Quantum cosmology of scalar-tensor theories and self-adjointness. (English) Zbl 1365.83033
Summary: In this paper, the problem of the self-adjointness for the case of a quantum mini-superspace Hamiltonian retrieved from a Brans-Dicke action is investigated. Our matter content is presented in terms of a perfect fluid, onto which Schutz’s formalism will be applied. We use the von Neumann theorem and the similarity with the Laplacian operator in one of the variables to determine the cases where the Hamiltonian is self-adjoint and if it admits self-adjoint extensions. For the latter, we study which extension is physically more suitable.
©2017 American Institute of Physics
©2017 American Institute of Physics
MSC:
| 83F05 | Relativistic cosmology |
| 83D05 | Relativistic gravitational theories other than Einstein’s, including asymmetric field theories |
| 83C45 | Quantization of the gravitational field |
| 47B25 | Linear symmetric and selfadjoint operators (unbounded) |
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
Keywords:
Quantum cosmology; scalar-tensor theories; quantum mini-superspace; Hamiltonian; Brans-Dicke actionReferences:
| [1] | DeWitt, B. S., Phys. Rev., 160, 1113 (1967) · Zbl 0158.46504 · doi:10.1103/physrev.160.1113 |
| [2] | 2.C. W.Misner, Phys. Rev.186, 1319 (1969)10.1103/physrev.186.13192.186, 1328 (1969). · Zbl 0186.28604 |
| [3] | Ashtekar, A.; Lewandowski, J., Classical Quantum Gravity, 21, R53 (2004) · Zbl 1077.83017 · doi:10.1088/0264-9381/21/15/r01 |
| [4] | Rovelli, C., Quantum Gravity (2004) |
| [5] | Green, M. B.; Schwarz, J. H.; Witten, E., Superstring Theory (1987) · Zbl 0619.53002 |
| [6] | Polchinski, J., String Theory, Volumes 1 and 2 (1998) · Zbl 1006.81521 |
| [7] | Kiefer, C., Quantum Gravity (2012) · Zbl 1243.83002 |
| [8] | Kuchar, K., Time and interpretation of quantum gravity (1992) |
| [9] | Isham, C. J., Canonical quantum gravity and the problem of time (1992) · Zbl 0773.53010 |
| [10] | Gotay, M. J.; Demaret, J., Phys. Rev. D, 28, 2402 (1983) · doi:10.1103/physrevd.28.2402 |
| [11] | Lapchinskii, V. G.; Rubakov, V. A., Theor. Math. Phys., 33, 1076 (1977) · doi:10.1007/bf01036991 |
| [12] | Alvarenga, F. G.; Lemos, N. A., Gen. Relativ. Gravitation, 30, 681 (1998) · Zbl 0922.58093 · doi:10.1023/a:1018896900336 |
| [13] | Alvarenga, F. G.; Fabris, J. C.; Lemos, N. A.; Monerat, G. A., Gen. Relativ. Gravitation, 34, 651 (2002) · Zbl 0998.83076 · doi:10.1023/a:1015986011295 |
| [14] | Acacio de Barros, J.; Pinto-Neto, N.; Sagioro-Leal, M. A., Phys. Lett. A, 241, 229 (1998) · doi:10.1016/s0375-9601(98)00169-8 |
| [15] | 15.B. F.Schutz, Phys. Rev. D2, 2762 (1970)10.1103/physrevd.2.276215.4, 3559 (1971). · Zbl 1227.83022 |
| [16] | Everett, H., Rev. Mod. Phys., 29, 454 (1957) · doi:10.1103/revmodphys.29.454 |
| [17] | Omnès, R., The Interpretation of Quantum Mechanics (1994) · Zbl 0848.46047 |
| [18] | Bohm, D.; Hiley, B. J., The Undivided Universe: An Ontological Interpretation of Quantum Theory (1993) |
| [19] | Holland, P. R., The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics (1993) |
| [20] | Pinto-Neto, N.; Fabris, J. C., Classical Quantum Gravity, 30, 143001 (2013) · Zbl 1273.83003 · doi:10.1088/0264-9381/30/14/143001 |
| [21] | Pal, S.; Banerjee, N., Phys. Rev. D, 90, 104001 (2014) · doi:10.1103/physrevd.90.104001 |
| [22] | Pal, S.; Banerjee, N., Phys. Rev. D, 91, 044042 (2015) · doi:10.1103/physrevd.91.044042 |
| [23] | Almeida, C. R.; Batista, A. B.; Fabris, J. C.; Moniz, P. R. L. V., Gravitation Cosmol., 21, 191 (2015) · Zbl 1327.83257 · doi:10.1134/s0202289315030020 |
| [24] | Valiki, B., Phys. Lett. B, 718, 34 (2012) · doi:10.1016/j.physletb.2012.10.036 |
| [25] | Reed, M.; Simon, B., Fourier Analysis, Self-Adjointness (1975) · Zbl 0308.47002 |
| [26] | Bulla, W.; Gesztesy, F., Deficiency indices and singular boundary conditions in quantum mechanics, J. Math. Phys., 26, 2520 (1985) · Zbl 0583.35029 · doi:10.1063/1.526768 |
| [27] | Reed, M.; Simon, B., Functional Analysis (1972) · Zbl 0242.46001 |
| [28] | Pal, S.; Banerjee, N., J. Math. Phys., 57, 122502 (2016) · Zbl 1432.83066 · doi:10.1063/1.4972292 |
| [29] | Chirenti, C.; Skákala, J.; Saa, A., Quasinormal modes from a naked singularity, Relativity and Gravitation, 339-346 (2014) · Zbl 1327.83195 |
| [30] | Gazeau, J. P., Acta Polytech., 56, 173-179 (2016) · doi:10.14311/ap.2016.56.0173 |
| [31] | Pal, S., Phys. Rev. D, 94, 084023 (2016) · doi:10.1103/physrevd.94.084023 |
| [32] | Fabris, J. C.; Falciano, F. T.; Marto, J.; Pinto-Neto, N.; Moniz, P. R. L. V., Braz. J. Phys., 42, 475 (2012) · doi:10.1007/s13538-012-0105-y |
| [33] | Essin, A. M.; Griffiths, D. J., Am. J. Phys., 74, 109 (2006) · doi:10.1119/1.2203362 |
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