Model P (φ) 4 quantum field theory. A nonstandard approach based on nonstandard pointwise-defined quantum fields. The field operators and the approximate vacuum

A new non-Archimedean approach to interacting quantum fields is presented. In the proposed approach, a field operator φ ( x , t ) is no longer a standard tempered operator-valued distribution but a nonclassical operator-valued function. We prove using this novel approach that a quantum field theory with a Hamiltonian P φ 4 exists and that the corresponding C * - algebra of bounded observables satisfies all the Haag‐Kastler axioms except for the Lorentz covariance. We prove that the λ φ 4 4 quantum field theory model is Lorentz covariant. In this paper, we consider a somewhat different hyperfinite cutoff theory, namely, the λ : φ 4 4 : theory in a periodic box. This gives a cutoff interaction that is translation invariant, and therefore, it is useful for the study of the vacuum state. In a hyperfinite interval, we prove that the total Hamiltonian is self-#-adjoint and has a complete set of normalizable eigenstates.