# Conformally covariant distributions

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 $x=y$.

• The short answer is you add $i \varepsilon$ to $x-y$. In higher dimensions one has to go to minkowksi space Jun 25, 2015 at 3:58
• @MarcelBischoff, can you please expand on the role of $i\epsilon$? I mean, if I wanted to define $1/x$, I could just use the principal value and that would be just fine, or add some delta functions to agree with $i\epsilon$. But I guess I do not quite understand the role of $i\epsilon$ in $1/|x|^\Delta$. Also, how does Minkowski space help? Jun 25, 2015 at 4:03

So for 1D example the two-point function gets: $$W(x,y) = \lim_{\varepsilon\to 0^+} \frac1{(x-y+i\varepsilon)^{2d}}$$ where the limit is in a weak sense. But the fields don't commute anymore, for example if $d=1$ $$\langle [\phi(x),\phi(y)]\rangle=W(x,y)-W(y,x)\sim \delta'(x-y)$$ so the commutator is supported on the diagonal. But it is a perfect conformal covariant distribution, which is conformally covariant. So in your example you could take $$W(f)=\lim_{\varepsilon\to 0^+} \int_\mathbb{R} \frac{f(x)}{x+i\varepsilon}\,dx$$ as an extension to the Schwartz functions.
• ...We cannot really give sign-alternating values to Schwinger functions because of $SO(d)$ invariance. In Wightman functions the phase changes across null cones, which are $SO(1,d-1)$-invariant. The rigorous argument is basically the OS papers. Oct 31, 2017 at 5:04