|
Abstract |
Recently, Di Valentino, Melchiorri and Silk argued that the Planck
satelite data provide evidence for a closed universe. In this
context we recall the derivation of the Quantum Hamilton--Jacobi
Equation (QHJE) from a fundamental geometrical principle. Underlying the
formalism there exist a basic cocycle condition, which is invariant
under $D$--dimensional finite M\"obius transformations. The invariance
of the cocycle condition under M\"obius transformations can only
be implemented consistently if spatial space is compact. Thus, the geometrical
formulation of the QHJE predicts that spatial space is compact. Additionally,
it implies energy quatisation and the undefinability of quantum trajectories.
Evidence for the compactness of the universe may therefore exist in the
Cosmic Microwave Background Radiation. |
