Conformally compactified homogeneous spaces (Possible Observable Consequences)

摘要

Some arguments based on the possible spontaneous violation of the Cosmological Principles (represented by the observed large-scale structures of galaxies), the Cartan-geometry of simple spinors and on the Fock-formulation of hydrogen-atom wave-equation in momentum-space, are presented in favour of the hypothesis that space-time and momentum-space should be both conformally compactified and represented by the two four-dimensional homogeneous spaces of the conformal group, both isomorphic to (S(sup 3) X S(sup 1))/Z(sub 2) and correlated by conformal inversion. Within this framework, the possible common origin for the S0(4) symmetry underlying the geometrical structure of the Universe, of Kepler orbits and of the H-atom is discussed. On of the consequences of the proposed hypothesis could be that any quantum field theory should be naturally free from both infrared and ultraviolet divergences. But then physical spaces defined as those where physical phenomena may be best described, could be different from those homogeneous spaces. A simple, exactly soluble, toy model, valid for a two-dimensional space-time is presented where the conjecture conformally compactified space-time and momentum-space are both isomorphic to (S(sup 1) X S(sup 1))/Z(sub 2), while the physical spaces are two finite lattice which are dual since Fourier transforms, represented by finite, discrete, sums may be well defined on them. Furthermore, a q-deformed SU(sub q)(1,1) may be represented on them if q is a root of unity. (author). 22 refs, 3 figs. (Atomindex citation 26:038651)