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I always thought that these two terms are some kind of synonyms, meaning that if you have a self-similar or scale invariant system, you can zoom in or out as you like and you will always see the same picture (physics).

But now I have just read in the context of a lattice of spins model for example, that if the system is at its critical point and therefore scale invariant, it does not mean that it is self-similar as naively discribed in the first paragraph of this question. The author of the paper I am reading even calls it a false picture on p. 9. Later on on p. 24 he explains that poles on the positive real axis on the so called Borel plane break self-similarity because lead to tha fact that to obtain the effective action (for a $\lambda\phi^4$ bare action) the scale dependent non perturbative power correction terms have to be kept. So if scale invariance and self-similarity are not exactly the same, the explanation for breaking of scale invariance should be (slightly and subtly?) different?

Now I am confused and my question simply is: what exactly is the difference between scale invariance and self-similarity (if any ...)?

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Extremely interesting paper! Btw, check out the last paragraph on Is it disagreeing with the paper's claim on renormalons in the $\lambda \phi^4$ theory :-? (if I understand the comments on p9 and p24 correctly). – Siva May 24 '13 at 17:52
@Siva yeah, I have just started to read the paper, but I already like it. Hm yes the last paragraph seems to disagree with what is said in the paper. I dont know how recent the recent suggested proof for the non existance of renormalons actually is? The papers mentioned in the Wiki article are all older than the ERG paper, so maybe this proof was wrong, who knows...? About those renormalons I have not yet heard before, maybe I'll have to ask about them anyway. – Dilaton May 24 '13 at 22:09
I first heard about renormalons in a nice talk by Argyres, based on his work with Unsal. You can look on the arXiv for their 2012 papers. Fwiw, a little googling turns up this interesting talk by Unsal. – Siva May 24 '13 at 23:51
The clarification is in ref [6] Section 1.4.2, I presume. – Kostya Jun 5 '13 at 10:20
up vote 5 down vote accepted

From the point of view of non linear dynamics where self similarity plays an important role if the attractor is a fractal I would say that the difference is one between continuous and discrete transformations.

A self similar transformation like the one producing the Cantor set or the Sierpinski triangle proceeds by discrete stages. The fractal which is the limit when the number of stages N tends to infinity shows self similarity (e.g is identical to itself) only for a discrete number of stages.

For instance when zooming on the Sierpinski triangle, one may not zoom anywhere and by any zooming factor. One has to zoom only with a factor 1/3 and center the zoom on the symmetry axis of the triangle. So basically the number of self similar objects is an integer and has for characteristic the self similarity dimension which is a number D such as N=L^D where N is the number of copies produced by changing the size by L.

As for scale invariance which is not so largely used, it is a statement that f(µx) = µ^D.f(x) with some constant D. The property is continuous and true for every x. Fractal attractors are generally not exactly scale invariant - they have often 2 or several different scalings.

Hence from this point of view the self similarity and scale invariance may only be identical in a discrete number of points for simple fractals which have a unique scaling factor. I am aware that this does not adress spin lattices but it answers the question in the frame of the chaos theory.

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