Does the renormalisation group flow actually involve a group?

Question

I've come across the notion of a renormalisation group flow in a number of places; for example, in Dr. Tongs lecture notes in QFT:

it will turn out that that the possible interactions in QFT will be governed by a few basic principles: locality, symmetry & renormalisation group flow (the decoupling of short scale phenomena from physics at larger scales)

The wiki article whilst informative, doesn't actually say; if a group is not involved, then presumably some group like thing is; also what exactly is 'flowing' in the renormalisation group flow?

I guess RG has all the properties of a group except having an inverse element. That is, once you integrate over the high-energy modes, you cannot get them back. — Sayan Mandal, Oct 02 '17 at 01:42
Possible duplicate of In what sense is the renormalization group equation a group? — Rococo, Oct 02 '17 at 01:52

score 2 · Answer 1 · answered Oct 02 '17 at 01:53

2

RG flow doesn't have a group structure. Here flow refers to the change in couplings in the coupling space with changes in the length or energy scale, which is induced by the beta function of the theory.

answered Oct 02 '17 at 01:53

Bruce Lee

5,237

While the renormalization "group" is not a group, it is almost a group. As mentioned by @Sayan Mandal above, the only thing that is missing is the inverse element. Thus, the renormalization "group" is called a semi-group – jak Oct 02 '17 at 07:20
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@JakobH agreed, and that is the precise reason why I said it doesn't have a group structure, whether you choose to call it a "semi-group" or something else. – Bruce Lee Oct 02 '17 at 07:54

Does the renormalisation group flow actually involve a group?

1 Answers1