Quote by strangerep
Could you please give me a specific reference where these statements are derived rigorously?
(Or are they easy to derive but I'm still missing something?)

Oh, they're certainly not easy to derive. The fact that Wick products give densely defined operators was first proved by Jaffe in 1966 [1]. However I personally fined a later derivation by Segal in 1967 to be much clearer [2]. Segal has a very erudite way of writing, which you will either love or find very difficult to read.
I should also say the theorem is much harder to prove in the Hamiltonian approach that I'm discussing. In the FunctionalIntegral (PathIntegral) approach it's just a matter of evaluating a single Feynman diagram. See Glimm and Jaffe's book Section 8.5, Proposition 8.5.1.
[1] Jaffe, A. : Wick polynomials at a fixed time.
J. Math. Phys. 7, 1250 — 1255
[2] Segal, I. "Notes toward the construction of nonlinear relativistic quantum
fields, I. The Hamiltonian in two spacetime dimensions as the generator
of a C*automorphism group."
Proc. Natl. Acad. Sci. U. S. 57, p.1178—1183
[Edit: I sense a note of frustration in your post #241, so I just like to say two things:
a) THANK YOU for going to the effort in those earlier posts, and THANK YOU in advance
for (hopefully) future episodes of the climbingtheladder saga.
b) I do want to understand these things rigorously, including how one goes about
proving convergence since (among other things) acquiring such functionalanalytic
skill is clearly valuable in any other nonWightman approach that one might wish to
investigate.

You're more than welcome, I will be glad to continue the series of posts.