In this question I asked wether the definition of the momentum operator as an operator that has to generate translations by satisfying the canonical commutation relations was ambiguous. The answer to that was that if I require the canonical commutation relations to hold in an exponentiated way, then the Stone-Von-Neumann-Theorem states that the momentum operator for one choosen position operator is unique (and that all pairs of momentum and position operators are connected via a unitary transformation). Correct me if I'm wrong.
In quantum field theory however, the Stone-Von-Neumann-Theorem doesn't work anymore - Does that also mean that now given one field-operator, there are different choices for the field-momentum-operator? (I already strongly presume the answer is "yes").
In case the field-momentum is in fact not uniquely defined anymore - how is the field-momentum usually chosen? Does fixing an operator ordering in that case fix the field-momentum as well, for example?