Questions tagged [scale-invariance]

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Experimental Test for the cyclic $G_{earth}$ prediction of a Cosmological Model

Can anyone suggest a way to measure or rule out a tiny cyclic variation in the earth’s gravitational constant $G_{earth}$, predicted by an alternative cosmological model? It’s an annual cyclic ...
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5answers
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Does a slowed down version of small stone falling in water look the same as a big rock falling in real time?

I was wondering: If you let a small stone drop on a body of water, record it on film, and replay the scene in slow motion, will it be possible to see the difference with a huge rock that falls, in ...
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0answers
41 views

Monte Carlo approach to determining mean free path of Lévy dust

Problem statement I am trying to determine the mean free path $\lambda$ of a so-called Lévy dust, i.e. $M$ points in a square $L\times L$ environment with distances between subsequent points ...
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84 views

Can scale invariance symmetry in a conformal field theory (a theory with beta function=0) be localized?

Imagine one has a truly Scale-invariant theory (I mean not classical scale invariance but quantum mechanical, beta function vanishing one), can this symmetry become localized? If yes, what can be the ...
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1answer
66 views

Definition of conformal flatness

I was trying to prove that the Ricci scalar $R$ is not invariant under conformal transformations and when we talk about conformal transformations we have the relation: $$ \hat g_{\mu\nu} = \Omega(x) \...
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3answers
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What are the implications of the scale invariance of the geodesic equation?

The geodesic equation in general relativity is famously invariant under affine reparametrization, i.e., under the reparametrization $\tau \to a\tau + b$ where $\tau $ is the proper time. This can be ...
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1answer
50 views

Why is an RG fixed point scale invariant?

I cannot understand why people say RG fixed point is scale invariant. Scale invariant means the action $S[\phi]$ of the theory is invariant under scale transformation like $\phi(x)\to\lambda^{-\Delta}...
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1answer
125 views

What sets the scale of a free Maxwell theory in $d\neq 4$?

The action for the free Maxwell theory is given by $$S=\int d^dx\sqrt{-g}\bigg(-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}\bigg)$$ The theory is invariant under conformal transformations $g_{\mu\nu}\to\Omega^2(x)...
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1answer
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What is the physical significance of scale invariance (under appropriate boundary conditions) in the 1D Helmholtz equation?

So we were given this problem in mathematical physics (Context is that we're learning about Sturm-Liouville): Consider the 1D Helmholtz equation with $k^2>0$ $$\frac{\partial ^2y}{\partial x^2}+k^...
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1answer
83 views

Interacting CFT fixed point of an RG flow

Suppose we have a gauge theory defined in the UV and it flows to an interacting CFT in the IR, i.e. the beta function vanishes for some finite value of the coupling. I am confused about the meaning of ...
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1answer
113 views

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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40 views

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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52 views

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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1answer
80 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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1answer
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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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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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34 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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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
82 views

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
115 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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1answer
148 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
53 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
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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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0answers
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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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0answers
89 views

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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368 views

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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1answer
408 views

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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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
36 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
115 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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0answers
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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
768 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
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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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1answer
186 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
169 views

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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0answers
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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
30 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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406 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
44 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
873 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
131 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
419 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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1answer
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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 ...