Renormalisation, as used in Quantum Field Theory, we can take a minimum length $\delta x$ and then the coupling constants must depend on this length. So we have an action something like:
$$\int \left(\cdots+\overline{\psi}(x) \gamma^\mu \frac{\psi(x+\delta x) - \psi(x)}{\delta x} + \overline{\psi}(x) e(\delta x) A_\mu(x)\psi(x) + \cdots \right) \, \delta x$$
where the coupling constant $e$ depends on the value we choose for $\delta x$. Which corresponds to some discretisation of space-time.
Now, the language of renormalisation does not seem to fit into the normal language of differential calculus or even functional calculus.
Therefor, is it right to say that renormalisation theory is a new kind of differential calculus? And if so, has there been any new notation to express this calculus (e.g. normal calculus has the integral and differential symbols). Most articles I've seen on renormalisation seem to have very clumsy notation. Or on the other hand very abstract notation.
Even trying to express renormalisation in a compact mathematical notation seems to be very tricky.