quantum field theory - What is the Wilsonian definition of renormalizability?

In chapter 23.6, Schwartz's quantum field theory book defines renormalizability as follows, paraphrasing a bit for brevity:

Consider a given subset $S$ of the operators and its complement $\bar{S}$. Choose coefficients for the operators in $S$ to be fixed at a scale $\Lambda_L \ll \Lambda_H$. If there is any way to choose the coefficients of the operators in $\bar{S}$ as a function of $\Lambda_H$ so that in the limit $\Lambda_H \to \infty$ all operators have finite coefficients at $\Lambda_L$, the theory restricted to the set $S$ is renormalizable.

I'm very confused about what Schwartz is saying here. The RG flow equations are just differential equations that run backward just as well as they run forward. Thus you can choose any couplings at $\Lambda_L$ whatsoever for all of the operators and simply run the RG flow backwards to see what the couplings at $\Lambda_H$ should be.

I also don't see how this is equivalent to the usual definition of 'no irrelevant operators in the Lagrangian'. Moreover, I'm not sure what 'the theory restricted to the set $S$' means. Does this mean we are supposed to forcibly set the coefficients for $\bar{S}$ to zero at $\Lambda_L$?

Could somebody shine some light on this passage?

Answer

There are two kinds of renormalization groups. Lots of pointers to the literature are given here.

The most common renormalization group definition is in the spirit of Kadanoff and Wilson. But this ''group'' is in spite of the name only a semigroup: The renormalization is not invertible, and in general one cannot run the equations backward. Thus being able to continue backwards (in this case this means to arbitrarily high energies) is a very stringent additional requirement.

This is already the rule for simpler systems, such as parabolic partial differential equations. For example, the initial value problem for the heat equation is well-posed, while that for the reverse heat equation is not. Most IVPs have no solution at all, and when there is a solution it is infinitely sensitive to changes in the initial conditions - arbitrarily small changes can be found with arbitrarily large consequences after arbitrarily tiny times. Thus nothing at all can be concluded from the initial conditions unless they are exact to infinitely many digits.

The other renormalization group definition is in the spirit of Bogoliubov & Stückelberg and is a true group.