M. Luescher in his talk on p.6 writes that the 2-point correlation function of a Hermitian local field $O_k$ of scaling dimension $d=3-k$ looks like $$ \langle 0| O_k(x) O_k(y) |0\rangle = A_k (x-y-i \epsilon)^{2k-6},\quad A_k\in\mathbb{C}. $$ I am not really sure why is it not just $$ A_k(x-y)^{2k-6}\quad? $$ Is $-i\epsilon$ simply added to show that one needs to integrate over the upper half-plane?
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First it means that it is a distribution, which you have to smear with test functions and then take the limit $\varepsilon \to 0$. Without the $\varepsilon$ description it is not well defined.
Now the precise $i\varepsilon$ comes from the positivity of energy, i.e. the spectrum of the generator of translations $P$ is positive.
In any book on Wightman field theory it is proven that the correlation function are boundary values of holomorphic functions in the upper plane (or more general, some tube).
Roughly speaking, in your case if you calculate the Fourier transformation $\hat W(p)$ of $W(x-y)=\langle \Omega,\phi(x)\phi(y)\Omega\rangle$ then the $i\varepsilon$ gives you that this is a function with support on the positive real line. The support of $\hat W(p)$ is the spectrum of $P$. Namely, $\hat W(p) \sim \Theta(p) p^{2d-1}$ where $d$ is the scaling dimension.
The general Fourier transformation is formula (2.13) in https://arxiv.org/abs/hep-th/0009004 Note that for higher dimensions it must be $i\varepsilon e_0$ where $e_0$ is a timelike vector.
Marcel
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