Quantizing de Sitter Spacetime in Heisenberg Picture

概要

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We consider the spatially flat Friedman-Robertson-Walker Universe with positive comological constant. By a suitable redefinition of the essential dynamical variable, the metric function f, we perform a non-perturbateve canonical quantization within the framework of the Heisenberg picture. Our approach is different from the usual ones in that it imposes the Wheeler-DeWitt equation as a constaint at the end of the quantization procedure. It turns out that there are no operator-ordering problems, and all the Heisenberg dynamical operators can be expressed in terms of two (time-independent) lowering and raising operators that generate and act on a Fock space spanned by the eigenstates of an associated harmonic oscillator. As expected, the Wheeler-DeWitt equation generalizing the G_<44>-Einstein equation for the classical de Sitter spacetime selects the physical states of the quantum de Sitter Universe. We show that there exist two orthogonal solutions (to the derived Wheeler-DeWitt equation) which are explicitly worked out as mixed quantum states. As a result of the exponential universal expansion, the usual Fock states (defined as the eigenstates of the number-operator) are no longer invariant under the derived Hamiltonian. They exhibit energy and metric quantum fluctuations which lead to a (geometrical) volume quantization.
理論物理学刊行会の論文
1996-06-25