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"In the quantum mechanics of closed universes we do not expect to find a notion of ground states as a state of lowest energy. [...]It is still reasonable, however, , to expect to be able to define a state of minimum excitation corresponding to the classical notion of a geometry of high symmetry. This paper contains a proposal for the definition of such a ground-state wave function for closed universes. The proposal is to extend to gravity the Euclidean-functional-integral construction of nonrelativistic quantum mechanics and field theory. Thus we write for the ground-sate wave function
Ψ0[hij] = N ∫ δg exp(-IE[g]),
where IE is the Euclidean action for gravity including a cosmological constant. The Euclidean four-geometries summed over must have a boundary on which the induced metric is hij. The remaining specification of the class of geometries which are summed over determines the ground state. Our proposal is that the sum should be over compact geometries. This means that the Universe does not have any boundaries in space or times (at least in the Euclidean regime)[...] One can interpret the functional integral over all compact four-geometries bounded by a given three-geometry as giving the amplitude for that three-geometry to arise from zero three-geometry, i.e., a single point. In other words, the ground state is the amplitude for the Universe to appear from nothing." (J.B. Hartle; S. W. Hawking, Wave function of the Universe. Physical Review, 15 December 1983)
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