No Access

TIME AND INTERPRETATIONS OF QUANTUM GRAVITY

KAREL V. KUCHAŘ

Blackett Laboratory, Imperial College of Science and Technology, South Kensington, London SW7 2BZ, UK

https://doi.org/10.1142/S0218271811019347Cited by:152 (Source: Crossref)

Abstract

In canonical quantization of gravity, the state functional does not seem to depend on time. This hampers the physical interpretation of quantum gravity. I critically examine ten major attempts to circumvent this problem and discuss their shortcomings.

Originally published in the Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics, eds. G. Kunstatter, D. Vincent and J. Williams (World Scientific, Singapore, 1992).

You currently do not have access to the full text article.
Recommend the journal to your library today!

References

A. Gryphius, Gesamtausgabe der deutschsprachigen Werke 2, eds. M. Szyrocki and H. Powell (Niemeyer, Tübingen, 1964) pp. 182–183. Google Scholar
R. F. Baierlein, D. H. Sharp and J. A. Wheeler, Phys. Rev. 126, 1864 (1962), DOI: 10.1103/PhysRev.126.1864. Crossref, Web of Science, Google Scholar
K. V. Kuchař, J. Math. Phys. 13, 768 (1972). Crossref, Web of Science, ADS, Google Scholar
A. Peres, Nuovo Cimento 26, 53 (1962). Crossref, Web of Science, Google Scholar
U. Gerlach, Phys. Rev. 177, 1929 (1969), DOI: 10.1103/PhysRev.177.1929. Crossref, Web of Science, Google Scholar
C. Lanczos, The Variational Principles of Mechanics, 4th edn. (Univ. of Toronto Press, Toronto, 1970) pp. 264–276. Google Scholar
C. Lanczos, The Variational Principles of Mechanics, 4th edn. (Univ. of Toronto Press, Toronto, 1970) pp. 262–264. Google Scholar
C. Lanczos, The Variational Principles of Mechanics, 4th edn. (Univ. of Toronto Press, Toronto, 1970) pp. 185–192. Google Scholar
P. Hájíček, Phys. Rev. D 34, 1040 (1986). Crossref, Web of Science, Google Scholar
P. Hájíček, J. Math. Phys. 30, 2488 (1989). Crossref, Web of Science, ADS, Google Scholar
P. Hájíček, Class. Quantum Grav. 7, 871 (1990). Crossref, Web of Science, ADS, Google Scholar
M. Schön and P. Hájíček, Class. Quantum Grav. 7, 861 (1990). Crossref, Web of Science, ADS, Google Scholar
P. Hájíček, Lecture Notes on Quantum Cosmology (University of Bern, Bern, 1990); unpublished . Google Scholar
C. Carathéodory , Variationsrechnung und partielle Differentialgleichungen erster Ordnung I ( Teubner , Leipzig , 1956 ) . Google Scholar
J. L. Synge , Handbuch der Physik III , ed. S. Flügge ( Springer , Berlin , 1960 ) . Google Scholar
A. Peres and N. Rosen, Nuovo Cimento 13, 430 (1959). Crossref, Web of Science, Google Scholar
A. Peres, Nuovo Cimento 18, 32 (1960). Crossref, Web of Science, Google Scholar
J. A. Wheeler , Relativity, Groups, and Topology , eds. ( Gordon and Breach , New York , 1964 ) . Google Scholar
J. A. Wheeler , Gravitation and Relativity , eds. H.-Y. Chiu and W. F. Hoffmann ( Benjamin , New York , 1964 ) . Google Scholar
E. P. Belasco and H. Ohanian, J. Math. Phys. 10, 1503 (1969), DOI: 10.1063/1.1664998. Crossref, Web of Science, ADS, Google Scholar
P. G. Bergmann , Relativity , eds. M. Carmeli , S. Fickler and L. Witten ( Plenum , New York , 1980 ) . Google Scholar
A. Komar, J. Math. Phys. 11, 820 (1970), DOI: 10.1063/1.1665216. Crossref, Web of Science, ADS, Google Scholar
C. M. Pereira, J. Math. Phys. 14, 1498 (1973), DOI: 10.1063/1.1666215. Crossref, Web of Science, ADS, Google Scholar
D. Christodoulou and M. Francaviglia , Enrico Fermi Proceedings , ed. J. Ehlers ( North Holland , New York , 1979 ) . Google Scholar
C. M. Pereira, J. Math. Phys. 22, 1064 (1980), DOI: 10.1063/1.525013. Crossref, Web of Science, Google Scholar
P. A. M. Dirac , Lectures on Quantum Mechanics ( Yeshiva Univ. , New York , 1964 ) . Google Scholar
P. A. M. Dirac, Can. J. Math. 3, 1 (1951), DOI: 10.4153/CJM-1951-001-2. Crossref, Web of Science, Google Scholar
K. V. Kuchař and C. G. Torre, Phys. Rev. D 43, 419 (1990). Crossref, Web of Science, ADS, Google Scholar
C. W. Misner, Rev. Mod. Phys. 29, 497 (1957), DOI: 10.1103/RevModPhys.29.497. Crossref, Web of Science, ADS, Google Scholar
P. W. Higgs, Phys. Rev. Lett. 1, 373 (1958), DOI: 10.1103/PhysRevLett.1.373. Crossref, Web of Science, ADS, Google Scholar
J. A. Wheeler , Batelle Rencontres: 1967 Lectures in Mathematics and Physics , eds. C. DeWitt and J. A. Wheeler ( Benjamin , New York , 1968 ) . Google Scholar
B. S. DeWitt, Phys. Rev. 160, 1113 (1967), DOI: 10.1103/PhysRev.160.1113. Crossref, Web of Science, ADS, Google Scholar
K. V. Kuchař , Quantum Gravity 2: A Second Oxford Symposium , eds. C. J. Isham , R. Penrose and D. W. Sciama ( Clarendon , Oxford , 1981 ) . Google Scholar
C. J. Isham , Relativity, Groups and Topology II: Les Houches 1983 , eds. B. S. DeWitt and R. Stora ( North-Holland , Amsterdam , 1984 ) . Google Scholar
K. V. Kuchař, Phys. Rev. D 4, 955 (1971), DOI: 10.1103/PhysRevD.4.955. Crossref, Web of Science, ADS, Google Scholar
R. H. Gowdy, Ann. Phys. (N.Y.) 83, 203 (1974), DOI: 10.1016/0003-4916(74)90384-4. Crossref, Web of Science, ADS, Google Scholar
C. W. Misner, Phys. Rev. D 8, 3271 (1973), DOI: 10.1103/PhysRevD.8.3271. Crossref, Web of Science, ADS, Google Scholar
B. Berger, Phys. Rev. D 11, 2770 (1975), DOI: 10.1103/PhysRevD.11.2770. Crossref, Web of Science, ADS, Google Scholar
B. Berger, Phys. Lett. B 108, (1982). Google Scholar
B. Berger, Ann. Phys. (N.Y.) 156, 155 (1984), DOI: 10.1016/0003-4916(84)90214-8. Crossref, Web of Science, ADS, Google Scholar
V. Husain, Class. Quantum Grav. 4, 1587 (1987), DOI: 10.1088/0264-9381/4/6/017. Crossref, Web of Science, ADS, Google Scholar
M. O. Green , J. Schwarz and E. Witten , Superstring Theory ( Cambridge Univ. Press , Cambridge , 1987 ) . Google Scholar
L. Brink and M. Henneaux , Principles of String Theory ( Plenum , New York , 1988 ) . Crossref, Google Scholar
K. V. Kuchař and C. G. Torre , Conceptual Problems of Quantum Gravity , eds. A. Ashtekar and J. Stachel ( Birkhäuser , Boston , 1991 ) . Google Scholar
P. A. M. Dirac, Can. J. Math. 3, 1 (1951), DOI: 10.4153/CJM-1951-001-2. Crossref, Web of Science, Google Scholar
K. V. Kuchař, J. Math. Phys. 17, 792 (1976). Crossref, Web of Science, ADS, Google Scholar
K. V. Kuchař, J. Math. Phys. 17, 801 (1976). Crossref, Web of Science, ADS, Google Scholar
J. J. Halliwell and S. W. Hawking, Phys. Rev. D 31, 1777 (1985), DOI: 10.1103/PhysRevD.31.1777. Crossref, Web of Science, ADS, Google Scholar
C. W. Misner , Magic Without Magic: John Archibald Wheeler , A Collection of Essays in Honor of his 60th Birthday , ed. J. Klauder ( Freeman , San Francisco , 1972 ) . Google Scholar
M. Ryan , Hamiltonian Cosmology ( Springer , Berlin , 1972 ) . Google Scholar
M. A. H. MacCallum , Quant. Gravity , eds. C. J. Isham , R. Penrose and D. W. Sciama ( Clarendon , Oxford , 1975 ) . Google Scholar
J. J. Halliwell, Int. J. Mod. Phys. A 5, 2473 (1990). Link, Web of Science, ADS, Google Scholar
C. W. Misner, Phys. Rev. Lett. 22, 1071 (1969), DOI: 10.1103/PhysRevLett.22.1071. Crossref, Web of Science, Google Scholar
C. W. Misner, Phys. Rev. 186, 1328 (1969), DOI: 10.1103/PhysRev.186.1328. Crossref, Web of Science, ADS, Google Scholar
C. W. Misner , Relativity , eds. M. Carmeli , S. Fickler and L. Witten ( Plenum , New York , 1970 ) . Google Scholar
J. Abbot, S. Giddings and K. V. Kuchař, Gen. Rel. Grav. 16, 751 (1984). Web of Science, Google Scholar
E. Witten, Nucl. Phys. B 311, 46 (1988), DOI: 10.1016/0550-3213(88)90143-5. Crossref, Web of Science, ADS, Google Scholar
E. Witten, Nucl. Phys. B 323, 113 (1989), DOI: 10.1016/0550-3213(89)90591-9. Crossref, Web of Science, ADS, Google Scholar
V. Moncrief, J. Math. Phys. 30, 2907 (1989), DOI: 10.1063/1.528475. Crossref, Web of Science, Google Scholar
A. Hosoya and K. Nakao, Class. Quantum Grav. 7, 163 (1990), DOI: 10.1088/0264-9381/7/2/010. Crossref, Web of Science, ADS, Google Scholar
S. Carlip, Phys. Rev. D 42, 2647 (1990), DOI: 10.1103/PhysRevD.42.2647. Crossref, Web of Science, ADS, Google Scholar
S. Carlip, Preprint UCDPHYS-PUB-23-90 . Google Scholar
G. Nordström, Phys. Z 13, 1126 (1912). Google Scholar
G. Nordström, Ann. Phys. (Leipzig) 42, 533 (1913). Google Scholar
A. Einstein and A. D. Fokker, Ann. Phys. (Leipzig) 44, 321 (1914). ADS, Google Scholar
P. Hájíček and K. V. Kuchař, Phys. Rev. D 41, 1091 (1990), DOI: 10.1103/PhysRevD.41.1091. Crossref, Web of Science, Google Scholar
K. V. Kuchař, Phys. Rev. D 22, 1285 (1980), DOI: 10.1103/PhysRevD.22.1285. Crossref, Web of Science, ADS, Google Scholar
J. B. Hartle and K. V. Kuchař, J. Math. Phys. 25, 57 (1984), DOI: 10.1063/1.525998. Crossref, Web of Science, ADS, Google Scholar
J. B. Barbour, Nature 249, 328 (1974). Crossref, Web of Science, ADS, Google Scholar
J. B. Barbour, Nature 250, 606 (1974). Crossref, Web of Science, ADS, Google Scholar
J. B. Barbour, Nuovo Cimento B 26, 16 (1975), DOI: 10.1007/BF02755533. Crossref, Web of Science, ADS, Google Scholar
J. B. Barbour and B. Bertotti, Nuovo Cimento B 38, 1 (1977), DOI: 10.1007/BF02726208. Crossref, Web of Science, ADS, Google Scholar
J. B. Barbour and B. Bertotti, Proc. Roy. Soc. London A 382, 295 (1982). Crossref, Web of Science, ADS, Google Scholar
W. Pauli , Handbuch d. Physik V , ed. S. Flugge ( Springer , Berlin , 1958 ) . Google Scholar
J. B. Hartle, Phys. Rev. D 38, 2985 (1988), DOI: 10.1103/PhysRevD.38.2985. Crossref, Web of Science, Google Scholar
W. G. Unruh and R. M. Wald, Phys. Rev. D 40, 2598 (1989), DOI: 10.1103/PhysRevD.40.2598. Crossref, Web of Science, ADS, Google Scholar
K. V. Kuchař, Phys. Rev. D 39, 1579 (1989), DOI: 10.1103/PhysRevD.39.1579. Crossref, Web of Science, ADS, Google Scholar
K. V. Kuchař, Phys. Rev. D 39, 2263 (1989), DOI: 10.1103/PhysRevD.39.2263. Crossref, Web of Science, ADS, Google Scholar
K. V. Kuchař , Highlights in Gravitation and Cosmology , eds. B. R. Iyer et al. ( Cambridge Univ. Press , Cambridge , 1988 ) . Google Scholar
G. Torre, Class. Quantum Grav. 8, 1895 (1991), DOI: 10.1088/0264-9381/8/10/015. Crossref, Web of Science, Google Scholar
K. V. Kuchař and C. G. Torre, J. Math. Phys. 30, 1769 (1989). Crossref, Web of Science, ADS, Google Scholar
D. G. Boulware and S. Deser, J. Math. Phys. 8, 1468 (1967), DOI: 10.1063/1.1705368. Crossref, Web of Science, ADS, Google Scholar
J. Anderson, Phys. Rev. 114, 1182 (1959), DOI: 10.1103/PhysRev.114.1182. Crossref, Web of Science, ADS, Google Scholar
J. Anderson , Proceedings of the First Eastern Theoretical Physics Conference , ed. M. E. Ross ( Gordon and Breach , New York , 1962 ) . Google Scholar
J. Schwinger, Phys. Rev. 132, 1317 (1963), DOI: 10.1103/PhysRev.132.1317. Crossref, Web of Science, ADS, Google Scholar
P. A. M. Dirac , Contemporary Physics: Trieste Symposium 1968 1 , ed. L. Fonda ( IAEA , Vienna , 1969 ) . Google Scholar
A. Komar, Phys. Rev. D 19, 2908 (1979), DOI: 10.1103/PhysRevD.19.2908. Crossref, Web of Science, Google Scholar
A. Komar, Phys. Rev. D 20, 830 (1979), DOI: 10.1103/PhysRevD.20.830. Crossref, Web of Science, ADS, Google Scholar
A. Ashtekar, Phys. Rev. D 36, 1587 (1987), DOI: 10.1103/PhysRevD.36.1587. Crossref, Web of Science, ADS, Google Scholar
N. C. Tsamis and R. P. Woodard, Phys. Rev. D 36, 3641 (1987), DOI: 10.1103/PhysRevD.36.3641. Crossref, Web of Science, ADS, Google Scholar
J. L. Friedman and I. Jack, Phys. Rev. D 37, 3495 (1988), DOI: 10.1103/PhysRevD.37.3495. Crossref, Web of Science, ADS, Google Scholar
J. L. Friedman and I. Jack , Conceptual Problems of Quantum Gravity , eds. A. Ashtekar and J. Stachel ( Birkhäuser , Boston , 1991 ) . Google Scholar
P. Hájíček and K. V. Kuchař, J. Math. Phys. 31, 1723 (1990). Crossref, Web of Science, ADS, Google Scholar
P. A. M. Dirac, Phys. Rev. 114, 924 (1959), DOI: 10.1103/PhysRev.114.924. Crossref, Web of Science, ADS, Google Scholar
K. V. Kuchař , Conceptual Problems of Quantum Gravity , eds. A. Ashtekar and J. Stachel ( Birkhäuser , Boston , 1991 ) . Google Scholar
C. J. Isham , Recent Developments of Quantum Fields , eds. H. Mitter and H. Gaustener ( Springer Verlag , 1992 ) . Google Scholar
J. B. Barbour and L. Smolin, Yale University Preprint April 8, 1988 . Google Scholar
J. J. Halliwell, Phys. Rev. D 43, 2590 (1991), DOI: 10.1103/PhysRevD.43.2590. Crossref, Web of Science, ADS, Google Scholar
R. Arnowitt , S. Deser and C. W. Misner , Gravitation: An Introduction to Current Research , ed. L. Witten ( Wiley , New York , 1962 ) . Google Scholar
R. Armowitt, S. Deser and C. W. Misner, Phys. Rev. 117, 1595 (1960). Crossref, Web of Science, ADS, Google Scholar
K. V. Kuchař, J. Math. Phys. 11, 3322 (1970). Crossref, ADS, Google Scholar
K. V. Kuchař , Relativity, Astrophysics and Cosmology , ed. W. Israel ( Reidel , Dordrecht , 1973 ) . Crossref, Google Scholar
J. W. York, Phys. Rev. Lett. 28, 1082 (1972), DOI: 10.1103/PhysRevLett.28.1082. Crossref, Web of Science, Google Scholar
J. W. York , Sources of Gravitational Radiation , ed. L. Smarr ( Cambridge Univ. Press , Cambridge , 1979 ) . Google Scholar
Y. Choquet-Bruhat and J. W. York , General Relativity and Gravitation the Hundred Years After the Birth of Albert Einstein , eds. P. Bergmann , J. Goldberg and A. Held ( Plenum , New York , 1980 ) . Google Scholar
I. M. Singer, Commun. Math. Phys. 60, 7 (1978), DOI: 10.1007/BF01609471. Crossref, Web of Science, ADS, Google Scholar
D. N. Page , Quantum Concepts in Space and Time , eds. R. Penrose and C. J. Isham ( Clarendon , Oxford , 1986 ) . Google Scholar
Y. Choquet-Bruhat, C. R. Acad. Sci. (Paris) 280A, 169 (1975). Google Scholar
D. Brill and F. Flaherty, Commun. Math. Phys. 50, 157 (1976), DOI: 10.1007/BF01617993. Crossref, Web of Science, ADS, Google Scholar
A. J. Goddard, Commun. Math. Phys. 54, 279 (1977), DOI: 10.1007/BF01614089. Crossref, Web of Science, ADS, Google Scholar
L. L. Smarr and J. W. York, Phys. Rev. D 17, 2529 (1978), DOI: 10.1103/PhysRevD.17.2529. Crossref, Web of Science, ADS, Google Scholar
D. M. Eardley and L. L. Smarr, Phys. Rev. D 19, 2239 (1979), DOI: 10.1103/PhysRevD.19.2239. Crossref, Web of Science, ADS, Google Scholar
J. E. Marsden and F. J. Tipler, Physics Rep. 66, 109 (1980), DOI: 10.1016/0370-1573(80)90154-4. Crossref, Web of Science, ADS, Google Scholar
J. Isenberg and J. E. Marsden, J. Geom. Phys. 1, 85 (1984). Crossref, Web of Science, Google Scholar
W. Blyth and C. J. Isham, Phys. Rev. D 11, 768 (1975), DOI: 10.1103/PhysRevD.11.768. Crossref, Web of Science, ADS, Google Scholar
K. V. Kuchař, J. Math. Phys. 17, 777 (1976). Crossref, Web of Science, ADS, Google Scholar
K. V. Kuchař, J. Math. Phys. 17, 792 (1976). Crossref, Web of Science, ADS, Google Scholar
A. Einstein, Über die spezielle und die allgemeine Relativitätstheorie (Vieweg, Braunschweig, 1920), Sec. 28, p. 67; English transl. R. W. Lawson, Relativity: The Special and the General Theory, 17th edn. (Crown, New York, 1961) . Google Scholar
D. Hilbert, 2 Mitt., Nachr. Ges. Wiss. Göttinger 53 (1917); included in D. Hilbert, Gessamette Abhandlungen, Vol. 3, 2nd edn. (Springer, Berlin, 1970) . Google Scholar
B. S. DeWitt , Gravitation: An Introduction to Current Research , ed. L. Witten ( Wiley , New York , 1962 ) . Google Scholar
N. Bohr and L. Rosenfeld, Mat.-Fysé. Medd. Dan. vid. Selsk. 12(8) (1933), English transl. in Selected Papers by Léon Rosenfeld, eds. R. S. Cohen and J. Stachel, Boston Studies in the Philosophy of Science 21 (Reidel, Dordrecht, 1978) . Google Scholar
C. Rovelli, Sissa Preprint 101/89/ep. (1989) . Google Scholar
P. G. Bergmann and A. Komar, Int. J. Theor. Phys. 5, 15 (1972), DOI: 10.1007/BF00671650. Crossref, Web of Science, Google Scholar
D. C. Salisbury and K. Sundermeyer, Phys. Rev. D 27, 740 (1983), DOI: 10.1103/PhysRevD.27.740. Crossref, Web of Science, ADS, Google Scholar
J. Lee and R. M. Wald, J. Math. Phys. 31, 725 (1990), DOI: 10.1063/1.528801. Crossref, Web of Science, ADS, Google Scholar
R. M. Wald, Univ. of Chicago Preprint (1991) . Google Scholar
C. J. Isham and K. V. Kuchař, Ann. Phys. (N.Y.) 164, 288 (1985), DOI: 10.1016/0003-4916(85)90018-1. Crossref, Web of Science, ADS, Google Scholar
C. J. Isham and K. V. Kuchař, Ann. Phys. (N.Y.) 164, 316 (1985), DOI: 10.1016/0003-4916(85)90019-3. Crossref, Web of Science, ADS, Google Scholar
K. V. Kuchař, Found. Phys. 16, 193 (1986), DOI: 10.1007/BF01889472. Crossref, Web of Science, ADS, Google Scholar
J. B. Hartle , Proceedings of the Fifth Marcel Grossmann Meeting , eds. D. Blair and M. Buckingham ( World Scientific , Singapore , 1989 ) . Google Scholar
J. J. Halliwell and J. B. Hartle, Phys. Rev. D 43, 1170 (1991), DOI: 10.1103/PhysRevD.43.1170. Crossref, Web of Science, ADS, Google Scholar
K. V. Kuchař and C. G. Torre, Phys. Rev. D 44, 3116 (1991), DOI: 10.1103/PhysRevD.44.3116. Crossref, Web of Science, ADS, Google Scholar
C. L. Stone and K. V. Kuchař, Class. Quantum Grav. 91, 757 (1991). Google Scholar
K. V. Kuchař, Phys. Rev. D 44, 43 (1992). Web of Science, Google Scholar
C. G. Torre, Phys. Rev. D 40, 2588 (1989), DOI: 10.1103/PhysRevD.40.2588. Crossref, Web of Science, ADS, Google Scholar
K. K. Tam and J. O'Hanlon, Nuovo Cimento 62B, 351 (1969). ADS, Google Scholar
B. F. Schutz, Phys. Rev. D 4, 3559 (1971), DOI: 10.1103/PhysRevD.4.3559. Crossref, Web of Science, ADS, Google Scholar
C. Eckart, Phys. Rev. 58, 919 (1940), DOI: 10.1103/PhysRev.58.919. Crossref, Web of Science, ADS, Google Scholar
G. A. Kluitenberg and S. R. deGroot, Physica 21, 148 (1955). Crossref, Web of Science, ADS, Google Scholar
S. W. Hawking and G. F. R. Ellis , The Large Scale Structure of Space-Time ( Cambridge Univ. Press , Cambridge , 1973 ) . Crossref, Google Scholar
K. V. Kuchař and C. G. Torre, Phys. Rev. D 43, 419 (1990). Crossref, Web of Science, ADS, Google Scholar
A. Einstein, Ann. Phys. (Leipzig) 49, 769 (1916). ADS, Google Scholar
W. G. Unruh, Phys. Rev. D 40, 1048 (1989), DOI: 10.1103/PhysRevD.40.1048. Crossref, Web of Science, Google Scholar
A. Einstein, Sitzunsber. Preuss. Acad. Wiss., Phys.-Math. Kl. (1919); English transl. in The Principle of Relativity (Dover, New York, 1952) . Google Scholar
J. J. Van de Bij, H. van Dam and Y. J. Ng, Physica A 116, 307 (1982). Crossref, Web of Science, ADS, Google Scholar
F. Wilczek, Phys. Rep. 104, 143 (1984), DOI: 10.1016/0370-1573(84)90206-0. Crossref, Web of Science, ADS, Google Scholar
W. Buchmüller and N. Dragon, Phys. Lett. B 207, 292 (1988). Crossref, Web of Science, ADS, Google Scholar
S. Weinberg, Rev. Mod. Phys. 61, 1 (1989), DOI: 10.1103/RevModPhys.61.1. Crossref, Web of Science, ADS, Google Scholar
M. Henneaux and C. Teitelboim, Phys. Lett. B 222, 195 (1989). Crossref, Web of Science, ADS, Google Scholar
W. G. Unruh , Proceedings of the Fourth Seminar on Quantum Gravity , eds. M. A. Markov , V. A. Berezin and V. P. Frolov ( World Scientific , Singapore , 1988 ) . Google Scholar
R. D. Sorkin , History of Modern Gauge Theories , eds. M. Dresden and A. Rosenblum ( World Scientific , Singapore , 1990 ) . Google Scholar
R. D. Sorkin , Conceptual Problems of Quantum Gravity , eds. A. Ashtekar and J. Stachel ( Birkhäuser , Boston , 1991 ) . Google Scholar
J. D. Brown and J. W. York, Phys. Rev. D 40, 3312 (1989), DOI: 10.1103/PhysRevD.40.3312. Crossref, Web of Science, ADS, Google Scholar
K. V. Kuchař, Phys. Rev. D 43, 3332 (1991), DOI: 10.1103/PhysRevD.43.3332. Crossref, Web of Science, ADS, Google Scholar
A. Ashtekar and A. Magnon, Proc. Roy. Soc. London A 346, 375 (1975). Crossref, Web of Science, ADS, Google Scholar
A. Magnon-Ashtekar, Thesis (Département de Mathématiques Pures, Université de Clermont, Clermont, 1976); unpublished . Google Scholar
A. Ashtekar and A. Magnon-Ashtekar, C. R. Acad. Sci. (Paris) A 286, 531 (1978). Google Scholar
B. S. Kay, Commun. Math. Phys. 62, 55 (1978), DOI: 10.1007/BF01940330. Crossref, Web of Science, ADS, Google Scholar
K. V. Kuchař, J. Math. Phys. 22, 2640 (1981). Crossref, Web of Science, ADS, Google Scholar
K. V. Kuchař, J. Math. Phys. 25, 1647 (1982). Google Scholar
J. Friedman and A. Higuchi, Phys. Rev. D 41, 2479 (1990), DOI: 10.1103/PhysRevD.41.2479. Crossref, Web of Science, ADS, Google Scholar
K. V. Kuchař, J. Math. Phys. 18, 1589 (1977). Crossref, Web of Science, ADS, Google Scholar
T. P. Singh and T. Padhanabhan, Ann. Phys. (N.Y.) 196, 296 (1989), DOI: 10.1016/0003-4916(89)90180-2. Crossref, Web of Science, ADS, Google Scholar
T. Banks, Nucl. Phys. B 249, 332 (1985), DOI: 10.1016/0550-3213(85)90020-3. Crossref, Web of Science, ADS, Google Scholar
A. Vilenkin, Phys. Rev. D 39, 1116 (1989), DOI: 10.1103/PhysRevD.39.1116. Crossref, Web of Science, ADS, Google Scholar
R. Brout, Found. Phys. 17, 603 (1987), DOI: 10.1007/BF01882790. Crossref, Web of Science, ADS, Google Scholar
R. Brout and G. Venturi, Phys. Rev. D 39, 2436 (1989), DOI: 10.1103/PhysRevD.39.2436. Crossref, Web of Science, ADS, Google Scholar
R. Brout, G. Horowitz and D. Wiel, Phys. Lett. B 192, 318 (1987). Crossref, Web of Science, ADS, Google Scholar
J. B. Hartle , Gravitation in Astrophysics, Cargese, 1986 , eds. B. Carter and J. B. Hartle ( Plenum , New York , 1986 ) . Google Scholar
J. J. Halliwell , Quantum Cosmology and Baby Universes , eds. S. Coleman et al. ( World Scientific , Singapore , 1991 ) . Link, Google Scholar
T. Padhanabhan, Class. Quantum Grav. 6, 533 (1989). Crossref, Web of Science, ADS, Google Scholar
C. Møller , Les Théories Relativisties de la Gravitation , eds. A. Lichnerowicz and M. A. Tonnelat ( CNRS , Paris , 1962 ) . Google Scholar
L. Rosenfeld, Nucl. Phys. 40, 353 (1963). Crossref, Web of Science, Google Scholar
H. D. Zeh, Phys. Lett. A 116, 9 (1986), DOI: 10.1016/0375-9601(86)90346-4. Crossref, Web of Science, ADS, Google Scholar
H. D. Zeh, Phys. Lett. A 126, 311 (1988), DOI: 10.1016/0375-9601(88)90842-0. Crossref, Web of Science, ADS, Google Scholar
E. Joos, Phys. Lett. A 116, 6 (1986), DOI: 10.1016/0375-9601(86)90345-2. Crossref, Web of Science, ADS, Google Scholar
C. Kiefer, Class. Quantum Grav. 4, 1369 (1987), DOI: 10.1088/0264-9381/4/5/031. Crossref, Web of Science, ADS, Google Scholar
J. J. Halliwell, Phys. Rev. D 39, 2912 (1989), DOI: 10.1103/PhysRevD.39.2912. Crossref, Web of Science, Google Scholar
V. Lapchinski and V. A. Rubakov, Acta Phys. Pol. B 10, 1041 (1979). Web of Science, Google Scholar
J. J. Halliwell, Phys. Rev. D 36, 3626 (1987), DOI: 10.1103/PhysRevD.36.3626. Crossref, Web of Science, ADS, Google Scholar
J. B. Barbour, Preprint 1991 . Google Scholar
A. Vilenkin, Phys. Rev. D 37, 888 (1988), DOI: 10.1103/PhysRevD.37.888. Crossref, Web of Science, ADS, Google Scholar
T. Padhanabhan, Pramana. J. Phys. 35, L199 (1990). Crossref, Web of Science, ADS, Google Scholar
J. Greensite, Nucl. Phys. B 342, 409 (1990), DOI: 10.1016/0550-3213(90)90196-K. Crossref, Web of Science, ADS, Google Scholar
C. Kiefer, Zürich University Preprint ZU-TH 10/91. To appear in the Proceedings of the Journées Relativistes, Cargèse, 2.5.–4.5 1991. "How does Quantum Gravity Modify the Schrödinger Equation for Matter Fields?" C. Kiefer and T. P. Singh, Preprint 1990. "Quantum Gravitational Corrections to the Functional Schrödinger Equation." . Google Scholar
G. T. Horowitz and R. M. Wald, Phys. Rev. D 21, 1462 (1980), DOI: 10.1103/PhysRevD.21.1462. Crossref, Web of Science, ADS, Google Scholar
S. Danjbar-Daemi, B. S. Kay and T. W. B. Kibble, Phys. Lett. 91B, 417 (1980). Google Scholar
T. W. B. Kibble , Quantum Gravity 2: A Second Oxford Symposium , eds. C. J. Isham , R. Penrose and D. W. Sciama ( Clarendon , Oxford , 1981 ) . Google Scholar
M. J. Duff , Quantum Gravity 2: A Second Oxford Symposium , eds. C. J. Isham , R. Penrose and D. W. Sciama ( Clarendon , Oxford , 1981 ) . Google Scholar
W.-M. Suen, Phys. Rev. Lett. 62, 2217 (1989), DOI: 10.1103/PhysRevLett.62.2217. Crossref, Web of Science, ADS, Google Scholar
W.-M. Suen, Phys. Rev. D 40, 315 (1989), DOI: 10.1103/PhysRevD.40.315. Crossref, Web of Science, ADS, Google Scholar
S. Giddings and A. Strominger, Nucl. Phys. B 321, 481 (1988), DOI: 10.1016/0550-3213(89)90353-2. Crossref, Web of Science, ADS, Google Scholar
S. Coleman, Nucl. Phys. B 310, 643 (1988), DOI: 10.1016/0550-3213(88)90097-1. Crossref, Web of Science, ADS, Google Scholar
M. McGuigan, Phys. Rev. D 38, 3031 (1988), DOI: 10.1103/PhysRevD.38.3031. Crossref, Web of Science, ADS, Google Scholar
M. McGuigan, Phys. Rev. D 39, 2229 (1989), DOI: 10.1103/PhysRevD.39.2229. Crossref, Web of Science, ADS, Google Scholar
L. Rosenfeld , Niels Bohr and the Development of Physics ( Pergamon , London , 1955 ) . Google Scholar
C. Møller , The Theory of Relativity , 2nd edn. ( Clarendon , Oxford , 1972 ) . Google Scholar
L. D. Landau and E. M. Lifshitz, Teorija polja (Gosudarstvennoe izdatel'stvo fizikomatematiceskoj literatury, Moskva, 1962). English transl. M. Hamermesh, The Classical Theory of Fields (Pergamon, Oxford, 1962), Sec. 84 . Google Scholar
J. B. Hartle and S. W. Hawking, Phys. Rev. D 28, 2960 (1983), DOI: 10.1103/PhysRevD.28.2960. Crossref, Web of Science, Google Scholar
S. W. Hawking, Nucl. Phys. B 239, 257 (1984), DOI: 10.1016/0550-3213(84)90093-2. Crossref, Web of Science, ADS, Google Scholar
S. W. Hawking , Relativity, Groups and Topology II , Les Houches Session XL , eds. B. DeWitt and R. Stora ( North Holland , Amsterdam , 1984 ) . Google Scholar
S. W. Hawking and D. N. Page, Nucl. Phys. B 264, 185 (1986). Crossref, Web of Science, ADS, Google Scholar
S. W. Hawking and D. N. Page, Nucl. Phys. B 298, 789 (1988), DOI: 10.1016/0550-3213(88)90008-9. Crossref, Web of Science, ADS, Google Scholar
D. N. Page, Phys. Rev. D 34, 2267 (1986), DOI: 10.1103/PhysRevD.34.2267. Crossref, Web of Science, ADS, Google Scholar
D. N. Page and W. K. Wootters, Phys. Rev. D 27, 2885 (1983), DOI: 10.1103/PhysRevD.27.2885. Crossref, Web of Science, Google Scholar
D. N. Page, ITP Preprint NSF-ITP-89-18 (1989) . Google Scholar
D. N. Page , Conceptual Problems of Quantum Gravity , eds. A. Ashtekar and J. Stachel ( Birkhäuser , Boston , 1991 ) . Google Scholar
W. K. Wootters, Int. J. Theor. Phys. 23, 701 (1984), DOI: 10.1007/BF02214098. Crossref, Web of Science, Google Scholar
C. Teitelboim, Phys. Lett. 96B, 77 (1980). ADS, Google Scholar
C. Teitelboim, Phys. Rev. D 25, 3159 (1982), DOI: 10.1103/PhysRevD.25.3159. Crossref, Web of Science, ADS, Google Scholar
M. Henneaux and C. Teitelboim, Ann. Phys. 143, 127 (1982). Crossref, Web of Science, ADS, Google Scholar
J. B. Hartle, Phys. Rev. D 37 (1988) 2818; Phys. Rev. D 43 (1991) 1434 . Google Scholar
J. B. Hartle, VCSB, preprint . Google Scholar
J. B. Hartle , The Proceedings of the 60th Birthday Celebration for Murray Gell-Mann , eds. J. Schwarz and F. Zachariasen ( Cambridge Univ. Press , Cambridge , 1990 ) . Google Scholar
M. Gell-Mann and J. B. Hartle , Complexity, Entropy, and the Physics of Information , Santa Fe Institute Studies in the Sciences of Complexity, VIII , ed. W. H. Zurek ( Addison-Wesley , Reading , 1990 ) . Google Scholar
J. B. Hartle , Quantum Cosmology and Baby Universes , eds. S. Coleman et al. ( World Scientific , Singapore , 1991 ) . Google Scholar
J. B. Hartle, Preprint UCSBTH91-02 . Google Scholar
P. G. Bergmann, Nuovo Cimento 3, 1956 (1956). Google Scholar
P. G. Bergmann, Helv. Phys. Acta Suppl. 4, 1956 (1956). Google Scholar
E. Newman and P. G. Bergmann, Rev. Mod. Phys. 29, 443 (1957), DOI: 10.1103/RevModPhys.29.443. Crossref, Web of Science, ADS, Google Scholar
A. Komar, Phys. Rev. 111, 1182 (1958), DOI: 10.1103/PhysRev.111.1182. Crossref, Web of Science, ADS, Google Scholar
B. S. DeWitt, Ref. 32, Section 4, p. 160 . Google Scholar
D. R. Brill and R. H. Gowdy, Rep. Progress Phys. 33, 413 (1970), DOI: 10.1088/0034-4885/33/2/301. Crossref, Web of Science, ADS, Google Scholar
C. Rovelli , Conceptual Problems of Quantum Gravity , eds. A. Ashtekar and J. Stachel ( Birkhäuser , Boston , 1991 ) . Google Scholar
C. Rovelli, Phys. Rev. D 42, 2638 (1990), DOI: 10.1103/PhysRevD.42.2638. Crossref, Web of Science, ADS, Google Scholar
C. Rovelli, Phys. Rev. D 43, 442 (1991), DOI: 10.1103/PhysRevD.43.442. Crossref, Web of Science, ADS, Google Scholar
P. Hájíček, Phys. Rev. D 44, 1337 (1991). Crossref, Web of Science, ADS, Google Scholar
L. Van Hove, Acad. Roy. Belg. Bull Cl. Sci. 37, 610 (1951). Google Scholar
C. Rovelli, Phys. Rev. D 44, 1339 (1991), DOI: 10.1103/PhysRevD.44.1339. Crossref, Web of Science, ADS, Google Scholar
J. B. Hartle, Private Communication (1991) . Google Scholar

Vol. 20, No. supp01

Metrics

Downloaded 391 times

History

PDF download

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

TIME AND INTERPRETATIONS OF QUANTUM GRAVITY

Abstract

Recommended