# What are the necessary or sufficient conditions for a renormalization group scheme to be “valid”?

Suppose I have a super operator $$G$$ which acts on Hamiltonians to produce a new Hamiltonian that is related somehow. For the purposes of this question, suppose that these Hamiltonians are defined on an $$L\times L$$ 2D lattice with spins at each point (but the question applies more generally). What are the conditions that $$G$$ implement a good renormalization group scheme (in real space)? Is there a rigorous definition?

We are typically interested in phases when applying the RG scheme, hence presumably we want to preserve whether the Hamiltonian is gapped or gapless. I suppose this requires preservation of the low energy subspace.

Typically we see that given a family of Hamiltonian with parameters $$a_1, a_2, \dots$$, then a good RG scheme must map to a Hamiltonian of the same form $$G:H(\vec{a}) \rightarrow H(\vec{a}').$$

But what can these parameters include? Can they include the local Hilbert space dimension (i.e. can the local Hilbert space dimension diverge), or are they limited to the just coupling parameters? If so, why?

Any references would be greatly appreciated! All I can find in the current literature are very "hand-wavey"/ intuitive notions of what a renormalization group is.