Self-Interacting Real Scalar Quantum Field Theories which Admit Stationary States and a Classical Limit

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Abstract
Conditions for a local, essentially nonlinear, real scalar field theory to admit vacuum, one-quantum, and quasiclassical stationary states are derived with the Rayleigh-Ritz procedure for functionalities. It is shown that only so-called Class A theories, for which the self-interaction energy density divided by the field squared is uniformly bounded for all values of the field, admit wholly acceptable one-quantum and quasi-classical stationary states. The Class-A theories appear to be effectively linear, with no macroscopically observable self-interaction, in the classical limit.