Questions tagged [scale-invariance]
The scale-invariance tag has no usage guidance.
111
questions
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0answers
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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}(\...
23
votes
11answers
5k views
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:
$$F(s\vec{x})=ms\vec{\ddot{x}} \Longleftrightarrow \frac{F(s\vec{x})}{s}=m\...
2
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0answers
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Scale invariance of lagrangians and transformation properties of fields under dilations
Consider a field theory, and a rescaling transformation of the coordinates
\begin{equation}
T_\epsilon[\phi(x)]=\phi((1+\epsilon)x).
\end{equation}
From what I understand, one usually requires that, ...
1
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0answers
27 views
Is a function of conformal ratios invariant under conformal transformations?
If I have a function $f:=f(r,s)$ a function of the conformal ratios $r$ and $s$ only, i.e. for example:
$$r := \frac{(x1-x2)^2(x3-x4)^2}{(x1-x3)^2(x2-x4)^2} \qquad, \qquad s := \frac{(x1-x4)^2(x2-x3)^...
4
votes
1answer
348 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
$$
x^\mu \...
1
vote
1answer
48 views
Scaling transformations, definitions and all that's not mentioned
If we transform the massless scalar field Lagrangian $$\mathcal{L}=\frac{1}{2}(\partial_\mu\varphi)^2-\frac{\alpha}{4!}\varphi^4$$ with the simultaneous transformations $$x\mapsto x^\prime= \lambda x,\...
2
votes
1answer
86 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 ...
0
votes
0answers
13 views
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 ...
3
votes
1answer
120 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 ...
1
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1answer
41 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,
$$\...
2
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2answers
915 views
A traceless stress energy tensor?
I'm trying to solve this exercise:
Suppose an arbitrary theory (Flat space-time?) with a single field (Is a scalar field?) invariant under dilations, i.e.
$x\mapsto b x$ and $\phi \mapsto \phi$. ...
2
votes
0answers
70 views
How does it make sense to talk about the size of a string if the string action is conformally invariant?
From what I understand the Polyakov action in string theory is essentially something like
$$S(\xi, g, G)=\kappa \int_{\Sigma} d \mu_{g} \operatorname{Tr}_{g} \xi^{*} G$$
where $\Sigma$ is a given ...
1
vote
0answers
72 views
Conformal transformation
I am reading some lecture notes on Conformal Field Theory by Joshua D. Qualls (https://arxiv.org/abs/1511.04074).
At the end of page 5 of these notes, it is stated that the four momentum transforms ...
0
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0answers
45 views
Derivation of the Ising free energy close to a critical point
In "Statistical physics of fields" Mehran Kardar states that the Ising free energy scales with,
$$
f(t,h)\sim t^\alpha g_f\left(\frac{h}{t^\Delta}\right),
$$
wherein $t=\vert T-T_c\vert/T_c$ ...
4
votes
0answers
210 views
Finite conformal transformations of fields from infinitesimal
I know that in conformal field theories conformal group acts not by pushforwards but (e.g. for scalar field $\phi$ with conformal dimension $\Delta$)
$$
\phi(x) \mapsto \phi'(x') = \left| \frac{\...
7
votes
1answer
389 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 $\...
0
votes
0answers
68 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 ...
0
votes
1answer
32 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 ...
4
votes
0answers
79 views
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
...
2
votes
1answer
88 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:
$$...
2
votes
0answers
127 views
Mismatch between conformal generators and conformal transformations as changes of variables
Introduction
It is known that under changes of coordinates different fields transform according to their tensorial nature (scalar, vector, etc.) like$^{[1]}$
\begin{equation}
\phi(x)\rightarrow\phi'...
1
vote
0answers
109 views
What do we mean by scale invariance in a classical field?
First of all, I read many questions but they don't seem to answer my specific question. So, here it goes
According to Francesco's Conformal Field Theory and many other books, a scale transformation
\...
1
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0answers
27 views
Scale invariance and state equation for fluid dynamics
I am trying to understand the example provided in this section of the Wikipedia article on scale invariance. In particular where it says
In order to deduce the scale invariance of these equations ...
2
votes
1answer
396 views
Noether's theorem for scale invariance [duplicate]
When we have the Lagrangian
$$\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.\...
11
votes
1answer
324 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:
...
3
votes
1answer
135 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{...
0
votes
1answer
92 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 ...
1
vote
0answers
83 views
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 ...
0
votes
1answer
26 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 ...
3
votes
1answer
153 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: "...
0
votes
1answer
41 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 ...
3
votes
1answer
548 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 $\...
1
vote
1answer
90 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 ...
2
votes
1answer
275 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 ...
asked Jun 9 '18 at 5:32
Quantum spaghettification
5,6617 gold badges34 silver badges106 bronze badges
9
votes
1answer
369 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 ...
3
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0answers
66 views
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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0answers
122 views
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.. ...
3
votes
1answer
65 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 ...
0
votes
2answers
130 views
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}$ ...
5
votes
2answers
247 views
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 ...
1
vote
2answers
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 ...
4
votes
0answers
162 views
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 ...
5
votes
1answer
446 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 ...
12
votes
1answer
444 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 ...
3
votes
1answer
180 views
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 ...
1
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1answer
136 views
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 ...
5
votes
2answers
668 views
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-...
5
votes
1answer
221 views
Most general Lagrangian in CFT in 0+1D
My question is about $CFT_1$. Page 18 of this says that $$L={\frac{\overset{.}{Q}^2}{2} - \frac{g}{2Q^2}}\tag{1.11}$$ is the most general Lagrangian that preserves time translation and scale ...
5
votes
1answer
541 views
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 ...
1
vote
3answers
340 views
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, ...
