Here's a question about inequivalent representations of the CCRs... For a given Hilbert space representation, what is it that determines which set of field operators \phi(x), or \phi(f) if we want to get rigorous a la Wightman, gives us THE field operators for that representation. For example, say I use a canonical transformation to go from the Fock representation to one of the "coherent representations," that is I transform \phi(f) -> \phi'(f) = \phi(f) + L(f) I for L some linear functional and I the identity operator. If L(f) is not bounded for normalized f, the resulting coherent representation is unitarily inequivalent to the Fock representation. But the field operators \phi'(f) are linear combinations of the Fock field operators and scalar multiples of the identity operator, so of course these are also operators in the Fock representation. And they satisfy the same commutation relations as the Fock field operators do. So what makes it the case that the operators \phi(f) are the Fock field operators, while the \phi'(f) are the field operators in the coherent representaiton, when both sets of operators are well-defined on both representations?