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I am reading a paper called "New symmetries of massless QED", written by Temple He, Prahar Mitra, Achilleas P. Porfyriadis and Andrew Strominger (https://arxiv.org/abs/1407.3789). At some point at the very beginning the authors state that

"This paper considers theories in which there are no stable massive charged particles, and the quantum state begins and ends in the vacuum at past and future timelike infinity.Of course, in real-world QED the electron is a stable massive charged particle, so it is highly desirable to generalize our analysis to this case.2 However, stable massive charges create technical complications because the charge current has no flux through future null infinity."

From this I am led to understand that the "in" and "out" states are defined in the $|t|\rightarrow\infty$ limit in scattering theory, regardless whether or not the theory is massless (just as the QED theory the authors consider) or massive (just as the real-world theory of QED). This is the reason, hence, the authors want to generalize their arguments to the massive case. However, in the lecture notes written by Strominger called "Lectures on the infrared structure of gravity and gauge theory" (https://arxiv.org/abs/1703.05448), and specifically in Chapter 2, the author states that in order to study scattering problems "whether in electrodynamics or gravity, classical or quantum, one starts by specifying initial data at the past null infinity"

So, which of the above am I supposed to believe and why? Does scattering theory involve definint in and out states at $|t|\rightarrow\infty$ both for massless and massive theories, or does the scattering regarding massless theories imply that we need to consider that the initial states are defined at the past null infinity and the final ones at the future null infinity?

Any help is appreciated.

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As I mentioned in a comment in your other question, null infinity is a Cauchy slice only in purely massless theories. If there are also massive excitations, then you must include $i^+$ as well. Alternatively, you can also use as a Cauchy slice a constant $t$ slice with $t \to \pm\infty$. The latter is what is typically done in standard QFT.

The crux of the paper you mentioned and of Strominger's lectures is to carefully analyze the effects of the boundary of the Cauchy slice. In traditional QFT, one assumes that all fields vanish at spatial infinity, but this assumption is wrong. Boundary contributions are responsible (as Strominger explains in his notes) for all the infrared effects that we see in the real world (such as infrared divergences and soft theorems). These boundary issues are best studied by looking at the Cauchy slices ${\cal I}^\pm \cup i^\pm$ instead of constant $t$ slices.

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