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.


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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