# How to properly understand the residue in the LSZ theorem?

The LSZ theorem for a scalar field reads $$\mathcal M=\lim_{p^2\to m^2}\left[\prod_{i=1}^n(p^2-m^2)\right]\tilde G(p_1,\dots,p_n)$$ where $G$ is the $n$-point function, to wit, $$G(x_1,\dots,x_n)=\langle\phi(x_1)\cdots \phi(x_n)\rangle$$

This object (and its Fourier transform) are tempered distributions. I'd like to understand what is the precise meaning of the $\lim$ operation above. Let me make three remarks to motivate the problem:

1. The singular support of $\tilde G$ agrees with the point spectrum of the mass operator $P^2$. But it is precisely at such points where we want to calculate the limit, so we cannot identify $\tilde G$ with a regular function. In other words, the $\lim$ operation is truly an operation on distributions, because we are calculating it at points where $\tilde G$ does not behave as a function.

2. That being said, we cannot regard $\lim$ as a limit in the space of distributions. Such limits are defined as $$\lim_{a\to b}T^a=T\quad\text{iff}\quad \lim_{a\to b}T^a[f]=T[f]\quad\forall f\in \mathcal D$$

Here, we want to take a limit with respect to the "integration variables" $\{p_1,\dots,p_n\}$, so the definition above makes no sense.

3. Finally, what makes this even more perplexing is that, once we take the limit, the resulting object $\mathcal M$ has empty singular support, so it is a regular function.

Summarising these points, $\lim$ is neither a map from $\mathcal D$ to $\mathcal D$, nor a map from $\mathcal D'$ to $\mathcal D'$. It seems to be a map from $\mathcal D'$ to $\mathcal D$, an operation I've never seen before, and that I have no idea how to understand. Thus, my question: what is the precise meaning of the $\lim$ operation? what is its rigorous definition?

• Note: this is not a purely mathematical question, because there is something special about the distributions $G$. Indeed, given an arbitrary distribution you wouldn't expect to observe the behaviour above. But I don't know what makes $G$ special, so it's impossible to formulate the problem purely in mathematical terms. An input from physics is required. It is the analytic properties of a local QFT what is making the problem non-trivial. – AccidentalFourierTransform Mar 29 '18 at 20:09
• There is a precise derivation in sections 14.1 and 14.2 of the book by Glimm and Jaffe (2nd edition). Note that its better to work with connected $n$-point function so there is only one momentum conservation delta function to factor out, i.e., seeing your distribution on a smaller subspace. – Abdelmalek Abdesselam Mar 29 '18 at 21:49
• @AbdelmalekAbdesselam Ah, yes, I was implicitly talking about the connected $n$-point function. Thank you for pointing it out. I'll check G&J. Cheers! – AccidentalFourierTransform Mar 30 '18 at 0:20
• projecteuclid.org/download/pdf_1/euclid.cmp/1103841074 might be useful – Akoben Mar 31 '18 at 21:13