Consistent factor ordering of constraints may be ambiguous

Karel V. Kucha

doi:10.1103/PhysRevD.35.596

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Consistent factor ordering of constraints may be ambiguous

Karel V. Kucha

Department of Physics, University of Utah, Salt Lake City, Utah 84112

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Phys. Rev. D 35, 596 – Published 15 January, 1987

DOI: https://doi.org/10.1103/PhysRevD.35.596

Abstract

It is shown in a simple model that consistency requirements of the Dirac constraint quantization may be satisfied by more than one, essentially different, factor orderings of the constraints and the Hamiltonian.

References (5)

It was first noticed by J. Anderson, in Eastern Theoretical Conference, edited by M. E. Rose (Gordon and Breach, New York, 1963), that the standard form of the super-Hamiltonian and supermomentum constraints in canonical gravity cannot be turned into Hermitian operators which would satisfy the consistency requirements. A solution was offered by J. Schwinger, Phys. Rev. 130, 1253 (1963); 132, 1317 (1963), and criticized by P. A. M. Dirac, in Contemporary Physics: Trieste Symposium 1968, edited by A. Salam (International Atomic Energy Agency, Vienna, 1969), p. 539. The best, and certainly the shortest, exposition of Schwinger's solution may be found in a footnote of the paper by B. S. DeWitt, Phys. Rev. A. Ashtekar160, 1113 (1967), who himself made a different proposal on how to remove the problem by letting any two field operators taken at the same point formally commute. Recently, , Phys. Rev. Lett. 57, 2244 (1986), has reexpressed the Hamiltonian theory of gravity in a new set of fundamental canonical variables and found a simple factor ordering of the constraints which (disregarding the regularization difficulties) satisfies the consistency requirements.
Basic properties of the helical model are mentioned in the footnote on p. 729 of the article by B. S. DeWitt, in General Relativity: An Einstein Centenary Survey, edited by S. W. Hawking and W. Israel (Cambridge University Press, Cambridge, 1979). The model is studied in detail by K. Kuchař, Phys. Rev. D 34, 3031 (1986).
K. Kuchař, Phys. Rev. D 34, 3044 (1986).
See, e.g., A. Messiah, Quantum Mechanics I (North-Holland, Amsterdam, 1958).
I. Bialynicki-Birula (private communication).

[c1] It was first noticed by J. Anderson, in Eastern Theoretical Conference, edited by M. E. Rose (Gordon and Breach, New York, 1963), that the standard form of the super-Hamiltonian and supermomentum constraints in canonical gravity cannot be turned into Hermitian operators which would satisfy the consistency requirements. A solution was offered by J. Schwinger, Phys. Rev. 130, 1253 (1963); 132, 1317 (1963), and criticized by P. A. M. Dirac, in Contemporary Physics: Trieste Symposium 1968, edited by A. Salam (International Atomic Energy Agency, Vienna, 1969), p. 539. The best, and certainly the shortest, exposition of Schwinger's solution may be found in a footnote of the paper by B. S. DeWitt, Phys. Rev. A. Ashtekar160, 1113 (1967), who himself made a different proposal on how to remove the problem by letting any two field operators taken at the same point formally commute. Recently, , Phys. Rev. Lett. 57, 2244 (1986), has reexpressed the Hamiltonian theory of gravity in a new set of fundamental canonical variables and found a simple factor ordering of the constraints which (disregarding the regularization difficulties) satisfies the consistency requirements.

[c2] Basic properties of the helical model are mentioned in the footnote on p. 729 of the article by B. S. DeWitt, in General Relativity: An Einstein Centenary Survey, edited by S. W. Hawking and W. Israel (Cambridge University Press, Cambridge, 1979). The model is studied in detail by K. Kuchař, Phys. Rev. D 34, 3031 (1986).

[c3] K. Kuchař, Phys. Rev. D 34, 3044 (1986).

[c4] See, e.g., A. Messiah, Quantum Mechanics I (North-Holland, Amsterdam, 1958).

[c5] I. Bialynicki-Birula (private communication).

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