Abstract
A generally covariant theory, written in the spirit of Bohm's theory of quantum potentials, which applies to spinless, non interacting, gravitating systems, is formulated. In this theory the quantum state ψ is coupled to the metric tensor g, and the effect of the “quantum potential” is absorbed in the geometry. At the same time, ψ satisfies a covariant wave equation with respect to the very same g. This provides sufficient constraints to derive 11 coupled equations in the 11 unknowns: ψ and the components of the metric tensor gµv. The states of stable localized particles are identified, and vacuum-state solutions for both the Euclidean and the Lorentzian case are explicitly presented.
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Notes and references
D. Bohm,Phys. Rev. 85, 166–193 (1952).
J. S. Bell, “Quantum mechanics for cosmologists,” inSpeakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1987), pp. 117–138.
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For the details of this unit system see C. W. Misner, K. S. Thorne, and J. A. Wheeler,Gravitation (Freeman, San Francisco, 1970), p. 36.
S. W. Hawking, in S. W. Hawking and E. Israel, eds.,General Relativity, an Einstein Centenary Survey (Cambridge University Press, Cambridge, 1979), pp. 746–789.
For the affine connection of that metric, see S. Weinberg,Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (Wiley, New York, 1972), p. 471.
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Pitowsky, I. Bohm's quantum potentials and quantum gravity. Found Phys 21, 343–352 (1991). https://doi.org/10.1007/BF01883639
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DOI: https://doi.org/10.1007/BF01883639