In Conformal Field Theory (in $D$ dimensions) one considers (in particular) correlation functions of the form
$$
\langle O(x)O(y)\rangle,
$$
where $O$ is a scalar primary field. Scale covariance demands
$$
\langle O(\lambda x)O(\lambda y)\rangle=\lambda^{-2\Delta}\langle O(x)O(y)\rangle,
$$
where $\Delta\geq\frac{D-2}{2}$ is the scaling dimension of $O$. Translation and rotation invariance require the correlator to be a function of $|x-y|$ only.

More generally for a conformal transformation $x\to x'$ one has
$$
\langle O(x)O(y)\rangle=\Omega(x')^{\Delta}\Omega(y')^{\Delta}\langle O(x')O(y')\rangle,
$$
where
$$
\frac{\partial {x'}^\mu}{\partial x^\nu}=\Omega(x')R_\nu^\mu(x'),
$$
and $R$ is an orthogonal matrix. For the two-point function considered above the unique non-zero solution of these constraints is, up to a normalization,
$$
\langle O(x)O(y)\rangle=\frac{1}{|x-y|^{2\Delta}}.
$$
This is in the class of functions defined for $x\neq y$, and can be seen by using a conformal transformation which brings $x$ and $y$ to some standard positions.

Now, Osterwalder-Schrader axioms (OS) require the correlation functions to be distributions defined on a suitable class of test functions. Clearly, if $\Delta$ is sufficiently large, one cannot *naively* interpret the above correlation function as a distribution on test functions with compact support due to the singularity at $x=y$. In fact, OS use a class of smooth functions which vanish with all derivatives at $x=y$. This clearly removes the singularity.

If we consider an analogous problem of turning $G(x)=|x|^{-\Delta}$ into a distribution in 1 dimension, one can try something like
$$
\langle G_\epsilon,f\rangle=\int_{|x|>\epsilon}G(x)f(x)dx+\int_{|x|<\epsilon}G(x)\left(f(x)-f(0)-f'(0)x-\ldots\right)dx,
$$
where in the second integral we subtract sufficiently many terms of Taylor expansion of $f$ at $0$ to make the integral convergent. $G_\epsilon$ defined this way is a distribution on compactly supported test functions without any requirement on behavior at $0$. Note that $G_\epsilon-G_{\epsilon'}$ is supported at $0$.

However, this definition breaks scale covariance, since we explicitly introduce a scale $\epsilon$. If we generalize to higher dimensions, then conformal covariance is also broken. My question is, is it possible to define the above correlation function as a distribution on a class of test functions larger than that of OS, while preserving conformal or scale covariance? I am particularly interested in test functions which would feel distributions supported at $x=y$.

This post imported from StackExchange MathOverflow at 2015-07-07 11:11 (UTC), posted by SE-user Peter Kravchuk