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 spectra, i.e., a set of eigenvalues of the Laplacian defined on the Universe. Here, 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 such a measure of closeness from a general discussion. First, we introduce a suitable functional P_G[⋅], where the geometrical information G (represented by the spectra) determines the detailed shape of the functional. Then, the overlapping functional integral between P_G[⋅] and P_G′[⋅] is taken, providing a measure of closeness between G and G′, d(G,G′). The basic properties of this distance (hereafter referred to as ‘‘spectral distance,’’ for brevity) are then investigated. First, it can be related to a reduced density-matrix element in quantum cosmology between G and G′. Thus, calculating the spectral distance d(G,G′) gives us insight into the quantum theoretical decoherence between two universes, corresponding to G and G′. Second, the spectral distance becomes divergent except for when G and G′ have the same dimension and volume. This is very suggestive if the above-mentioned density-matrix interpretation is taken into account. Third, d(G,G′) does not satisfy the triangular inequality, which illustrates clearly that the spectral distance and the distance defined by the DeWitt metric on the superspace are not equivalent. We then pose a question: Do two universes with different topologies interfere with each other quantum mechanically? In particular, we concentrate on the difference in the orientabilities. To investigate this problem, several concrete models in two dimensions are set up, and the spectral distances between them are investigated: distances between tori and Klein’s bottles, and those between 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. This may suggest that, without any other special mechanism, two such universes interfere with each other quite strongly, contrary to our intuition. We discuss some curious features of the heat kernel for tori and Klein’s bottles in terms of Epstein’s theta and zeta functions. Differences and parallelisms between the spectral distance and the DeWitt distance are also discussed. © 1996 The American Physical Society.