# Wilson's Renormalization Group and Lie's Third Theorem

If you think of a one-parameter group of transformations along a curve in the plane as a (Lie) group, and the tangent vector to the curve as a generator of the curve we can intuitively understand Lie's Third Theorem on how a local Lie algebra generates a global Lie group.

Does this theorem apply for semi-groups? More specifically, does it apply to Wilson's Renormalization group, in other words do the renormalization group equations generate a curve?

(It seems as thought they generate the flow of the coupling parameter so if so - is that all it is, a mathematical irrelevancy or is there physical meaning?)

I ask to understand whether one can give Wilson's renormalization (semi) group, in the context of the statistical-mechanical partition function, any kind of similar interpretation. If it helps, refer to the 1-dimensional Ising model as an example, thank you.

References

1. "Renormalization Group Theory", Yale 430b/530b "Statistical Methods and Thermodynamics" Lecture Notes http://xbeams.chem.yale.edu/~batista/vaa/
2. Bluman "Symmetry and Integration Methods for Differential Equations" P. 37
3. Wilson, "Renormalization Group and the Kadanoff Scaling Picture", P. 3177
4. Grossco, "Solid State Physics" 2nd Ed. P. 764 (For the Flow Diagram)
• I don't understand the question. Are you asking if the RG produces a flow of the parameters in the action ? – Adam Jun 8 '14 at 16:23
• @Dilaton: Yes sure, I know all that. I just don't understand if that's the question asked. – Adam Jun 8 '14 at 19:07
• I'm only working with a partition function as in that Ising model link. – bolbteppa Jun 8 '14 at 20:04
• What should be, in your question, the corresponding of the Lie algebra appearing in the Lie theorem? – Valter Moretti Jun 9 '14 at 9:51