Convergence of nonlinear massive quantum field theory in the Einstein Universe

Abstract

We treat as prototype for four-dimensional nonlinear quantum field theories the gϕ q theory in the Einstein Universe E = R 1 × S 3. The underlying free system is defined by the Klein-Gordon equation in E. We show rigorously, without the intervention of any cutoffs or perturbative renormalizations, that the interaction and total hamiltonians are self-adjoint operators in the free field Hilbert space that depend continuously on g. The boundary condition that the interacting field is asymptotically free in the infinite past is rigorously implemented, and a unitary S-matrix of Yang-Feldman type is given a finite expression. Our formalism agrees with that of conventional relativistic theory within terms of order 1 R , where R is the cosmic distance scale (radius of S 3) in laboratory units and 10 40 fm. Any mass packet in Minkowski space extends covariantly to the ambient Einstein Universe. The microscopic relevance of cosmic effects is discussed; e.g., Einstein gravity produces an effective cutoff of order 10 37 Gev on the energy of massive particle.