Compatibility of renormalisation with the quantum-classical correspondence principle We know that Quantum Theories obey the Heisenberg equations of the motion, taking the expected values of which gives us the classical equations.
Also, We replace the mass and coupling parameters of a Quantum Theory with limits like $\displaystyle\lim _{\Lambda \rightarrow 0} m(\Lambda) $. These limits are infinite quantities,  but plugging them into the probability calculations gives us finite predictions. This recipe is part of the definition of the Quantum Theory.
What I'm worried about is that the parameters of coupled classical fields are just normal finite numbers,  instead of limits. We do use infinite masses, but only in the case of field-point particle interaction theories. We don't use infinite parameters for coupled classical fields.
Then how would the correspondence principle still hold if the quantum and the classical theories have different parameters?
 A: I believe that there is no incompatibility. As Connor Behan's comment says, QED does not really exist in the ultraviolet. So to make sense of this question, we first have to work with a QED that is well-defined : Lattice QED.
Now, in lattice QED, our parameters won't be infinite but will just be very large depending on the lattice size. On this theory, we can take a classical limit to obtain a lattice classical electrodynamics theory with very large parameters. It seems like there is some incompatibility here because the classical electrodynamics that produces correct predictions in our daily lives does not have extremely large parameters.
But we have to remember that the classical field theory that the correspondence principle yielded is a theory of a spinor field coupled to the electromagnetic field. This isn't the classical electrodynamics that we're familiar with as we don't model the classical electrons as a spinor field.
To model an electron from Newtonian mechanics, we have to take another limit on the spinor-electromagnetic interaction theory that the correspondence principle yielded us. In this limit, we approximate the current density of the spinor field with an sharply peaked delta-like function.
After taking this limit, we end up with a theory of the electromagnetic coupled to almost point-like particle of extremely large negative mass. This theory reproduces the classical electrodynamics that we're amiliar with, as we know that we have to model point-like particles in classical electrodynamics with an extremely large negative mass.
