  The Fubini's idea to introduce a fundamental scale of hadron phenomena by
means of dilatation non-invariant vacuum state in the frame work of a scale
invariant Lagrangian field theory is recalled. The Fubini vacua is invariant
under the de Sitter subgroup of the full conformal group. We obtain a finite
entropy for the quantum state corresponding to the classical Fubini vacua in
Euclidean space-time resembeling the entropy of the de Sitter vacua. In
Minkowski space-time it is shown that the Fubini vacua is mainly a bath of
radiation with Rayleigh-Jeans distribution for the low energy radiation. In
four dimensions, the critical scalar theory is shown to be equivalent to the
Einstein field equation in the ansatz of conformally flat metrics and to the
SU(2) Yang-Mills theory in the 't Hooft ansatz. In D-dimensions, the Hitchin
formula for the information geometry metric of the moduli space of instantons
is used to obtain the information geometry of the free-parameter space of the
Fubini vacua which is shown to be a (D+1)-dimensional AdS space. Considering
the Fubini vacua as a de Sitter vacua, the corresponding cosmological constant
is shown to be given by the coupling constant of the critical scalar theory. In
Minkowski spacetime it is shown that the Fubini vacua is equivalent to an open
FRW universe.
