On the mathematical description of quantized fields

Abstract

We start from Haag's proposal to describe quantum fields at a point, corresponding to the heuristic description by means of their matrix elements (A(x)Φ‖Ψ) between vectors of a dense linear manifoldD of the Hilbert space. We particularize this idea, so that the sesquilinear functional that describes the field at a point may be considered as an element of the sequential completion of a space of operators, endowed with a suitable “D-weak” topology.

It is shown that any Wightman field may be described in this way, as a rather elementary consequence of the existence of a translation invariant cyclic vacuum. Furthermore the field turns out to be an infinitely differentiable function of space and time.