Klein-Gordon-Wheeler-DeWitt-Schrödinger equation |
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Authors: | Matej Pav&scaron i? |
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Affiliation: | Jo?ef Stefan Institute, Jamova 39, SI-1000, Ljubljana, Slovenia |
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Abstract: | We start from the Einstein-Hilbert action for the gravitational field in the presence of a “point particle” source, and cast the action into the corresponding phase space form. The dynamical variables of such a system satisfy the point particle mass shell constraint, the Hamilton and the momentum constraints of the canonical gravity. In the quantized theory, those constraints become operators that annihilate a state. A state can be represented by a wave functional Ψ that simultaneously satisfies the Klein-Gordon and the Wheeler-DeWitt-Schrödinger equation. The latter equation, besides the term due to gravity, also contains the Schrödinger like term, namely the derivative of Ψ with respect to time, that occurs because of the presence of the point particle. The particle?s time coordinate, X0, serves the role of time. Next, we generalize the system to p-branes, and find out that for a quantized spacetime filling brane there occurs an effective cosmological constant, proportional to the expectation value of the brane?s momentum, a degree of freedom that has two discrete values only, a positive and a negative one. This mechanism could be an explanation for the small cosmological constant that drives the accelerated expansion of the universe. |
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