Spectral representation of the spacetime structure: The ‘‘distance’’ between universes with different topologies

Masafumi Seriu
Phys. Rev. D 53, 6902 – Published 15 June 1996
PDFExport Citation

Abstract

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 PG[⋅], where the geometrical information G (represented by the spectra) determines the detailed shape of the functional. Then, the overlapping functional integral between PG[⋅] and PG[⋅] 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.

  • Received 20 November 1995

DOI:https://doi.org/10.1103/PhysRevD.53.6902

©1996 American Physical Society

Authors & Affiliations

Masafumi Seriu

  • Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411007, India
  • Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606, Japan

References (Subscription Required)

Click to Expand
Issue

Vol. 53, Iss. 12 — 15 June 1996

Reuse & Permissions
Access Options
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review D

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×