Abstract
The small algebra of loop functionals, defined by Rovelli and Smolin, on the Ashtekar phase space of general relativity is studied. Regarded as coordinates on the phase space, the loop functionals become degenerate at certain points. All the degenerate points are found and the corresponding degeneracy is discussed. The intersection of the set of degenerate points with the real slice of the constraint surface is shown to correspond precisely the Goldberg-Kerr solutions. The evolution of the holonomy group of Ashtekar's connection is examined, and the complexification of the holonomy group is shown to be preserved under it. Thus, an observable of the gravitational field is constructed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ashtekar, A.: Phys. Rev. Lett.57, 2244–2247 (1986)
Ashtekar, A.: New perspectives in canonical gravity. (Naples: Bibliopolis 1988)
Ashtekar, A.: Lectures on nonperturbative canonical gravity. (Notes prepared in collaboration with R. Tate). Singapore: World Scientific 1981
Ashtekar, A.: Phys. Rev. D36, 1587–1603 (1987)
Jacobson, T., Smolin, L.: Nucl. Phys. B229, 295 (1988)
Rovelli, C., Smolin, L.: Nucl. Phys. B331, 80–152 (1990)
Ashtekar, A., Husain, V., Rovelli, C., Samuel, J., Smolin, L.: Class. Quantum Grav.6, L185 (1989)
Ashtekar, A., Rovelli, C.: Loop representation for the Maxwell field: quantum Faraday lines. Submitted to J. Math. Phys.
Ashtekar, A., Rovelli, C., Smolin, L.: Gravitons and loops. Preprint (1991)
Gambini, R., Brugmann, B., Pullin, J.: Knot invariants as nondegenerate states of four dimensional quantum gravity. Syracuse University Preprint, SU-GP-91/8-4 (1991)
Samuel, J.: Private communication (1991)
Schell, J.F.: J. Math. Phys.2, 202 (1962)
Goldberg, J.N., Kerr, R.P.: J. Math. Phys.2, 327 (1961)
Kerr, R.P., Goldberg, J.N.: J. Math. Phys.2, 332 (1961)
Kramer, D. et al.: Exact solutions of Einstein's field equations. Deutscher Verlag der Wissenschaften 1980
Goldberg, J.N., Lewandowski, J.: In preparation
Author information
Authors and Affiliations
Additional information
Communicated by S.-T. Yau
Rights and permissions
About this article
Cite this article
Goldberg, J.N., Lewandowski, J. & Stornaiolo, C. Degeneracy in loop variables. Commun.Math. Phys. 148, 377–402 (1992). https://doi.org/10.1007/BF02100867
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02100867