# Link between anomalous dimensions and fractal dimensions

I just realized that anomalous dimensions in quantum/statistical field theory is not that different from fractal dimensions of objects. They both describe how quantitaive objects transform under a scale transformation (renormalization group transformation in the QFT case and dilation of the mesh/ruler when computing the perimeter of a fractal). Is there a more profound link between the two? I have't read much about the subject but it seems that for any statistical model at criticality the fractal dimension of the clusters becomes a function of the full dimension of the field. Is it a general rule?

$$S=\int d\omega dk\, \psi^\dagger(\omega, k) [i\omega - \epsilon(k)] \psi(\omega, k)$$
It is easy to verify the scaling dimension of $$\psi$$ is $$-3/2$$, which is fractional. A field scaling with $$-3/2$$ power of the linear size is much less exotic than a shape doing so. The former does not have a direct correspondence with any fractal.