

    Google Scholar 

  2. 2.

    J. M. Pons, D. C. Salisbury, and L. C. Shepley, “Gauge transformations in Einstein– Yang–Mills theories,” J. Math. Phys. 41, 5557–5571 (2000), gr-qc/9912086.

    Google Scholar 

  3. 3.

    J. M. Pons, D. C. Salisbury, and L. C. Shepley, “The gauge group in the real triad for-mulation of general relativity,” Gen. Rel. Grav. 32, 1727–1744 (2000), gr-qc/9912087.

    Google Scholar 

  4. 4.

    J. M. Pons, D. C. Salisbury, and L. C. Shepley, “Gauge group and reality conditions in Ashtekar's formulation of canonical gravity,” Phys. Rev. D 62, 064026–064040 (2000), gr-qc/9912085.

    Google Scholar 

  5. 5.

    J. Lee and R. M. Wald, “Local symmetries and constraints,” J. Math. Phys. 31, 725–743 (1990).

    Google Scholar 

  6. 6.

    P. G. Bergmann and A. Komar, Int. J. Theor. Phys. 5, 15 (1972).

    Google Scholar 

  7. 7.

    D. C. Salisbury and K. Sundermeyer, “The realization in phase space of general coor-dinate transformations,” Phys. Rev. D 27, 740–756 (1983).

    Google Scholar 

  8. 8.

    D. C. Salisbury and K. Sundermeyer, “The local symmetries of the Einstein Yang–Mills theory as phase space transformations,” Phys. Rev. D 27, 757–763 (1983).

    Google Scholar 

  9. 9.

    D. C. Salisbury, J. M. Pons, and L. C. Shepley, “Gauge symmetries in Ashtekar's formula-tion of general relativity, Nucl. Phys. B (Proceedings Supplement) 88, 314–317 (2000).

    Google Scholar 

  10. 10.

    C. J. Isham, Imperial College preprint /TP/91-92/25, gr-qc/9210011.

  11. 11.

    C. Rovelli and L. Smolin, Nucl. Phys. B 442, 593 (1995), gr-qc/9411005.

    Google Scholar 

  12. 12.

    C. Rovelli, Phys. Rev. D 59, 104015 (1999), gr-qc/9806121.

    Google Scholar 

  13. 13.

    M. Gaul and C. Rovelli, “Loop quantum gravity and the meaning of diffeomorphism invariance,” gr-qc/9910079.

  14. 14.

    C. Rovelli and L. Smolin, “Spin networks and quantum gravity,” Phys. Rev. D 52, 5743–5759 (1995).

    Google Scholar 

  15. 15.

    R. Penrose, “Angular momentum: An approach to combinatorial spacetime,” in Quantum Theory and Beyond, T. Bastin, ed. (Cambridge University Press, Cambridge, 1971), pp. 151_180.

    Google Scholar 

  16. 16.

    T. Thiemann, Class. Quant. Grav. 15, 1281–1314 (1998).

    Google Scholar 

  17. 17.

    M. P. Reisenberger and C. Rovelli, “ ‘sum over surfaces’ form of loop quantum gravity,” Phys. Rev. D 56, 3490–3508 (1997).

    Google Scholar 

  18. 18.

    J. M. Pons and D. C. Salisbury, “The gauge group in Ashtekar's real formulation of canonical gravity,” in preparation.

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Salisbury, D.C. Quantum General Invariance and Loop Gravity. Foundations of Physics 31, 1105–1118 (2001). https://doi.org/10.1023/A:1017530508201

Download citation

Keywords

  • General Relativity
  • Gauge Symmetry
  • Volume Operator
  • Quantum General
  • Symmetry Transformation