Exact Results in Quantum Theory

We develop a framework in which quantum field theory is formulated on afinite, context-dependent region of phase space characterized by adimensionless cardinality N, rather than by an independentlyprescribed ultraviolet cutoff. In the gravitational de Sitter problem thegravitational cardinality N0 is identified with theGibbons--Hawking entropy, relating the small cosmological constant to thelarge phase-space cardinality of the equilibrium state. At fixedcardinality, variations of the spacetime and momentum scales are linked,so that the gravitational stationarity condition can be written as ascale-stationarity condition, which we denote by RG=GR. Including a one-loop massive scalar determinant,we find finite quantum-corrected de Sitter solutions, with heavy fields decoupling from the cosmological curvature. We also emphasize that this resultapplies to the global equilibrium problem; local quantum processes generallyinvolve their own contextual phase-space regions and cardinalities.