For what values of $\lambda$ is the distribution $(x-i\varepsilon)^\lambda$ positive?

Question

I've been reading the famous unpublished paper by Luescher and Mack "The energy momentum tensor of critical quantum field theories in 1+1 dimensions". In the proof of their main theorem, page 7 of the manuscript, they write:

"$$ <0|O_k^\dagger(x) O_k(y)|0> = B_k (x-y-i\varepsilon)^{2k-6}, \quad B_k\in\mathbb{C},\; x \in\mathbb{R} $$

This distribution should be positive. As is well known [5], this implies $2k-6\le 0$. Thus $O_k = 0$ for $k>3$."

[5] is Gelfand, Shilov "Generalized functions, Vol I".
Here $k=0,1,2,\dots$

I found in this book that for $\lambda = -n$, where $n$ is a positive integer,

$$ (x-i0)^{-n}=x^{-n} + \frac{i\pi(-1)^{n-1}}{(n-1)!}\delta^{(n-1)}(x), $$

where the distribution $x^{-n}$ for $n=2m$, i.e. $n$ even, is defined

$$ (x^{-2m},\phi)=\int_0^\infty x^{-2m}\left(\phi(x)+\phi(-x)-2\left[\phi(0)+\frac{x^2}{2!}\phi''(0)+\dots+\frac{x^{2m-2}}{(2m-2)!}\phi^{(2m-2)}(0)\right]\right)dx $$

I tried calculating the inner product with $\phi(x)=\exp(-x^2)\ge 0$ for various $\lambda$'s. However, for $\lambda = -2$ I got $$ ((x-i0)^{-2},\phi)=(x^{-2},\phi)=\\ \int_0^\infty \frac{\phi(x)+\phi(-x)-2\phi(0)}{x^2}dx=\int_0^\infty \frac{2\exp(-x^2)-2}{x^2}dx=-2\sqrt\pi<0 $$ so that according to Lieb and Loss "Analysis" definition of a positive distribution, $(x-i0)^{-2}$ is not positive, whereas according to the paper the distribution should be positive at least for $\lambda=-6, -4, -2,0$.

Definition of positive distribution (Lieb and Loss)
A distribution $T\in \mathcal{D}'(\Omega)$ is positive if $T(\phi) \ge 0\; \forall \phi \ge 0$, where $\phi\in \mathcal{D}(\Omega)$. Here $\phi \ge 0$ means that $\phi(x) \ge 0 \;\, \forall x\in\Omega \subset \mathbb{R^n}$.

The paper can be found here, the operators $O_k$ are defined on page 6: Luescher and Mack "The energy momentum tensor of critical quantum field theories in 1+1 dimensions"

Answer 1

I am not sure if you mean $x$ one or two dimensional variable. I don't have the article at hand but I guess we probably mean $D=1$??

The argument goes as follows: you have to use Bochner's theorem, which says that $W(x-y):=W_2(x,y)$ is positive distribution if and only if the Fourier transformation yields is a positive measure $\tilde W(p) d^Dp$.

In this case the Fourier transformation is something like $$ \tilde W (p) \sim \frac{\theta(p_0)\theta(p^2)}{(p^2)^{\frac D2 +\lambda}} $$ see e.g. Remark 2.2. in Ref. 1.

Then you get for $\lambda \geq 0$ this is not integrable. Also $d=0$ is a trivial special case.

Reference:

1. N. M. Nikolov & I. T. Todorov, "Rationality of Conformally Invariant Local Correlation Functions on Compactified Minkowski Space", Commun. Math. Phys. 218 (2001) 417-436, arXiv:hep-th/0009004.