The spectral representation of the spacetime structure: The `distance' between universes with different topologies

We investigate the representation of the geometrical information of the universe in terms of the eigenvalues of the Laplacian defined on the universe. We concentrate only on one specific problem along this line: To introduce a concept of distance between universes in terms of the difference in the spectra. We can find out such a measure of closeness from a general discussion. The basic properties of this `spectral distance' are then investigated. It can be related to a reduced density matrix element in quantum cosmology. Thus, calculating the spectral distance gives us an insight for the quantum theoretical decoherence between two universes. The spectral distance does not in general satisfy the triangular inequality, illustrating that it is not equivalent to the distance defined by the DeWitt metric on the superspace. We then pose a question: Whether two universes with different topologies interfere with each other quantum mechanically? We concentrate on the difference in the orientabilities. Several concrete models in 2-dimension are set up, and the spectral distances between them are investigated: Tori and Klein's bottles, spheres and real projective spaces. Quite surprisingly, we find many cases of spaces with different orientabilities in which the spectral distance turns out to be very short. It may suggest that, without any other special mechanism, two such universes interfere with each other quite strongly.Comment: 47 page