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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.

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You need to make a distinction between two kinds of classical field theory. One is classical field theory in statistical mechanics (e.g. in John Cardy's book). That indeed has a lot to do with quantum field theory and renormalization.

What your question seems to be about is instead deterministic classical field theories like general relativity or a point particle interacting with an electromagnetic field. In classical field theories we can also introduce a cutoff scale. One reason we would do this is if we get divergent quantities and need to regularize. This is discussed in the references at the end of your question.

But another reason is if we don't care about the physics at short distances because it is too nonlinear. This is the case in gravitational physics when we are describing large scale structure. At very large scales in the universe we can use a linear approximation and treat any nonlinearities as perturbations. But at the scales of individual stars for instance matter is clumped and the gravitational equations are highly nonlinear (and even non-gravitational physics may come into play depending on scale).

So we consider smoothed out fields that get rid of modes higher than some cutoff. Interactions with higher modes instead take place through effective interactions like viscosity terms. Our choice of the exact cutoff was arbitrary, so in order to get the same large scale physics no matter what we choose we need to treat the 'bare' couplings like viscosity to be cutoff dependent, and there is an RG flow.

Some examples are arXiv:1310.2920 and arXiv:1206.2926.

(by the way, the acronym CFT is usually used for conformal field theory)

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