Questions tagged [scale-invariance]

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Is a scale-invariant QFT defined by its symmetries?

Do distinct scale-invariant quantum field theories necessarily have different symmetries?
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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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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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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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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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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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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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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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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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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: "...
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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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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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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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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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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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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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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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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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What's the Lie group generated only by dilation and Poincaré symmetry?

Given space $\mathbb{R}^{1,d-1}$($d\ge3$), the total conformal group is $SO(d,2)$ generated by $1$-dilation, $d$-translation, $d$-special conformal, $d(d-1)/2$-Lorentz transformation. But we know ...
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Physical interpretation of power law cluster size distribution in percolation problem

In the site percolation problem, when the occupation probability $p \rightarrow p_c$, where $p_c$ is the critical probability. The characteristic length diverges, and assuming the usual scaling ansatz ...
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Are fixed points of RG evolution really scale-invariant?

It is often stated that points in the space of quantum field theories for which all parameters are invariant under renormalisation – that is to say, fixed points of the RG evolution – are scale-...
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Scale invariance at phase transitions

The Wikipedia entry for scale invariance states In statistical mechanics, scale invariance is a feature of phase transitions. The key observation is that near a phase transition or critical ...
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How do fixed points of the RGEs get perturbed in QFTs?

Coming from the bottom up, we can use the renormalization group equations to calculate if there are any fixed points and if yes, where they lie. Fixed points correspond to scale invariant theories, ...
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Why does renormalizability mean that “ultimately locality will have to be abandoned”?

This is stated by Zinn-Justin in his paper Quantum Field Theory: renormalization and the renormalization group: Low energy physics does not depend on all the details of the microscopic model ...
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Is a universe without massive particles scale-invariant?

In a popular talk by Roger Penrose about spacetime geometry, when introducing his conformal cyclic cosmology starting at 17:15 I think he says that as soon as there are no massive particles left in ...
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What does “no characteristic length or time scale” mean?

When looking into the topic of "self-organized criticality," (SOC) one often comes across descriptions of SOC as a state where "the system has no characteristic length or time scale." (Examples here ...
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How to determine correlation length when the correlation function decays as a power law?

I am studying a system for which I observe a power-law decay in the correlation function: $\left\langle s\!\left(0\right)\cdot s\!\left(r\right) \right\rangle \propto r^{-\alpha}$ I am interested in ...
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Is the propagation speed of falling dominos scale-independent?

Suppose we scale all the linear space dimensions of falling dominos, like their thickness, width, height, distance (which implies that the volume, which is not a linear dimension, is not scaled by the ...
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Traceless energy-momentum tensor

I don't think it is clear to me what exactly is the physical meaning of the energy-momentum tensor being traceless.
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Does the Newton's law break scale invariance?

Under a scale transformation $$t\rightarrow \bar{t}=\mu t\hspace{0.3cm}\text{and}\hspace{0.3cm}\textbf{r}\to\bar{\textbf{r}}=\lambda\textbf{r},\tag{1}$$ Newton's law take the form $$m\frac{d^2\textbf{...
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Understanding the physical operations corresponding to unit conversion and scale transformation of time

By definition, under a unit conversion of time $$t\to\bar{t}=\alpha t\tag{1}$$ (i.e., measuring time in minutes rather than in seconds) the physics should not change, and therefore, the form of the ...
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Linear scale transformations for bosons and fermions

In the book from Coleman: The Aspects of Symmetry, p. 70; linear scale transformations or dilations are defined as $$ x \rightarrow e^\alpha x $$ with $\alpha$ being a real number. The fields change ...
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Identical games of billiards with one double the speed of the other

The situation: two friends play billiards and we should all hurry to the plane already but the game is still yet not finished. Time is about to run out and players decide to play the rest of the match ...
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The definition of the transformed field in CFT

I am a little puzzled by what people call "the transformed field" in CFT. The usual definition of the scale-invariant function is \begin{equation} \phi(\lambda z) = \lambda^\Delta \phi(z) \end{...
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Does scale invariance and R-invariance of Kähler potential imply superconformal symmetry?

Consider a four-dimensional $\mathcal{N} = 1$ field theory with Lagrangian: $ \mathcal{L} = \int d^4 \theta K(\Phi, \bar \Phi) $ and assume $K$ transforms well under dilations with scaling dimension ...
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Reference: Renormalization Group and scale invariance in statistical mechanics [duplicate]

Can anyone recommend a book/resource that succinctly explains how the Renormalization Group idea is applied in statistical mechanics? I have some background in undergraduate-level statmech, but none ...
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Is light scale invariant?

Are the equations that describe light's behavior unbounded to its energy and unrelated to the dimensions of the Universe (in which light exists)?