An unconventional canonical quantization of local scalar fields over quantum space‐time

The central issue addressed in this paper is the following: Is the conventional procedure the only one for extending the canonical quantization method for local field theories or does another way exist? Here an unconventional extension of the canonical quantization method is presented for a classical local field theory consisting of N real scalar fields. This approach is essentially a reconsideration of the conventional procedure in an alternative way offered by a recent new approach of classical local field theories. The proposed canonical commutation relations have a solution in the A‐valued Hilbert space H_A =H⊗A, unique up to A‐unitary equivalence, where A is the algebra of the bounded operators of the Hilbert space L²(R³). The canonical equations as operator equations are equivalent in form to the classical field equations, and are a priori well‐defined for interacting systems, too. This model of quantized fields lacks some of the difficulties of the conventional approach, e.g., the rigorous application of the interaction picture is not stemmed by Haag’s theorem and the ultraviolet catastrophe (there are no ultraviolet divergences in the S matrices of the model). Examples of the model satisfying the asymptotic condition provide examples for the axioms of Haag–Kastler while they satisfy the axioms Wightman only partially. The consistent interpretation of the model requires a new concept of space‐time, a quantum space‐time. The ‘‘local’’ state space H_A of the model is constructed over this quantum space‐time.