I had a hard time deciding whether this question is more appropriate for physicsSE or MO, so I have cross-listed it for the time being. The physicsSE post can be found [here](https://physics.stackexchange.com/q/830100/288281).

In the lecture [Is relativity compatible with quantum theory?](https://youtu.be/RgQixyA2Gcs?si=8KcGzIkQyIjsZuf0#t=49m10s) on constructive quantum field theory given by Arthur Jaffe (around the timestamp 49:10) he says:

>"Wavefunction renormalization entails a change of representation. In mathematical language a change to a unitarily inequivalent representation of the commutation relations or a change of Hilbert space, even in finite volume"

I found this a very interesting remark, but unfortunately could not find more about it anywhere in the literature. How does a change of representation of the commutation relations result in a kind of renormalization? I suspect the different Hilbert spaces referred to above has to do with the interacting theory being defined on a different Hilbert space as the bare theory and Haag's theorem, but how is this a (physical) renormalization?

On a related note, many introductory quantum field theory textbooks written by/for physicists use the words wavefunction renormalization and field strength renormalization interchangeably. In contrast, the constructive quantum field theory community seems to make a distinction between the two. Is there a mathematical definition that differentiates them?