Dilaton gravity in two dimensions. (English) Zbl 0998.83038

Grumiller, D.; Kummer, W.; Vassilevich, D. V.

Phys. Rep. 369, No. 4, 327-430 (2002).

Summary: The study of general two-dimensional models of gravity allows to tackle basic questions of quantum gravity, bypassing important technical complications which make the treatment in higher dimensions difficult. As the physically important examples of spherically symmetric Black Holes, together with string inspired models, belong to this class, valuable knowledge can also be gained for these systems in the quantum case. In the last decade, new insights regarding the exact quantization of the geometric part of such theories have been obtained. They allow a systematic quantum field theoretical treatment, also in interactions with matter, without explicit introduction of a specific classical background geometry. The present review tries to assemble these results in a coherent manner, putting them at the same time into the perspective of the quite large literature on this subject.

MSC:

83D05	Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
83C45	Quantization of the gravitational field

Keywords:

quantum gravity; geometric part quantization; quantum field theory

References:

[2]	Polchinski, J., String Theory, An Introduction to the Bosonic String, Vol. 1 (1998), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1006.81521
[5]	’t Hooft, G.; Veltman, M. J.G., Ann. Poincare Phys. Theor. A, 20, 69 (1974)
[6]	Goroff, M. H.; Sagnotti, A., Nucl. Phys. B, 266, 709 (1986)
[7]	Howe, P. S.; Stelle, K. S.; Townsend, P. K., Nucl. Phys. B, 236, 125 (1984)
[11]	Wheeler, J., Geometrodynamics and the issue of the final state, (DeWitt, C.; DeWitt, B., Relativity, Groups and Topology (1964), Gordon and Breach: Gordon and Breach London), 316 · Zbl 0148.46204
[13]	DeWitt, B. S., Phys. Rev., 160, 1113 (1967)
[15]	Ashtekar, A.; Geroch, R., Rep. Prog. Phys., 37, 1211 (1974)
[18]	Hawking, S. W., Commun. Math. Phys., 43, 199 (1975)
[19]	Unruh, W. G., Phys. Rev. D, 14, 870 (1976)
[22]	D’Hoker, E.; Jackiw, R., Phys. Rev. D, 26, 3517 (1982)
[23]	Teitelboim, C., Phys. Lett. B, 126, 41 (1983)
[24]	D’Hoker, E.; Freedman, D.; Jackiw, R., Phys. Rev. D, 28, 2583 (1983)
[25]	D’Hoker, E.; Jackiw, R., Phys. Rev. Lett., 50, 1719 (1983)
[26]	Jackiw, R., Nucl. Phys. B, 252, 343 (1985)
[27]	Katanaev, M. O.; Volovich, I. V., Phys. Lett. B, 175, 413 (1986)
[28]	Katanaev, M. O.; Volovich, I. V., Ann. Phys., 197, 1 (1990)
[29]	Mann, R. B.; Shiekh, A.; Tarasov, L., Nucl. Phys. B, 341, 134 (1990)
[30]	Sikkema, A. E.; Mann, R. B., Class. Quant. Grav., 8, 219 (1991)
[31]	Brown, J., Lower Dimensional Gravity (1988), World Scientific: World Scientific Singapore
[33]	Berger, B. K., Phys. Rev. D, 5, 2467 (1972)
[34]	Benguria, R.; Cordero, P.; Teitelboim, C., Nucl. Phys. B, 122, 61 (1977)
[35]	Thomi, P.; Isaak, B.; Hájíček, P., Phys. Rev. D, 30, 1168 (1984)
[36]	Hájíček, P., Phys. Rev. D, 30, 1178 (1984)
[37]	Mignemi, S.; Wiltshire, D. L., Class. Quant. Grav., 6, 987 (1989)
[42]	Mandal, G.; Sengupta, A. M.; Wadia, S. R., Mod. Phys. Lett. A, 6, 1685 (1991)
[43]	Elitzur, S.; Forge, A.; Rabinovici, E., Nucl. Phys. B, 359, 581 (1991)
[44]	Witten, E., Phys. Rev. D, 44, 314 (1991)
[45]	Dijkgraaf, R.; Verlinde, H.; Verlinde, E., Nucl. Phys. B, 371, 269 (1992)
[47]	Ishibashi, N.; Li, M.; Steif, A. R., Phys. Rev. Lett., 67, 3336 (1991)
[48]	de Alwis, S. P.; Lykken, J., Phys. Lett. B, 269, 264 (1991)
[56]	Odintsov, S. D.; Shapiro, I. L., Phys. Lett. B, 263, 183 (1991)
[59]	Polyakov, A. M., Mod. Phys. Lett. A, 2, 893 (1987)
[60]	Solodukhin, S. N., Class. Quant. Grav., 10, 1011 (1993)
[61]	Solodukhin, S. N., JETP Lett., 57, 329 (1993)
[65]	Kummer, W.; Schwarz, D. J., Phys. Rev. D, 45, 3628 (1992)
[94]	Haider, F.; Kummer, W., Int. J. Mod. Phys. A, 9, 207 (1994)
[98]	Batalin, I. A.; Vilkovisky, G. A., Phys. Lett. B, 69, 309 (1977)
[107]	Bekenstein, J., Lett. Nuovo Cimento, 11, 467 (1974)
[124]	Nakahara, M., Geometry, Topology and Physics (1990), IOP Publishing: IOP Publishing Bristol · Zbl 0764.53001
[125]	Einstein, A., Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), 1915, 778 (1915)
[126]	Einstein, A., Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), 1915, 844 (1915)
[127]	Palatini, A., Rend. Circ. Mat. Palermo, 43, 203 (1919)
[128]	Hehl, F. W., Rev. Mod. Phys., 48, 393 (1976)
[132]	Hawking, S.; Ellis, G., The Large Scale Structure of Space-Time (1973), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0265.53054
[133]	Wald, R. M., General Relativity (1984), The University of Chicago Press: The University of Chicago Press Chicago
[135]	Walker, M., J. Math. Phys., 11, 2280 (1970)
[137]	Kuchař, K., Phys. Rev. D, 4, 955 (1971)
[138]	Gowdy, R. H., Phys. Rev. Lett., 27, 826 (1971)
[139]	Geroch, R., J. Math. Phys., 13, 394 (1972)
[140]	Husain, V.; Smolin, L., Nucl. Phys. B, 327, 205 (1989)
[143]	Gibbons, G. W.; Maeda, K., Nucl. Phys. B, 298, 741 (1988)
[144]	Callan Jr, C. G., Nucl. Phys. B, 262, 593 (1985)
[152]	Banks, T.; O’Loughlin, M., Nucl. Phys. B, 362, 649 (1991)
[156]	Reissner, H., Ann. Phys., 50, 106 (1916)
[157]	Nordström, G., Proc. K. Ned. Akad. Wet., 20, 1238 (1916)
[158]	Schmidt, H. J., J. Math. Phys., 32, 1562 (1991)
[159]	Frolov, V. P., Phys. Rev. D, 46, 5383 (1992)
[160]	Odintsov, S. D.; Shapiro, I. L., Mod. Phys. Lett. A, 7, 437 (1992)
[164]	Louis-Martinez, D.; Kunstatter, G., Phys. Rev. D, 49, 5227 (1994)
[177]	Fierz, M., Helv. Phys. Acta, 29, 128 (1956)
[178]	Jordan, P., Schwerkraft und Weltall: Grundlagen der theoretischen Kosmologie (1955), Vieweg: Vieweg Braunschweig · Zbl 0048.21604
[179]	Jordan, P., Z. Phys., 157, 112 (1959)
[180]	Brans, C.; Dicke, R. H., Phys. Rev., 124, 925 (1961)
[183]	Dick, R., Gen. Rel. Grav., 30, 435 (1998)
[185]	Faraoni, V.; Gunzig, E., Int. J. Theor. Phys., 38, 217 (1999)
[192]	Schouten, J., Conv. Geom. Diff., 1, 7 (1954)
[193]	Nijenhuis, A., Proc. K. Ned. Akad. Wet. Amsterdam A, 58, 390 (1955)
[196]	Bayen, F., Ann. Phys., 111, 61 (1978)
[197]	Bayen, F., Ann. Phys., 111, 111 (1978)
[203]	Eddington, A. S., Nature, 113, 192 (1924)
[204]	Finkelstein, D., Phys. Rev., 110, 965 (1958)
[205]	Birmingham, D., Phys. Rept., 209, 129 (1991)
[206]	Schwarz, A. S., Lett. Math. Phys., 2, 247 (1978)
[207]	Schwarz, A. S., Commun. Math. Phys., 67, 1 (1979)
[208]	Witten, E., Commun. Math. Phys., 117, 353 (1988)
[209]	Witten, E., Commun. Math. Phys., 118, 411 (1988)
[210]	Witten, E., Commun. Math. Phys., 121, 351 (1989)
[211]	Carter, B., J. Math. Phys., 10, 70 (1969)
[212]	Penrose, R., Riv. Nuovo Cimento, 1, 252 (1969)
[213]	Katanaev, M. O., J. Math. Phys., 34, 700 (1993)
[215]	Israel, W., Phys. Rev., 143, 1016 (1966)
[218]	Mignemi, S., Phys. Rev. D, 50, 4733 (1994)
[221]	Kummer, W.; Widerin, P., Mod. Phys. Lett. A, 9, 1407 (1994)
[224]	Forgacs, P.; Manton, N. S., Commun. Math. Phys., 72, 15 (1980)
[225]	Witten, E., Phys. Rev. Lett., 38, 121 (1977)
[227]	Howe, P. S., J. Phys. A, 12, 393 (1979)
[228]	Brown, M.; Gates, J.; James, S., Ann. Phys., 122, 443 (1979)
[229]	Martinec, E. J., Phys. Rev. D, 28, 2604 (1983)
[230]	Rocek, M.; van Nieuwenhuizen, P.; Zhang, S. C., Ann. Phys., 172, 348 (1986)
[234]	Ikeda, N., Int. J. Mod. Phys. A, 9, 1137 (1994)
[236]	Duff, M. J.; Nilsson, B. E.W.; Pope, C. N., Phys. Rep., 130, 1 (1986)
[238]	Penrose, R.; Rindler, W., Spinors and Space-Time I (1984), Cambridge University Press: Cambridge University Press Cambridge
[240]	Riess, A. G., Publ. Astron. Soc. Pacific, 112, 1284 (2000)
[244]	Wetterich, C., Nucl. Phys. B, 302, 668 (1988)
[248]	Ratra, B.; Peebles, P. J.E., Phys. Rev. D, 37, 3406 (1988)
[252]	Polyakov, A. M., Phys. Lett. B, 103, 207 (1981)
[254]	Witten, E., Commun. Math. Phys., 92, 455 (1984)
[257]	Roberts, M. D., Gen. Rel. Grav., 21, 907 (1989)
[264]	Choptuik, M. W., Phys. Rev. Lett., 70, 9 (1993)
[265]	Christodoulou, D., Commun. Math. Phys., 106, 587 (1986)
[266]	Christodoulou, D., Commun. Math. Phys., 105, 337 (1986)
[267]	Christodoulou, D., Commun. Math. Phys., 109, 591 (1987)
[268]	Christodoulou, D., Commun. Math. Phys., 109, 613 (1987)
[271]	Arnowitt, R.; Deser, S.; Misner, C. W., (Witten, L., Gravitation: An Introduction to Current Research (1962), Wiley: Wiley New York)
[272]	Faddeev, L. D., Sov. Phys. Usp., 25, 130 (1982)
[276]	Gibbons, G. W.; Hawking, S. W., Phys. Rev. D, 15, 2752 (1977)
[277]	Brown, J. D.; York, J.; James, W., Phys. Rev. D, 47, 1407 (1993)
[278]	Abbott, L. F.; Deser, S., Nucl. Phys. B, 195, 76 (1982)
[296]	Frolov, V. P.; Novikov, I. D., Black Hole Physics: Basic Concepts and New Developments (1998), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Netherlands · Zbl 0978.83001
[299]	Christensen, S. M.; Fulling, S. A., Phys. Rev. D, 15, 2088 (1977)
[305]	Dowker, J. S.; Critchley, R., Phys. Rev. D, 13, 3224 (1976)
[306]	Hawking, S. W., Commun. Math. Phys., 55, 133 (1977)
[307]	Boulware, D. G., Phys. Rev. D, 11, 1404 (1975)
[308]	Israel, W., Phys. Lett. A, 57, 107 (1976)
[309]	Gibbons, G. W.; Perry, M. J., Phys. Rev. Lett., 36, 985 (1976)
[317]	Chiba, T.; Siino, M., Mod. Phys. Lett. A, 12, 709 (1997)
[348]	Novozhilov, Y. V.; Vassilevich, D. V., Phys. Lett. B, 220, 36 (1989)
[355]	Wipf, A. W., Nucl. Phys. B, 269, 24 (1986)
[360]	Gitman, D. M.; Tyutin, I. V., Quantization of Fields with Constraints (1990), Springer: Springer Berlin · Zbl 1028.83011
[361]	Henneaux, M.; Teitelboim, C., Quantization of Gauge Systems (1992), Princeton University Press: Princeton University Press Princeton, NJ, USA · Zbl 0838.53053
[363]	Ashtekar, A., Phys. Rev. Lett., 57, 2244 (1986)
[364]	Ashtekar, A., Phys. Rev. D, 36, 1587 (1987)
[367]	Fradkin, E. S.; Vilkovisky, G. A., Phys. Lett. B, 55, 224 (1975)
[368]	Fradkin, E. S.; Fradkina, T. E., Phys. Lett. B, 72, 343 (1978)
[369]	Toms, D. J., Phys. Rev. D, 35, 3796 (1987)
[370]	Fujikawa, K., Phys. Rev. D, 37, 391 (1988)
[372]	Barvinsky, A. O., Phys. Lett. B, 195, 344 (1987)
[376]	Esposito, G., Quantum Gravity in Four-Dimensions (2001), Nova Science: Nova Science New York
[393]	Aharonov, Y.; Casher, A.; Nussinov, S., Phys. Lett. B, 191, 51 (1987)
[395]	Hawking, S. W., Commun. Math. Phys., 87, 395 (1982)
[396]	Banks, T.; Susskind, L.; Peskin, M. E., Nucl. Phys. B, 244, 125 (1984)
[397]	Misner, C., (Klauder, J., Magic without Magic: John Archibald Wheeler, A Collection of Essays in Honor of his 60th Birthday (1972), Freeman: Freeman San Francisco, CA)
[398]	Lund, F., Phys. Rev. D, 8, 3247 (1973)
[409]	Beig, R.; ó Murchadha, N., Ann. Phys. (N.Y.), 174, 463 (1987)
[410]	Boulware, D.; Deser, S., J. Math. Phys., 8, 1468 (1967)
[411]	Kuchař, K., Phys. Rev. D, 39, 2263 (1989)
[412]	Kuchař, K., Phys. Rev. D, 39, 1579 (1989)
[414]	Faddeev, L. D.; Jackiw, R., Phys. Rev. Lett., 60, 1692 (1988)
[426]	Gilkey, P. B., Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem (1994), CRC Press: CRC Press Boca Raton, FL
[427]	Esposito, G.; Kamenshchik, A.; Pollifrone, G., Euclidean Quantum Gravity on Manifolds with Boundary (1997), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0905.53051
[428]	Avramidi, I. G., Heat Kernel and Quantum Gravity (2000), Springer: Springer Berlin · Zbl 0862.58054
[429]	Kirsten, K., Spectral Functions in Mathematics and Physics (2001), Chapman & Hall/CRC Press: Chapman & Hall/CRC Press Boca Raton, FL
[432]	DeWitt, B. S., Dynamical Theory of Groups and Fields (1965), Gordon and Breach: Gordon and Breach New York · Zbl 0169.57101
[433]	Fock, V. A., Izv. USSR Acad. Sci. (Phys.), 4-5, 551 (1937)
[434]	Gilkey, P. B., J. Differential Geom., 10, 601 (1975)
[435]	Branson, T. P.; Gilkey, P. B., Commun. Partial Differential Equations, 15, 245 (1990)
[438]	Vaz, C.; Witten, L., Nucl. Phys. B, 487, 409 (1997)
[439]	Kummer, W.; Schwarz, D. J., Nucl. Phys. B, 382, 172 (1992)