This paper explores in some detail a recent proposal (the Rieffel induction/refined algebraic quantization scheme) for the quantization of constrained gauge systems. Below, the focus is on systems with a single constraint and, in this context, on the uniqueness of the construction. While in general the results depend heavily on the choices made for certain auxiliary structures, an additional physical argument leads to a unique result for typical cases. We also discuss the 'superselection laws' that result from this scheme and how their existence also depends on the choice of auxiliary structures. Again, when these structures are chosen in a physically motivated way, the resulting superselection laws are physically reasonable.
@article{bwmeta1.element.bwnjournal-article-bcpv39z1p331bwm,
author = {Marolf, Donald},
title = {Refined Algebraic Quantization: Systems with a single constraint},
journal = {Banach Center Publications},
volume = {38},
year = {1997},
pages = {331-344},
zbl = {0887.46050},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv39z1p331bwm}
}
Marolf, Donald. Refined Algebraic Quantization: Systems with a single constraint. Banach Center Publications, Tome 38 (1997) pp. 331-344. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv39z1p331bwm/
[000] [1] A. Ashtekar, Non-Perturbative Canonical Gravity, Lectures notes prepared in collaboration with R. S. Tate, World Scientific, Singapore, 1991.
[001] [2] A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourão, and T. Thiemann, Quantization of diffeomorphism invariant theories of connections with local degrees of freedom, J. Math. Phys. 36 (1995), 6456-6493; gr-qc/9504018. | Zbl 0856.58006
[002] [3] A. Ashtekar and R. S. Tate, An algebraic extension of Dirac quantization: Examples, J. Math. Phys. 35 (1994), 6434-6470. | Zbl 0820.58058
[003] [4] B. DeWitt, Quantum Theory of Gravity. I: The Canonical Theory, Phys. Rev. (2) 160 (1967), 1113-1148. | Zbl 0158.46504
[004] [5] P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science, Yeshiva University, New York, 1964.
[005] [6] I. M. Ge{l'fand, N. Ya. Vilenkin, Generalized Functions: vol. 4, Applications of Harmonic Analysis, Academic Press, New York, London, 1964.
[006] [7] P. Hájíček, Quantization of Systems with Constraints, in: Canonical Gravity: from classical to quantum, J. Ehlers, H. Friedrich (eds.), Lecture Notes in Phys. 434, Springer, Berlin, 1994, 113-149. | Zbl 0820.58059
[007] [8] A. Higuchi, Quantum linearization instabilities of de Sitter spacetime: II, Classical Quantum Gravity 8 (1991), 1983-2004.
[008] [9] A. Higuchi, Linearized quantum gravity in flat space with toroidal topology, Classical Quantum Gravity 8 (1991), 2023-2034.
[009] [10] C. Isham, Canonical Gravity and the Problem of Time, Imperial College, preprint TP/91-92/25; gr-qc/9210011, 1992.
[010] [11] K. Kuchař, Time and Interpretations of Quantum Gravity, in: Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics, G. Kunstatter et al. (eds.), World Scientific, New Jersey, 1992, 211-314.
[011] [12] N. Landsman, Rieffel induction as generalized quantum Marsden-Weinstein reduction, J. Geom. Phys. 15 (1995), 285-319; hep-th/9305088. Erratum: ibid. 17 (1995), 298. | Zbl 0819.58012
[012] [13] N. Landsman, Classical and quantum representation theory, in: Proceedings Seminar Mathematical Structures in Field Theory, E. A. de Kerf and H. G. J. Pijls (eds.), CWI-syllabus, CWI, Amsterdam, to appear.
[013] [14] N. Landsman and U. Wiedemann, Massless Particles, Electromagnetism, and Rieffel Induction, Rev. Math. Phys. 7 (1995), 923-958; hep-th/9411174. | Zbl 0833.58051
[014] [15] D. Marolf, The spectral analysis inner product for quantum gravity, preprint gr-qc/9409036, to appear in the Proceedings of the VIIth Marcel-Grossman Conference, R. Ruffini and M. Keiser (eds.), World Scientific, Singapore, 1995.
[015] [16] D. Marolf, Green's Bracket Algebras and their Quantization, Ph. D. Dissertation, The University of Texas at Austin, 1992.
[016] [17] D. Marolf, Quantum observables and recollapsing dynamics, Classical Quantum Gravity 12 (1995), 1199-1220; gr-qc/9404053. | Zbl 0839.47048
[017] [18] D. Marolf, Observables and a Hilbert Space for Bianchi IX, Classical Quantum Gravity 12 (1995), 1441-1454; gr-qc/9409049.
[018] [19] D. Marolf, Almost Ideal Clocks in Quantum Cosmology: A Brief Derivation of Time, Classical Quantum Gravity 12 (1995), 2469-2486; gr-qc/9412016. | Zbl 0840.53064
[019] [20] B. Simon, Quantum Mechanics for Hamiltonians defined as quadratic forms, Princeton Univ. Press, Princeton, 1971, p. 120. | Zbl 0232.47053