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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?

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

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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.

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