A toy model of quantum gravity with a quantized cosmological constant∗

The path integral for the hydrogen atom, for example, is formulated as a onedimensional quantum gravity coupled to matter fields representing the electron coordinates. The (renormalized) cosmological constant, which corresponds to the energy eigenvalue, is thus quantized in this model. Possible implications of the quantized cosmological constant are briefly discussed. The problem of the cosmological constant in quantum gravity has a long history. For a review of this subject, see ref.[1] and papers cited therein. Recently, we pointed out that the path integral of the hydrogen atom, for example, in non-relativistic quantum mechanics is formulated as a one-dimensional quantum gravity coupled to matter fields representing the electron coordinates. The basic idea of this formulation is to use the Jacobi’s principle of least action for the evaluation of the Green’s function with a given energy[2]. In this formulation, the (renormalized) cosmological constant corresponds to the energy eigenvalue of the Schroedinger equation, and thus the cosmological constant is quantized. Since the notion of a quantized cosmological constant appears to be something novel, at least to my knowledge, we would like to present a brief description of the model. The ∗To be published in Progress of Theoretical Physics(Kyoto)