This is a follow-up question to my question about the uniqueness of the field-momentum operator. The answer suggested that (partially) because the operators can act on different hilbert-spaces, there isn't a choice for the field operator and the field-momentum that is unique up to unitary equivalent choices. However, since the answer stated one of the problems was that those operators can lie in different hilbert-spaces, I was wondering:
When I restrict the pairs of field and field-momentum to those cases where the field-operators are unitary equivalent, does that mean that the operators act on the same hilbert-space, and following, that the field-momentum-operators are as well connected by the same unitary-transformation?
The train of thought here is as follows: Given two pairs of field and momentum, and requiering the fields to be connected by a unitary transformation, I place a condition to both pairs to act on the same hilbert-space. If the problem of having different hilbert spaces was the reason to not have uniqueness up to unitary-transformations, then this uniqueness is given now.