Does QFT renormalization preserve Hilbert space? In the Wilsonian picture, a renormalized theory is about change of scale. As we change scale for a quantum field theory, does the Hilbert space of a theory remain unchanged?
 A: We can think of a quantum field theory as an association between observables and regions of spacetime: for each spacetime region $\cal{O}$, we have a collection $\cal{A}(\cal{O})$ of observables. Together with a representation of all of these observables as operators on a single Hilbert space, this is the only data that needs to be specified. (The data must satisfy some conditions, but there is no additional data.)
Conceptually, in the Wilson picture, renormalization amounts to a simple two-step process:


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*Step 1: For some $\lambda>0$, replace each $\cal{A}(\cal{O})$ with $\cal{A}(\lambda\cal{O})$, where $\lambda\cal{O}$ is the image of $\cal{O}$ under a scale transformation of spacetime.

*Step 2: Re-scale the units so that $\lambda\cal{O}$ has the same size in the new units that $\cal{O}$ had in the original units.
Does this change the Hilbert space? For a typical quantum field theory in continuous spacetime, the original Hilbert space is infinite-dimensional (think about the effect of continuous translations), and so is the new one. Up to isomorphism, there is only one separable infinite-dimensional complex Hilbert space, so the Hilbert space must remain the same (for finite values of $\lambda$, at least).
There are two possible loopholes:


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*In the case of a lattice QFT, the Hilbert space may be finite-dimensional. Then the preceding argument fails.

*This one is a semantic loophole. In the quantum (field) theory literature, the expression "Hilbert space" sometimes implicitly refers to the net of observables as well as to the Hilbert space. For example, some authors may refer to the Hilbert space of square-integrable functions of a single real variable and the Hilbert space of square-integrable functions of two real variables as two different Hilbert spaces, when in fact they are the same Hilbert space in two different representations, each tailored to a different net of observables. The observables — at least their association to regions of spacetime — certainly do change under renormalization in the Wilson picture, and I suppose some authors might describe this as a change of the Hilbert space.
