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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70 views

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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RG fixed points and $T_{\mu\nu}$

It is common to refer to fixed points of the renormalization group as scale invariant theories. This statement can be formulated as $$ \beta(\mu) \Big |_{\mu^*} = 0 \; \; \Longrightarrow \; \; T^{\mu}...
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Can the breaking of a steel cable be considered as a crackling noise?

How strange are the ways for finding questions! I was searching for some information on the topic of topological quantum field theory (a question was asked in this context), which directed me to the ...
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59 views

Doesn't the massless $g\phi^4$ theory bound to have an infrared fixed point?

A free, massless scalar theory, $\mathcal{L}_1=\frac{1}{2}(\partial\phi)^2$, is scale-invariant both classically and quantum mechanically. However, a $g\phi^4$ theory, $\mathcal{L}_2=\frac{1}{2}(\...
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What's wrong with this argument that Newton's second law implies all potentials are quadratic?

Newton's second law states: $$F(\vec{x})=m\vec{\ddot{x}}$$ For $\vec{x}$ scaled by some arbitrary constant $s$, we obtain: $$F(s\vec{x})=ms\vec{\ddot{x}} \Longleftrightarrow \frac{F(s\vec{x})}{s}=m\...
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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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30 views

Is a function of conformal ratios invariant under conformal transformations?

If I have a function $f:=f(r,s)$ a function of the conformal ratios $r$ and $s$ only, i.e. for example: $$r := \frac{(x1-x2)^2(x3-x4)^2}{(x1-x3)^2(x2-x4)^2} \qquad, \qquad s := \frac{(x1-x4)^2(x2-x3)^...
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574 views

Noether's current for dilation transformation

Consider the Lagrangian of $\phi^4$ theory $$ \mathcal{L} = \frac{1}{2}\partial_\mu\phi \partial^\mu\phi - \frac{\lambda}{4!}\phi^4. $$ We define the following dilation transformation $$ x^\mu \...
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1answer
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Scaling transformations, definitions and all that's not mentioned

If we transform the massless scalar field Lagrangian $$\mathcal{L}=\frac{1}{2}(\partial_\mu\varphi)^2-\frac{\alpha}{4!}\varphi^4$$ with the simultaneous transformations $$x\mapsto x^\prime= \lambda x,\...
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1answer
95 views

Most general Lagrangian in Conformal Quantum Mechanics

This question has already been asked and answered in Most general Lagrangian in CFT in 0+1D. However I am just partially convinced with the answer. The idea is to construct the most general ...
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Is it possible to construct a real-space renormalization group for a system with limited self-similarity?

I'm trying to understanding multiscale dynamics in a system which can be loosely mapped to a lattice model embedded in real space, in which the types of entities at the lattice points, and their ...
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1answer
140 views

Do all the classical critical lattice models have emergent conformal invariance?

I understand that any quantum lattice model at the critical point which can be described by a massless relativistic quantum field theory has emergent conformal invariance. My question is what about ...
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1answer
44 views

Scale invariance in (2+1)D nonrelativistic field theory

Context: I am reading a paper named 'Nonrelativistic field-theoretic scale anomaly' on scale invariance in nonrelativistic field theory. The Lagrangian density for the scalar field is given by, $$\...
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2answers
1k views

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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76 views

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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48 views

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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Finite conformal transformations of fields from infinitesimal

I know that in conformal field theories conformal group acts not by pushforwards but (e.g. for scalar field $\phi$ with conformal dimension $\Delta$) $$ \phi(x) \mapsto \phi'(x') = \left| \frac{\...
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Why isn't scaling space and time considered the 11th dimension of the Galilean group?

Galilean transformations are said to have 10 degrees of freedom. Four for translation in space and time, three for rotation, and three for direction of the uniform motion. If I scale space axis by $\...
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75 views

How scale invariance is broken in nature?

By definition a system will exhibit scale invariance at low energies if it has an IR fixed point. I am having some doubts on how to interpret this fact in terms of quantum field theory and to ...
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1answer
35 views

Transformation of position operator

Consider a dilation of space $x\mapsto ax$ for some non-vanishing number $a$. Let $Q$ be the position operator defined by $(Q\psi)(x)=x\psi(x)$ on function $\psi$ of space. Suppose $\psi$ transforms ...
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Why is it that the equation of a massless scalar field *must* be conformal invariant?

I'm reading a paper [1], p.111 where it is said that: However, the equation of scalar field with zero mass must be conformal invariant while equation $\square\varphi=0$ does not satisfy this ...
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1answer
95 views

Invariance of Liouville action under rescaling

I was studying the Liouville action $$S=\frac{1}{8\pi} \int d^2 x\ \left[ \partial_\mu \phi \partial^\mu \phi + e^{\beta\phi} \right] \tag{1}$$ under the following general form of transformation: $$\...
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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$^{[1]}$ \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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1answer
489 views

Noether's theorem for scale invariance [duplicate]

When we have the Lagrangian $$\mathcal{L} = \frac{1}{2} \partial _\mu \phi\partial^\mu \phi \tag{1} $$ We have a symmetry given by $$x^\mu\mapsto e^\alpha x^\mu, \qquad\phi\mapsto e^{-\alpha} \phi.\...
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1answer
355 views

What is the actual definition of conformal invariance?

I've seen a large variety of slightly different definitions of conformal invariance. For simplicity I'll only consider scale invariance, which is already confusing enough. Some of the definitions are: ...
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145 views

Primary field in CFT and path integral

I should feel ashamed to ask such a naive question, but anyway let me start with the $\phi^4$ theory in the Minkowski spacetime, which has a Lagrangian of the form $$\frac{1}{2}(\partial\phi)^2-\frac{...
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1answer
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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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1answer
27 views

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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350 views

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: "...
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1answer
43 views

Dilatations and action on derivatives of fields

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 ...
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1answer
668 views

Show that a theory is scale invariant

I'm a bit new to this invariant transformations for fields so I've been having trouble manipulating them and I would appreciate any guidance. I saw in this wikipedia article that, for example, a $\...
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1answer
100 views

Apparition of scale invariance

When did "scale invariance" started to be seen as an important concept in the theory of phase transition? Phase transition and critical points started to be investigated in earnest in the middle of ...
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1answer
322 views

Correlation length at low temperatures?

The correlation length gives (approximately) the distance over which a spin flip has an effect. For systems with ordered phases, at low temperatures the correlation length is then small (since a ...
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380 views

Conformal vs. scale invariance of ${\cal N} = 4$ Supersymmetric Yang-Mills theory

I will quote the following from the Wikipedia article on Supersymmetry Nonrenormalization theorems. "In ${\cal N} = 4$ super Yang–Mills the $\beta$-function is zero for all couplings, meaning that ...
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Supersymmetric scale invariant non-conformal QFT in 4d

The fact that a QFT in 3+1d is scale invariant does not automatically imply that the QFT is also invariant under the full conformal group, cf. e.g. this Phys.SE post. Counterexamples are known, but as ...
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Restoring Scale Symmetry

To comprehend more about Scale Symmetry.. I need to know what it would take to restore Scale Symmetry that would make mass and length vanish. For example.. to restore Electroweak symmetry breaking.. ...
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1answer
70 views

Scales in Logarithmic CFTs

Logarithmic CFTs have OPEs (and operators) with logarithms. But to have logarithms one needs to have some scale to make the argument of the log a dimensionless quantity. But if the theory has a scale ...
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Is there a natural scale associated with polynomials?

This question is related to a previous question asked here. Power laws are scale invariant. They don't have a built-in or characteristic scale associated with them. Exponentials such as $e^{-x/\xi}$ ...
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Is the Landau free energy scale-invariant at the critical point?

My question is different but based on the same quote from Wikipedia as here. According to Wikipedia, In statistical mechanics, scale invariance is a feature of phase transitions. The key ...
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87 views

Energy scales and Lorentz Transformations

There are many particle physics processes where the initial particles must have some minimum energy in order to create the final ones. However, since I could just run through the lab really fast in ...
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Dilaton coupling to CFT

I am studying this paper of Luty, Polchinski and Rattazzi about the $a-$theorem in $d=4$ and the possibly allowed RG flow between fixed points of a theory with metric $g_{\mu\nu}$. First of all, they ...
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1answer
480 views

How is scale invariance broken in QCD?

It is generally believed that for the pure QCD, the classical scale invariance is broken at the quantum level (therefore anomaly rather than SSB). This breaking of scale invariance may be used to ...
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483 views

Identically vanishing trace of $T^{\mu\nu}$ and trace anomaly

Let us consider a theory defined by an action on a flat space $S[\phi]$ where $\phi$ denotes collectively the fields of the theory. We will study the theory on a general background $g_{\mu\nu}$ and ...