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1. Define microscopic system with some Hamiltonian H.

2. Coarse grain this system in some (arbitrary) fashion, which gives rise to a system with a new H in terms of the coarse grained variables.

3. Now we use some input from experiment: near the critical point the system appears to be self similar at all scales, so we impose the requirement that H is of the same form in the new system so that the coarse grained system behaves as it should; the new H has some different set of constants in general given by a transformation.

4. We can then go on to find critical exponents and the critical coupling.

The problem comes when I read or hear statements such as in Leo Kadanoff's book where he says that "each phase of the system can be described by a special set of couplings, K*, which are invariant under the RG transform". The procedure outlined above (if i've got it right!) only applies around the critical point from step 3 onwards, as elsewhere the system certainly doesn't look the same on all scales. Why does the recursion relation we previously obtained have any meaning away from the critical point?

Thanks in advance!