Abstract
The aim of this paper is to provide a general introduction to the problem of time in quantum gravity. This problem originates in the fundamental conflict between the way the concept of ‘time’ is used in quantum theory, and the role it plays in a diffeomorphism-invariant theory like general relativity. Schemes for resolving this problem can be sub-divided into three main categories: (I) approaches in which time is identified before quantising; (II) approaches in which time is identified after quantising; and (III) approaches in which time plays no fundamental role at all. Ten different specific schemes are discussed in this paper which also contain an introduction to the relevant parts of the canonical decomposition of general relativity.
Lectures presented at the NATO Advanced Study Institute “Recent Problems in Mathematical Physics”, Salamanca, June 15–27, 1992.
Research supported in part by SERC grant GR/G60918.
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References
Albrecht, A. (1990), ‘Identifying decohering paths in closed quantum systems’. preprint.
Albrecht, A. (1991), ‘Investigating decoherence in a simple system’. preprint.
Albrecht, A. (1992), Two perspectives on a decohering spin, in J. Halliwell, J. Perez Mercader & W. Zurek, eds, ‘Physical Origins of Time Asymmetry’, Cambridge University Press, Cambridge.
Alvarez, E. (1989), ‘Quantum gravity: an introduction to some recent results’, Rev. Mod. Phys. 61, 561 - 604.
Arnowitt, R., Deser, S. & Misner, C. (1959a), ‘Dynamical structure and definition of energy in general relativity’, Phys. Rev. 116, 1322–1330.
Arnowitt, R., Deser, S. & Misner, C. (1959b), ‘Quantum theory of gravitation: General formalism and linearized theory’, Phys. Rev. 113, 745 - 750.
Arnowitt, R., Deser, S. & Misner, C. (1960a), ‘Canonical variables for general relativity’, Phys. Rev. 117, 1595–1602.
Arnowitt, R., Deser, S. & Misner, C. (1960b), ‘Consistency of the canonical reduction of general relativity’, J. Math. Phys. 1, 434 - 439.
Arnowitt, R., Deser, S. & Misner, C. (1960c), ‘Energy and the criteria for radiation in general relativity’, Phys. Rev. 118, 1100–1104.
Arnowitt, R., Deser, S. & Misner, C. (1960d), ‘Finite self-energy of classical point particles’, Phys. Rev. Lett. 4, 375–377.
Arnowitt, R., Deser, S. & Misner, C. (1961a), ‘Coordinate invariance and energy expressions in general relativity’, Phys. Rev. 122, 997–1006.
Arnowitt, R., Deser, S. & Misner, C. (1961b), ‘Wave zone in general relativity’, Phys. Rev. 121, 1556–1566.
Arnowitt, R., Deser, S. & Misner, C. (1962), The dynamics of general relativity, in L. Witten, ed., ‘Gravitation: An Introduction to Current Research’, Wiley, New York, pp. 227–265.
Ashtekar, A. (1986), ‘New variables for classical and quantum gravity’, Phys. Rev. Lett. 57, 2244–2247.
Ashtekar, A. (1987), ‘New Hamiltonian formulation of general relativity’, Phys. Rev. D36, 1587–1602.
Ashtekar, A. (1991), Lectures on Non-Perturbative Canonical Gravity, World Scientific Press, Singapore.
Ashtekar, A. & Stachel, J., eds (1991), Conceptual Problems of Quantum Gravity, Birkhäuser, Boston.
Baierlein, R., Sharp, D. Si Wheeler, J. (1962), ‘Three-dimensional geometry as carrier of information about time’, Phys. Rev. 126, 1864–1865.
Banks, T. (1985), ‘TCP, quantum gravity, the cosmological constant and all that’, Nucl. Phys. B249, 332–360.
Barbour, J. (1990), ‘Time, gauge fields, and the Schrädinger equation in quantum cosmology’. Preprint.
Barbour, J. & Smolin, L. (1988), ‘Can quantum mechanics be sensibly applied to the universe as a whole?’. Yale University preprint.
Barvinsky, A. (1991), ‘Unitarity approach to quantum cosmology’. University of Alberta preprint.
Bergmann, P. & Komar, A. (1972), ‘The coordinate group of symmetries of general relativity’, Int. J. Mod. Phys. 5, 15–28.
Blencowe, M. (1991), ‘The consistent histories interpretation of quantum fields in curved spacetime’, Ann. Phys. (NY) 211, 87–111.
Blyth, W. Si Isham, C. (1975), ‘Quantisation of a Friedman universe filled with a scalar field’, Phys. Rev. D11, 768–778.
Bohr, N. & Rosenfeld, L. (1933), ‘Zur frage der messbarkeit der elektromagnetischen feldgrossen’, Kgl. Danek Vidensk. Selsk. Math.-fys. Medd. 12, 8.
Bohr, N. & Rosenfeld, L. (1978), On the question of the measurability of electromagnetic field quantities (English translation), in R. Cohen & J. Stachel, eds, ‘Selected Papers by Léon Rosenfeld’, Reidel, Dordrecht.
Brout, R. (1987), ‘On the concept of time and the origin of the cosmological temperature’, Found. Phys. 17, 603 - 619.
Brout, R. & Venturi, G. (1989), ‘Time in semiclassical gravity’, Phys. Rev. D39, 2436–2439.
Brout, R., Horowitz, G. & Wiel, D. (1987), ‘On the onset of time and temperature in cosmology’, Phys. Lett. B192, 318–322.
Brown, J. & York, J. (1989), ‘Jacobi’s action and the recovery of time in general relativity’, Phys. Rev. D40, 3312–3318.
Caldeira, A. & Leggett, A. (1983), ‘Path integral approach to quantum Brownian motion’, Physica Al21, 587–616.
Carlip, S. (1990), ‘Observables, gauge invariance, and time in 2 + 1 dimensional quantum gravity’, Phys. Rev. D42, 2647–2654.
Carlip, S. (1991), Time in 2 + 1 dimensional quantum gravity, in ‘Proceedings of the Banff Conference on Gravitation, August 1990’.
Castagnino, M. (1988), ‘Probabilistic time in quantum gravity’, Phys. Rev. D39, 2216–2228.
Coleman, S. (1988), ‘Why is there something rather than nothing: A theory of the cosmological constant’, Nucl. Phys. B310, 643 - 668.
Collins, P. & Squires, E. (1992), ‘Time in a quantum universe’. In press, Foundations of Physics.
Deutsch, D. (1990), ‘A measurement process in a stationary quantum system’. Oxford University preprint.
DeWitt, B. (1962), The quantization of geometry, in L. Witten, ed., ‘Gravitation: An Introduction to Current Research’, Wiley, New York.
DeWitt, B. (1965), Dynamical Theory of Groups and Fields, Wiley, New York.
DeWitt, B. (1967a), ‘Quantum theory of gravity. I. The canonical theory’, Phys. Rev. 160, 1113–1148.
DeWitt, B. (1967b), ‘Quantum theory of gravity. II. The manifestly covariant theory’, Phys. Rev. 160, 1195–1238.
DeWitt, B. (1967c), ‘Quantum theory of gravity. III. Applications of the covariant theory’, Phys. Rev. 160, 1239–1256.
Dirac, P. (1958a), ‘Generalized Hamiltonian dynamics’, Proc. Royal Soc. of London A246, 326–332.
Dirac, P. (1958b), ‘The theory of gravitation in Hamiltonian form’, Proc. Royal Soc. of London A246, 333–343.
Dirac, P. (1965), Lectures on Quantum Mechanics, Academic Press, New York.
Dowker, H. & Halliwell, J. (1992), ‘The quantum mechanics of history: The decoherence functional in quantum mechanics’, Phys. Rev.
Duff, M. (1981), Inconsistency of quantum field theory in a curved spacetime, in C. Isham, R. Penrose &D. Sciama, eds, ‘Quantum Gravity 2: A Second Oxford Symposium’, Clarendon Press, Oxford, pp. 81–105.
Earman, J. & Norton, J. (1987), ‘What price spacetime substantialism?’, Brit. Jour. Phil. Science 38, 515–525.
Fredenhagen, K. & Haag, R. (1987), ‘Generally covariant quantum field theory and scaling limits’, Comm. Math. Phys 108, 91–115.
Englert, F. (1989), ‘Quantum physics without time’, Phys. Lett. B228, 111–114.
Eppley, K. & Hannah, E. (1977), ‘The necessity of quantizing the gravitational field’, Found. Phys. 7, 51–68.
Fischer, A. & Marsden, J. (1979), The initial value problem and the dynamical formulation of general relativity, in S. Hawking & W. Israel, eds, ‘General Relativity: An Einstein Centenary Survey’, Cambridge University Press, Cambridge, pp. 138–211.
Friedman, J. Si Higuchi, A. (1989), ‘Symmetry and internal time on the superspace of asymptotically flat geometries’, Phys. Rev. D41, 2479–2486.
Fulling, S. (1990), ‘When is stability in the eye of the beholder? Comments on a singular initial value problem for a nonlinear differential equation arising in semiclassical cosmology’, Phys. Rev. D42, 4248 - 4250.
Gell-Mann, M. (1987), ‘Superstring theory—closing talk at the 2nd Nobel Symposium on Particle Physics’, Physica Scripta T15, 202–209.
Gell-Mann, M. Si Hartle, J. (1990a), Alternative decohering histories in quantum mechanics, in K. Phua Si Y. Yamaguchi, eds, ‘Proceedings of the 25th International Conference on High Energy Physics, Singapore, August, 2–8, 1990’, World Scientific, Singapore.
Gell-Mann, M. Si Hartle, J. (1990b), Quantum mechanics in the light of quantum cosmology, in S. Kobayashi, H. Ezawa, Y. Murayama & S. Nomura, eds, ’Proceedings of the Third International Symposium on the Foundations of Quantum Mechanics in the Light of New Technology’, Physical Society of Japan, Tokyo, pp. 321–343.
Gell-Mann, M. Si Hartle, J. (1990c), Quantum mechanics in the light of quantum cosmology, in W. Zurek, ed., ‘Complexity, Entropy and the Physics of Information, SFI Studies in the Science of Complexity, Vol. VIII’, Addison-Wesley, Reading, pp. 425–458.
Gell-Mann, M. Si Hartle, J. (1992), ‘Classical equations for quantum systems’. UCSB preprint UCSBTH-91–15.
Gerlach, U. (1969), ‘Derivation of the ten Einstein field equations from the semiclassical approximation to quantum geometrodynamics’, Phys. Rev. 117, 1929–1941.
Giddings, S. Si Strominger, A. (1988), ‘Baby universes, third quantization and the cosmological constant’, Nucl. Phys. B231, 481–508.
Greensite, J. (1990), ‘Time and probability in quantum cosmology’, Nucl. Phys. B342, 409–429.
Greensite, J. (1991a), ‘Ehrenfests’ principle in quantum-gravity’, Nucl. Phys. B351, 749–766.
Greensite, J. (1991b), ‘A model of quantum gravitational collapse’, Int. J. Mod. Phys. A6, 2693–2706.
Griffiths, R. (1984), ‘Consistent histories and the interpretation of quantum mechanics’, J. Stat. Phys. 36, 219–272.
Groenwold, H. (1946), ‘On the principles of elementary quantum mechanics’, Physica 12, 405–460.
Hajicek, P. & Kuchar, K. (1990a), ‘Constraint quantization of parametrized relativistic gauge systems in curved spacetimes’, Phys. Rev. D41, 1091–1104.
Hajicek, P. & Kuchaf, K. (1990b), ‘Transversal affine connection and quantisation of constrained systems’, J. Math. Phys. 31, 1723–1732.
Hâjfcek, P. (1986), ‘Origin of nonunitarity in quantum gravity’, Phys. Rev. D34, 1040–1048.
Hâjfcek, P. (1988), ‘Reducibility of parametrised systems’, Phys. Rev. D38, 3639–3647.
Hâjfcek, P. (1989), ‘Topology of parametrised systems’, J. Math. Phys. 20, 2488–2497.
Hâjfcek, P. (1990a), ‘Dirac quantisation of systems with quadratic constraints’, Class. Quan. Gray. 7, 871–886.
Hâjfcek, P. (1990b), ‘Topology of quadratic super-Hamiltonians’, Class. Quan. Gray. 7, 861–870.
Hâjfcek, P. (1991), ‘Comment on ‘Time in quantum gravity: An hypothesis“, Phys. Rev. D44, 1337–1338.
Halliwell, J. (1987), ‘Correlations in the wave function of the universe’, Phys. Rev. D36, 3626–3640.
Halliwell, J. (1988), ‘Derivation of the Wheeler-DeWitt equation from a path integral for minisuperspace models’, Phys. Rev. D38, 2468–2481.
Halliwell, J. (1989), ‘Decoherence in quantum cosmology’, Phys. Rev. D39, 2912–2923.
Halliwell, J. (1990), ‘A bibliography of papers on quantum cosmology’, Int. J. Mod. Phys. A5, 2473–2494.
Halliwell, J. (1991a), Introductory lectures on quantum cosmology, in S. Coleman, J. Hartle, T. Piran & S. Weinberg, eds, ‘Proceedings of the Jerusalem Winter School on Quantum Cosmology and Baby Universes’, World Scientific, Singapore.
Halliwell, J. (1991b), The Wheeler-DeWitt equation and the path integral in minisuperspace quantum cosmology, in A. Ashtekar & J. Stachel, eds, ‘Conceptual Problems of Quantum Gravity’, Birkhäuser, Boston, pp. 75–115.
Halliwell, J. (1992a), The interpretation of quantum cosmology models, in’Proceedings of the 13th International Conference on General Relativity and Gravitation, Cordoba, Argentina’.
Halliwell, J. (1992b), ‘Smeared Wigner functions and quantum-mechanical histories’, Phys. Rev.
Halliwell, J. Si Hawking, S. (1985), ‘Origin of structure in the universe’, Phys. Rev. D31, 1777–1791.
Halliwell, J., Perez-Mercander, J. Si Zurek, W., eds (1992), Physical Origins of Time Asymmetry, Cambridge University Press, Cambridge.
Hartle, J. (1986), Prediction and observation in quantum cosmology, in B. Carter & J. Hartle, eds, ‘Gravitation and Astrophysics, Cargese, 1986’, Plenum, New York.
Hartle, J. (1988a), ‘Quantum kinematics of spacetime. I. Nonrelativistic theory’, Phys. Rev. D37, 2818–2832.
Hartle, J. (1988b), ‘Quantum kinematics of spacetime. II. A model quantum cosmology with real clocks’, Phys. Rev. D38, 2985–2999.
Hartle, J. (1991a), The quantum mechanics of cosmology, in S. Coleman, P. Hartle, T. Piran Si S. Weinberg, eds, ‘Quantum Cosmology and Baby Universes’, World Scientific, Singapore.
Hartle, J. (1991b), ‘Spacetime grainings in nonrelativistic quantum mechanics’, Phys. Rev. D44, 3173–3195.
Hartle, J. & Hawking, S. (1983), ‘Wave function of the universe’, Phys. Rev. D28, 2960–2975.
Hawking, S. (1982), ‘The unpredictability of quantum gravity’, Comm. Math. Phys 87, 395–416.
Hawking, S. (1984a), Lectures in quantum cosmology, in B. DeWitt Si R. Stora, eds, ‘Relativity, Groups and Topology II’, North-Holland, Amsterdam, pp. 333–379.
Hawking, S. (1984b), ‘The quantum state of the universe’, Nucl. Phys. B239, 257–276.
Hawking, S. Si Ellis, G. (1973), The Large Scale Structure of Space-Time, Cambridge University Press, Cambridge.
Hawking, S. Si Page, D. (1986), ‘Operator ordering and the flatness of the universe’, Nucl. Phys. B264, 185–196.
Hawking, S. & Page, D. (1988), ‘How probable is inflation?’, Nucl. Phys. B298, 789–809.
Henneaux, M. & Teitelboim, C. (1989), ‘The cosmological constant and general covariance’, Phys. Lett. B222, 195–199.
Higgs, P. (1958), ‘Integration of secondary constraints in quantized general relativity’, Phys. Rev. Lett. 1, 373–375.
Hojman, S., Kuchar, K. Si Teitelboim, C. (1976), ‘Geometrodynamics regained’, Ann. Phys. (NY) 96, 88–135.
Horowitz, G. (1980), ‘Semiclassical relativity: The weak-field limit’, Phys. Rev. D21, 1445–1461.
Horowitz, G. (1981), Is flat space-time unstable?, in C. Isham, R. Penrose & D. Sciama, eds, ‘Quantum Gravity 2: A Second Oxford Symposium’, Clarendon Press, Oxford, pp. 106–130.
Horowitz, G. & Wald, R. (1978), ‘Dynamics of Einstein’s equations modified by a higher-derivative term’, Phys. Rev. D17, 414–416.
Horowitz, G. & Wald, R. (1980), ‘Quantum stress energy in nearly conformally flat spacetimes’, Phys. Rev. D21, 1462 - 1465.
Isenberg, J. & Marsden, J. (1976), ‘The York map is a canonical transformation’, Ann. Phys. (NY) 96, 88–135.
Isham, C. (1984), Topological and global aspects of quantum theory, inB. DeWitt & R. Stora, eds, ‘Relativity, Groups and Topology II’, North-Holland, Amsterdam, pp. 1062–1290.
Isham, C. (1985), Aspects of quantum gravity, inA. Davies &D. Sutherland, eds, ‘Superstrings and Supergravity: Proceedings of the 28th Scottish Universities Summer School in Physics, 1985’, SUSSP Publications, Edinburgh, pp. 1–94.
Isham, C. (1987), Quantum gravity—grgll review talk, inM. MacCallum, ed., ‘General Relativity and Gravitation: Proceedings of the 11th International Conference on General Relativity and Gravitation’, Cambridge University Press, Cambridge.
Isham, C. (1992), Conceptual and geometrical problems in quantum gravity, in H. Mitter & H. Gausterer, eds, ‘Recent Aspects of Quantum Fields’, Springer-Verlag, Berlin, pp. 123–230.
Isham, C. & Kakas, A. (1984a), ‘A group theoretical approach to the canonical quantisation of gravity: I. Construction of the canonical group’, Class. Quan. Gray. 1, 621–632.
Isham, C. Si Kakas, A. (1984b), ‘A group theoretical approach to the canonical quantisation of gravity: II. Unitary representations of the canonical group’, Class. Quan. Gray. 1, 633–650.
Isham, C. & Kuchat, K. (1985a), ‘Representations of spacetime diffeomorphisms. I. Canonical parametrised spacetime theories’, Ann Phys. (NY) 164, 288–315.
Isham, C. Si Kuchai, K. (1985b), ‘Representations of spacetime diffeomorphisms. II. Canonical geometrodynamics’, Ann. Phys. (NY) 164, 316–333.
Isham, C. Si Kuchar, K. (1994), Quantum Gravity and the Problem of Time. In preparation.
Jackiw, R. (1992a), Gauge theories for gravity on a line, in ‘Proceedings of NATO Advanced Study Institute ‘Recent Problems in Mathematical Physics, Salamanca, Spain, 1992“, Kluwer, The Netherlands.
Jackiw, R. (1992b), Update on planar gravity (ti physics of infinite cosmic strings), in H. Sato, ed., ‘Proceedings of the Sixth Marcel Grossman Meeting on General Relativity’, World Scientific, Singapore.
Joos, E. (1986), ‘Why do we observe a classical spacetime?’, Phys. Lett. A116, 6–8.
Joos, E. (1987), Quantum theory and the emergence of a classical world, in D. Greenberger, ed., ‘New Techniques and Ideas in Quantum Measurement’, New York Academic of Sciences, New York.
Joos, E. Si Zeh, H. (1985), ‘The emergence of classical properties through interaction with the environment’, Zeitschrift für Physik B59, 223–243.
Kaup, D. & Vitello, A. (1974), ‘Solvable quantum cosmological models and the importance of quantizing in a special canonical frame’, Phys. Rev. D9, 1648–1655.
Kibble, T. (1981), Is a semi-classical theory of gravity viable?, in C. Isham, R. Penrose &D. Sciama, eds, ‘Quantum Gravity 2: A Second Oxford Symposium’, Clarendon Press, Oxford, pp. 63–80.
Kiefer, C. (1987), ‘Continuous measurement of mini-superspace variables by higher multipoles’, Class. Quan. Gray. 4, 1369 - 1382.
Kiefer, C. (1989a), ‘Continuous measurement of intrinsic time by fermions’, Class. Quan. Gray. 6, 561–566.
Kiefer, C. (1989b), ‘Quantum gravity and Brownian motion’, Phys. Lett. A139, 201–203.
Kiefer, C. (1991), ‘Interpretation of the decoherence functional in quantum cosmology’, Class. Quan. Gray. 8, 379 - 392.
Kiefer, C. (1992), Decoherence in quantum cosmology, in ‘Proceedings of the Tenth Seminar on Relativistic Astrophysics and Gravitation, Postdam, 1991’, World Scientific, Singapore.
Kiefer, C. & Singh, T. (1991), ‘Quantum gravitational corrections to the functional Schrödinger equation’, Phys. Rev. D44, 1067–1076.
Klauder, J. (1970), Soluble models of guantum gravitation, in M. Carmeli, S. Flicker & Witten, eds, ‘Relativity’, Plenum, New York.
Komar, A. (1979a), ‘Consistent factoring ordering of general-relativistic constraints’, Phys. Rev. D19, 830–833.
Komar, A. (1979b), ‘Constraints, hermiticity, and correspondence’, Phys. Rev. D19, 2908–2912.
Kuchar, K. (1970), ‘Ground state functional of the linearized gravitational field’, J. Math. Phys. 11, 3322–3344.
Kuchar, K. (1971), ‘Canonical quantization of cylindrical gravitational waves’, Phys. Rev. D4, 955–985.
Kuchar, K. (1972), ‘A bubble-time canonical formalism for geometrodynamics’, J. Math. Phys. 13, 768–781.
Kuchar, K. (1973), Canonical quantization of gravity, in ‘Relativity, Astrophysics and Cosmology’, Reidel, Dordrecht, pp. 237–288.
Kuchar, K. (1974), ‘Geometrodynamics regained: A Lagrangian approach’, J. Math. Phys. 15, 708–715.
Kuchat, K. (1976a), ‘Dynamics of tensor fields in hyperspace III.’, J. Math. Phys. 17, 801–820.
Kuchar, K. (1976b), ‘Geometry of hyperspace I.’, J. Math. Phys. 17, 777–791.
Kuchar, K. (1976c), ‘Kinematics of tensor fields in hyperspace II.’, J. Math. Phys. 17, 792–800.
Kuchar, K. (1977), ‘Geometrodynamics with tensor sources IV’, J. Math. Phys.18, 1589–1597.
Kuchar, K. (1981a), Canonical methods of quantisation, in C. Isham, R. Penrose & D. Sciama, eds, ‘Quantum Gravity 2: A Second Oxford Symposium’, Clarendon Press, Oxford, pp. 329–374.
Kuchat, K. (1981b), ‘General relativity: Dynamics without symmetry’, J. Math. Phys. 22, 2640–2654.
Kuchat, K. (1982), ‘Conditional symmetries in parametrized field theories’, J. Math. Phys. 25, 1647–1661.
Kuchai, K. (1986a), ‘Canonical geometrodynamics and general covariance’, Found. Phys. 16, 193–208.
Kuchai, K. (1986b), ‘Covariant factor ordering for gauge systems’, Phys. Rev. D34, 3044–3057.
Kuchai, K. (1986c), ‘Hamiltonian dynamics of gauge systems’, Phys. Rev. D34, 3031–3043.
Kuchai, K. (1987), ‘Covariant factor ordering of constraints may be ambiguous’, Phys. Rev. D35, 596–599.
Kuchai, K. (1991a), ‘Does an unspecified cosmological constant solve the problem of time in quantum gravity?’, Phys. Rev. D43, 3332–3344.
Kuchai, K. (1991b), The problem of time in canonical quantization, in A. Ashtekar & J. Stachel, eds, ‘Conceptual Problems of Quantum Gravity’, Birkhäuser, Boston, pp. 141–171.
Kuchat, K. (1992a), ‘Extrinsic curvature as a reference fluid in canonical gravity’, Phys. Rev. D45, 4443–4457.
Kuchai, K. (1992b), Time and interpretations of quantum gravity, in‘Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics’, World Scientific, Singapore.
Kuchai, K. & Ryan, M. (1989), ‘Is minisuperspace quantization valid? Taub in Mixmaster’, Phys. Rev. D40, 3982–3996.
Kuchai, K. Si Torre, C. (1989), ‘World sheet diffeomorphisms and the cylindrical string’, J. Math. Phys. 30, 1769–1793.
Kuchai, K. & Torre, C. (1991a), ‘Gaussian reference fluid and the interpretation of geometrodynamics’, Phys. Rev. D43, 419–441.
Kuchat, K. & Torre, C. (1991b), ‘Harmonic gauge in canonical gravity’, Phys. Rev. D44, 3116–3123.
Kuchai, K. & Torre, C. (1991c), Strings as poor relatives of general relativity, in A. Ashtekar Si J. Stachel, eds, ‘Conceptual Problems of Quantum Gravity’, Birkhäuser, Boston, pp. 326–348.
Lapchinski, V. & Rubakov, V. (1979), ‘Canonical quantization of gravity and quantum field theory in curved spacetime’, Acta. Phys. Pol. B10, 1041.
Lee, J. & Wald, R. (1990), Local symmetries and constraints’, J. Math. Phys. 31, 725–743.
Linden, N. & Perry, M. (1991), ‘Path integrals and unitarity in quantum cosmology’, Nucl. Phys. B357, 289–307.
Manojlović & Miković (1992), ‘Gauge fixing and independent canonical variables in the ashtekar formulation of general relativity’.
McGuigan, M. (1988), ‘Third quantization and the Wheeler-DeWitt equation’, Phys. Rev. D38, 3031–3051.
McGuigan, M. (1989), ‘Universe creation from the third quantized vacuum’, Phys. Rev. D39, 2229–2233.
Mellor, F. (1991), ‘Decoherence in quantum Kaluza-Klein theories’, Nucl. Phys. B353, 291–301.
Misner, C. (1957), ‘Feynman quantization of general relativity’, Rev. Mod. Phys. 29, 497–509.
Misner, C. (1992), Minisuperspace, inJ. Klauder, ed., ‘Magic without Magic: John Archibald Wheeler, A Collection of Essays in Honor of his 60th Birthday’, Freeman, San Francisco.
Moller, C. (1962), The energy-momentum complex in general relativity and related problems, in A. Lichnerowicz & M. Tonnelat, eds, ‘Les. Théories Relativistes de la Gravitation’, CNRS, Paris.
Morikawa, M. (1989), ‘Evolution of the cosmic density matrix’, Phys. Rev. D40, 4023–4027.
Newman, E. & Rovelli, C. (1992), ‘Generalized lines of force as the gauge invariant degrees of freedom for general relativity and Yang-Mills theory’. University of Pittsburgh preprint.
Omnès, R. (1988a), ‘Logical reformulation of quantum mechanics. I. Foundations’, J. Stat. Phys. 53, 893–932.
Omnès, R. (1988b), ‘Logical reformulation of quantum mechanics. II. Interferences and the Einstein-Podolsky-Rosen experiment’, J. Stat. Phys. 53, 933–955.
Omnès, R. (1988c), ‘Logical reformulation of quantum mechanics. III. Classical limit and irreversibility’, J. Stat. Phys. 53, 957–975.
Omnès, R. (1989), ‘Logical reformulation of quantum mechanics. III. Projectors in semiclassical physics’, J. Stat. Phys. 57, 357–382.
Omnès, R. (1990), ‘From Hilbert space to common sense: A synthesis of recent progress in the interpretation of quantum mechanics’, Ann. Phys. (NY) 201, 354–447.
Omnès, R. (1992), ‘Consistent interpretations of quantum mechanics’, Rev. Mod. Phys. 64, 339–382.
Padmanabhan, T. (1989a), ‘Decoherence in the density matrix describing quantum three-geometries and the emergence of classical spacetime’, Phys. Rev. D39, 2924–2932.
Padmanabhan, T. (1989b), ‘Semiclassical approximations to gravity and the issue of back-reaction’, Class. Quan. Gray. 6, 533–555.
Padmanabhan, T. (1990), ‘A definition for time in quantum cosmology’, Pramana Jour. Phys. 35, L199–L204.
Page, D. (1986a), ‘Density matrix of the universe’, Phys. Rev. D34, 2267–2271.
Page, D. (1986b), Hawking’s wave function for the universe, in R. Penrose &C. Isham, eds, ‘Quantum Concepts in Space and Time’, Clarendon Press, Oxford, pp. 274–285.
Page, D. (1989), ‘Time as an inaccessible observable’. ITP preprint NSF-ITP-89–18.
Page, D. (1991), Intepreting the density matrix of the universe, in A. Ashtekar & J. Stachel, eds, ‘Conceptual Problems of Quantum Gravity’, Birkhäuser, Boston, pp. 116–121.
Page, D. & Geilker, C. (1981), ‘Indirect evidence for quantum gravity’, Phys. Rev. Lett. 47, 979–982.
Page, D. Si Hotke-Page, C. (1992), Clock time and entropy, in J. Halliwell, J. PerezMercander & W. Zurek, eds, ‘Physical Origins of Time Asymmetry’, Cambridge University Press, Cambridge.
Page, D. & Wooters, W. (1983), ‘Evolution without evolution: Dynamics described by stationary observables’, Phys. Rev. D27, 2885–2892.
Pilati, M. (1982), ‘Strong coupling quantum gravity I: solution in a particular gauge’, Phys. Rev. D26, 2645–2663.
Pilati, M. (1983), ‘Strong coupling quantum gravity I: solution without gauge fixing’, Phys. Rev. D28, 729–744.
Randjbar-Daemi, S., Kay, B. Si Kibble, T. (1980), ‘Renormalization of semi-classical field theories’, Phys. Rev. Lett. 91B, 417–420.
Rosenfeld, L. (1963), ‘On quantization of fields’, Nucl. Phys. 40, 353–356.
Rovelli, C. (1990), ‘Quantum mechanics without time: A model’, Phys. Rev. D42, 2638–2646.
Rovelli, C. (1991a), ‘Ashtekar formulation of general relativity and loop-space nonperturbative quantum gravity: A report.’. Pittsburgh University preprint.
Rovelli, C. (1991b), Is there incompatibility between the ways time is treated in general relativity and in standard quantum theory?, in A. Ashtekar Si J. Stachel, eds, ‘Conceptual Problems of Quantum Gravity’, Birkhäuser, Boston, pp. 126–140.
Rovelli, C. (1991c), ‘Quantum evolving constants. Reply to ‘Comments on time in quantum gravity: An hypothesis“, Phys. Rev. D44, 1339–1341.
Rovelli, C. (1991d), ‘Statistical mechanics of gravity and the problem of time’. University of Pittsburgh preprint.
Rovelli, C. (1991e), ‘Time in quantum gravity: An hypothesis’, Phys. Rev. D43, 442–456.
Rovelli, C. & Smolin, L. (1990), ‘Loop space representation of quantum general relativity’, Nucl. Phys. B331, 80–152.
Salisbury, D. & Sundermeyer, K. (1983), ‘Realization in phase space of general coordinate transformations’, Phys. Rev. D27, 740–756.
Simon, J. (1991), ‘The stability of flat space, semiclassical gravity and higher derivatives’, Phys. Rev. D43, 3308–3316.
Singer, I. (1978), ‘Some remarks on the Gribov ambiguity’, Comm. Math. Phys 60, 7–12.
Singh, T. Si Padmanabhan, T. (1989), ‘Notes on semiclassical gravity’, Ann. Phys. (NY) 196, 296–344.
Smarr, L. Si York, J. (1978), ‘Kinematical conditions in the construction of space-time’, Phys. Rev. D17, 2529–2551.
Smolin, L. (1992), ‘Recent c gvelopments in nonperturbative quantum gravity’. Syracuse University Preprint.
Squires, E. (1991), ‘The dynamical role of time in quantum cosmology’, Phys. Lett. A155, 357–360.
Stachel, J. (1989), Einstein’s search for general covariance, 1912–1915, in D. Howard & J. Stachel, eds, ‘Einstein and the History of General Relativity: Vol I, Birkhäuser, Boston, pp. 63–100.
Stone, C. Si Kuchar, K. (1992), ‘Representation of spacetime diffeomorphisms in canonical geometrodynamics under harmonic coordinate conditions’, Class. Quan. Gray. 9, 757–776.
Suen, W. (1989a), ‘Minkowski spacetime is unstable in semi-classical gravity’, Phys. Rev. Lett. 62, 2217–2226.
Suen, W. (1989b), ‘Stability of the semiclassical Einstein equation’, Phys. Rev. D40, 315–326.
Tate, R. (1992), ‘An algebraic approach to the quantization of constrained systems: Finite dimensional examples’. PhD. Dissertation.
Teitelboim, C. (1982), ‘Quantum mechanics of the gravitational field’, Phys. Rev. D25, 3159–3179.
Teitelboim, C. (1983a), ‘Causality versus gauge invariance in quantum gravity and supergravity’, Phys. Rev. Lett. 50, 705–708.
Teitelboim, C. (1983b), ‘Proper time gauge in the quantum theory of gravitation’, Phys. Rev. D28, 297–309.
Teitelboim, C. (1983c), ‘Quantum mechanics of the gravitational field’, Phys. Rev. D25, 3159–3179.
Teitelboim, C. (1983d), ‘Quantum mechanics of the gravitational field in asymptotically flat space’, Phys. Rev. D28, 310–316.
Teitelboim, C. (1992), in‘Proceedings of NATO Advanced Study Institute ‘Recent Problems in Mathematical Physics, Salamanca, Spain, 1992“, Kluwer, The Netherlands.
Torre, C. (1989), ‘Hamiltonian formulation of induced gravity in two dimensions’, Phys. Rev. D40, 2588–2597.
Torre, C. (1991), ‘A complete set of observables for cylindrically symmetric gravitational fields’, Class. Quan. Gray. 8, 1895–1911.
Torre, C. (1992), ‘Is general relativity an “already parametrised” theory?’.
Unruh, W. (1988), Time and quantum gravity, in M. Markov, V. Bérezin Si V. Frolov, eds, ‘Proceedings of the Fourth Seminar on Quantum Gravity’, World Scientific, Singapore, pp. 252–268.
Unruh, W. (1989), ‘Unimodular theory of canonical quantum gravity’, Phys. Rev. D40, 1048–1052.
Unruh, W. (1991), Loss of quantum coherence for a damped oscillator, in A. Ashtekar Si J. Stachel, eds, ‘Conceptual Problems of Quantum Gravity’, Birkhäuser, Boston, pp. 67–74.
Unruh, W. & Wald, R. (1989), ‘Time and the interpretation of quantum gravity’, Phys. Rev. D40, 2598–2614.
Unruh, W. & Zurek, W. (1989), ‘Reduction of the wave-packet in quantum Brownian motion’, Phys. Rev. D40, 1071–1094.
Valentini, A. (1992), ‘Non-local hidden-variables and quantum gravity’. SISSA preprint 105/92/A.
Van Hove, L. (1951), ‘On the problem of the relations between the unitary transformations of quantum mechanics and the canonical transformations of classical mechanics’, Acad. Roy. Belg. 37, 610–620.
Vilenkin, A. (1988), ‘Quantum cosmology and the initial state of the universe’, Phys. Rev. D39, 888–897.
Vilenkin, A. (1989), ‘Interpretation of the wave function of the universe’, Phys. Rev. D39, 1116–1122.
Vink, J. (1992), ‘Quantum potential interpretation of the wavefunction of the universe’, Nucl. Phys. B369, 707–728.
Wheeler, J. (1962), Neutrinos, gravitation and geometry, in‘Topics of Modern Physics. Vol 1’, Academic Press, New York, pp. 1–130.
Wheeler, J. (1964), Geometrodynamics and the issue of the final state, inC. DeWitt & B. DeWitt, eds, ‘Relativity, Groups and Topology’, Gordon and Breach, New York and London, pp. 316–520.
Wheeler, J. (1968), Superspace and the nature of quantum geometrodynamics, in C. DeWitt Si J. Wheeler, eds, ‘Batelle Rencontres: 1967 Lectures in Mathematics and Physics’, Benjamin, New York, pp. 242–307.
Wooters, W. (1984), ‘Time replaced by quantum corrections’, Int. J. Theor. Phys. 23, 701–711.
York, J. (1972a), ‘Mapping onto solutions of the gravitational initial value problem’, J. Math. Phys. 13, 125–130.
York, J. (1972b), ‘Role of conformal three-geometry in the dynamics of gravitation’, Phys. Rev. Lett.28, 1082–1085.
York, J. (1979), Kinematics and dynamics of general relativity, inL. Smarr, ed., ‘Sources of Gravitational Radiation’, Cambridge University Press, Cambridge, pp. 83–126.
Zeh, H. (1986), ‘Emergence of classical time from a universal wave function’, Phys. Lett.A116, 9–12.
Zeh, H. (1988), ‘Time in quantum gravity’, Phys. Lett. Al26, 311–317.
Zurek, W. (1981), ‘Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?’, Phys. Rev.D24, 1516–1525.
Zurek, W. (1982), ‘Environment-induced superselection rules’, Phys. Rev. D26, 1862–1880.
Zurek, W. (1983), Information transer in quantum measurements:Irreversibility and amplification, in P. Meystre Si M. Scully, eds, ‘Quantum Optics, Experimental Gravity, and Measurement Theory’, Plenum, New York, pp. 87–116.
Zurek, W. (1986), Reduction of the wave-packet: How long does it take?, in G. Moore Si M. Scully, eds, ‘Frontiers of Nonequilibrium Statistical Physics’, Plenum, New York, pp. 145–149.
Zurek, W. (1991), Quantum measurements and the environment-induced transition from quantum to classical, in A. Ashtekar &J. Stachel, eds, ‘Conceptual Problems of Quantum Gravity’, Birkhäuser, Boston, pp. 43–66.
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Isham, C.J. (1993). Canonical Quantum Gravity and the Problem of Time. In: Ibort, L.A., Rodríguez, M.A. (eds) Integrable Systems, Quantum Groups, and Quantum Field Theories. NATO ASI Series, vol 409. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-1980-1_6
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