# Difference between the idea of renormalization in quantum and classical field theories

In this answer, AccidentalFourierTransform explains that

Renormalisation has nothing to do with Classical vs Quantum. Any theory, classical or quantum mechanical, needs renomalisation if and only if it is non-linear.

Can anyone help me understand the idea of renormalization in classical field theory (CFT) and how is it similar/different from the idea of renormalization in QFT?

To be concrete, consider the Lagrangian of QED. But let us treat it classically i.e., the Dirac field and the electromagnetic fields as classical fields. How is the renormalization carried out in this CFT?

In QFT, renormalization has to do with integrating out high frequency Fourier modes in the partition function defined in terms the path-integral as $$Z[J]=\int D\phi \exp[i\int d^4x(\mathcal{L}+J\phi)].$$ The Wilsonian picture in QFT quantities get renormalized in the process of integrating out the short-distance physics. But no such path-integral exist in CFT, this picture of renormalization (and RG flow) is lost.

Is there a similar "Wilsonian-like" perspective to understand the necessity and meaning of renormalization in CFT? Or is it that the idea of renormalization is completely different in classical physics and has nothing to do with integrating out short-distance physics?

I have looked at the references 1 and 2 , none of which address the question I'm interested in.