# Questions tagged [scale-invariance]

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### Why does the Ising model at the critical point have scale invariance?

If my current understanding of phase transitions and the renormalization group (RG) method is true, RG is a kind of 'zooming out' process, since this procedure makes a block of neighboring spins and ...
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### Scaling invariance of fluid energy

I have read that the incompressible Navier Stokes equation is preserved by the scaling $$x',y',z'=\lambda x, \lambda y, \lambda z$$ $$t'=\lambda^2 t$$ $$u'=(1/\lambda) u$$ As I understand it, fluid ...
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### Scaling of correlation functions

In book Renormalization Group and Fixed Points there is formula for scaling of correlation function: But this formula contradict my understanding of scaling of correlation functions. For free theory ...
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### Importance of Tracelessness of Tensor?

What makes the trace-free tensor (or part of it) so important? As in trace-free Ricci tensor or Weyl tensor.
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### Scale invariance of lagrangians and transformation properties of fields under dilations

Consider a field theory, and a rescaling transformation of the coordinates \begin{equation} T_\epsilon[\phi(x)]=\phi((1+\epsilon)x). \end{equation} From what I understand, one usually requires that, ...
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### A traceless stress energy tensor?

I'm trying to solve this exercise: Suppose an arbitrary theory (Flat space-time?) with a single field (Is a scalar field?) invariant under dilations, i.e. $x\mapsto b x$ and $\phi \mapsto \phi$. ...
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### How does it make sense to talk about the size of a string if the string action is conformally invariant?

From what I understand the Polyakov action in string theory is essentially something like $$S(\xi, g, G)=\kappa \int_{\Sigma} d \mu_{g} \operatorname{Tr}_{g} \xi^{*} G$$ where $\Sigma$ is a given ...
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### Conformal transformation

I am reading some lecture notes on Conformal Field Theory by Joshua D. Qualls (https://arxiv.org/abs/1511.04074). At the end of page 5 of these notes, it is stated that the four momentum transforms ...
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### Derivation of the Ising free energy close to a critical point

In "Statistical physics of fields" Mehran Kardar states that the Ising free energy scales with, $$f(t,h)\sim t^\alpha g_f\left(\frac{h}{t^\Delta}\right),$$ wherein $t=\vert T-T_c\vert/T_c$ ...
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### Mismatch between conformal generators and conformal transformations as changes of variables

Introduction It is known that under changes of coordinates different fields transform according to their tensorial nature (scalar, vector, etc.) like$^{}$ \begin{equation} \phi(x)\rightarrow\phi'...
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### What do we mean by scale invariance in a classical field?

First of all, I read many questions but they don't seem to answer my specific question. So, here it goes According to Francesco's Conformal Field Theory and many other books, a scale transformation \...
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### Scale invariance and state equation for fluid dynamics

I am trying to understand the example provided in this section of the Wikipedia article on scale invariance. In particular where it says In order to deduce the scale invariance of these equations ...
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### Constraints on correlation functions of Quasi Primary Fields

I have problems understanding constraints on correlation functions of quasi primary fields (QPF) following DiFrancesco's Conformal field theory book. In chapter 4, section 4.2.1, a QFP is defined as a ...
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### Why is vanishing beta function associated with scale-invariance?

Why is vanishing beta function associated with scale-invariance? Coupling constants have change rate of zero at some scale, but how is that related to scale-invariance? Association of vanishing beta ...
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### Rotational symmetry $\leftrightarrow$ isotropy, dilation symmetry $\leftrightarrow$ ________?

Symmetries correspond to specific properties of the space in question. translation symmetry $\leftrightarrow$ homogeneity, rotational symmetry $\leftrightarrow$ isotropy What property is related to ...
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### Why does Critical Points have fluctuations on all scales (Infinite correlation length?

I have been studying statistical field theory for a while and I still haven't found a physical explanation for this question. Every answer seems to be kind of circular. Basically something like this: "...
How do derivatives of fields transform under dilatations? Specifically I am interested on what I misunderstand with the example: Consider a theory that has a field $A_\mu$ that transforms under ...