Crossposted from physics stackexchange:
I am interested in what is known about the possibility of smearing interacting quantum fields on a Cauchy slice. This is easy to do for free fields and their conjugate momentum, and indeed this is necessary to impose the CCR's. However, in *Local Quantum Physics* Haag makes a claim that "renormalization theory *suggests* that it is essential to smear out in both space and time, in contrast to the case of free fields... Due to the stronger singularities one cannot assume well-defined commutation relations at equal times."[Haag 1996, p.59]
I don't know what logic exactly he is drawing on here, but one can at least see that unlike a smearing in spacetime, or even a smearing on a timelike curve, spatial smearings do not *have* to map states of bounded energy to each other. It is a happy accident that spatially smeared free fields do, and since the set of states of bounded energy is typically dense in the Hilbert space, this means that they are well-defined unbounded operators. However, from abstract principles alone it's hard to rule out that this is also true for general interacting quantum field theories in the continuum - and even if it isn't, and spatial smearings *don't* preserve the set of bounded energy states, it still doesn't mean they aren't well-defined on some dense subset. I haven't found any further arguments in the literature, except for a remark in Halvorson's notes on AQFT that Haag's claim is speculative, and no proofs are known.[Halvorson 2006, p.53]
What *is* known about this question? For example, are there concrete interacting models where the Wightman field cannot be smeared in space? What wisdom does rigorous causal perturbation theory suggest? If you adiabatically switch on an interaction in a compact region of spacetime, can you see at any order whether or not the field can be smeared in space?