The Born-Oppenheimer approach to the matter-gravity system and unitarity

Classical and Quantum Gravity

Volumn 13, Issue 9, 1996, Pages 2375-2383

  Bertoni, C ^a   Finelli, F ^b   Venturi, G ^b  

^a INAF ISTITUTO DI RADIOASTRONOMIA   (Italy)

^b UNIVERSITY OF BOLOGNA   (Italy)

Author keywords

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Indexed keywords

EID: 21444445467     PISSN: 02649381     EISSN: None     Source Type: Journal    
DOI: 10.1088/0264-9381/13/9/005     Document Type: Article

References (23)

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5. Banks T 1985 Nucl. Phys. B 249 332
6. Kiefer C and Singh T P 1991 Phys. Rev. D 44 1067
7. Kiefer C 1992 Class. Quantum Grav. 9 147;
8. 1994 Canonical Gravity-From Classical to Quantum ed J Ehlers and H Friedrich (Berlin: Springer)
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10. Phys. Rev. D 52 3382
11. 4.
12. DeWitt B S 1967 Phys. Rev. 160 113
13. ↔/∂a|χ〉N which is explicitly Hermitian.
14. 2 is strongly peaked on the classical trajectory a(η). For a molecule this will correspond to considering the motion of the nuclei to be quasi-classical while that of the electrons is quantum mechanical. Clearly the semiclassical limit is an addition to the Born-Oppenheimer factorization and is necessary for time to emerge. In the semiclassical limit, in a path integral representation for the wavefunction, neighbouring paths will tend to yield cancelling contributions on account of the rapid variation of the phase associated the exponential of the (effective) action. An exception to this rule occurs at stationary points of the exponent and the associated paths are related to classical trajectories. It is clear that for this limit to exist the fluctuations about the solutions to the classical equations of motions must be small and the integral over them finite. In general this leads to a constraint on the effective potential associated with the fluctuations, should this not be satisfied one could have for example fluctuations which increase exponentially in time which, of course, signal an instability.
15. M〉).
16. Venturi G 1993 Astropart. Phys. 1 417
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18. 1995 Gen. Rel. Grav. 27 341
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20. See for example Casadio R and Venturi G 1995 Class. Quantum Grav. 12 1267
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23. That is eigenstates of explicitly time-dependent non-trivial Hermitian operators Î(η) satisfying dÎ/dη ≡ ∂Î/∂η - i/ℏ[Î, Ĥ] = 0. See Lewis H R Jr and Riesenfeld W B 1969 J. Math. Phys. 10 1458