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- Is renormalization group a true group for renormalizable theories?

It is often said that the renormalization group (RG) is not a true group but only a semi-group, because the RG transformation is not invertible. But for renormalizable theories, the renormalized Hamiltonian has the same form as the original Hamiltonian, only with some different values of the finite number of parameters (coupling constants). Denoting the set of these "coupling constants" as ##g=(g_1,\ldots,g_n)##, for a renormalizable theory we have a set of coupled equations of the form

$$\frac{dg_i(t)}{dt}=\beta_i(g(t))$$

where ##t## is the scale parameter. Such equations can be solved uniquely in both forward and backward directions of ##t##, so in this sense the RG transformation is invertible. Does it mean that RG is a true group for renormalizable theories?

$$\frac{dg_i(t)}{dt}=\beta_i(g(t))$$

where ##t## is the scale parameter. Such equations can be solved uniquely in both forward and backward directions of ##t##, so in this sense the RG transformation is invertible. Does it mean that RG is a true group for renormalizable theories?

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