I'm currently studying the Landau-Wilson model for critical phenomena (Statistical Mechanics, Kerson Huang) where the renormalization group is a central object. In the end, the calculations lead to a set of coupled differential equations that describe the (metaphorical) evolution of the parameters in the Landau energy functional as a function of the logarithm ##\tau## of the uv-cutoff. For a system with two such parameters the vector flow of the solutions can be plotted on a 2D-diagram. Now, I have two issues with this 1) The renormalization group is not a true group because there are no inverse operations. If you zoom out, you lose information and there is no unique way to zoom in again. But with these diferential equations, you can just take ##\tau\rightarrow -\tau## to invert the operations it seems to me? Is this apparent paradox a consequense of some approximations maybe? 2) It looks as if renormalization changes the temperature of the system (it corresponds to one of the eigenvalues of the linearized equations), is this correct? How is this consistent with temperature as an objective macroscopic quantity? for the latter question, I asked a similar question a while ago (https://www.physicsforums.com/threads/is-rest-mass-a-subjective-quantity.866691/) in the context of quantum field theories, before I started studying the renormalization in more detail for critical phenomena. Is the answer similar: is there some limit that defines "the" temperature? Since most paths of the flow go to infinity, this would seem strange to me.