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Convergence of nonlinear massive quantum field theory in the Einstein universe. (English) Zbl 0875.53017
Summary: We treat as prototype for four-dimensional nonlinear quantum field theories the $$g\varphi^q$$ theory in the Einstein universe $$E= R^1\times S^3$$. 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 selfadjoint 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 $$\gtrsim 10^{40} \text{ fm}$$. 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 mass particle.

##### MSC:
 53Z05 Applications of differential geometry to physics 83C47 Methods of quantum field theory in general relativity and gravitational theory 81T20 Quantum field theory on curved space or space-time backgrounds 81V17 Gravitational interaction in quantum theory
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