# What are the scaling variables in the RG approach?

I am reading Cardy's "Scaling and Renormalization in Statistical Physics" and I am a little confused about a concept he introduces in section $$3.8$$.

He starts off with a theory which has some coupling constants $$K_a$$ which under RG flow to $$K'_a$$. Near a fixed point $$K^*_a$$ we get $$K_a' - K_a^* = T_{ab}(K_b-K_b^* )$$ He then introduces left eigenvectors of $$T_{ab}$$ which he calls $$e^i$$ so that $$\sum_a e^i_a T_{ab} = \lambda^i e^i_b$$

He then introduces scaling variables $$u_i = \sum_a e^i_a (K_a' - K_a^*)$$ which have the property that under $$RG$$ $$u_i' = \lambda^iu_i$$

I sort of understand what it means for the coupling constants $$K_a$$ to get bigger or smaller under $$RG$$ flow, I don't have an intuition for what it means for $$u_i$$ to get bigger or smaller. Can anyone give me a physical intuition for what these scaling variables correspond to, physically?

• Recall what a matrix does to the magnitude of its eigenvectors--- whose directions it preserves... Apr 23, 2020 at 19:06

The vector formed with $$u_i$$ as its components is the displacement vector from the fixed point in terms of the eigenbasis of $$T$$.

In other words, the $$u_i$$s represent the displacement from the fixed point in the direction of the eigenvector $$e^i$$; i.e. they tell you how far away you are from the fixed point in coupling constant space.