Questions tagged [scale-invariance]
The scale-invariance tag has no usage guidance.
145
questions
0
votes
0answers
54 views
Local scale invariance without conformal anomaly
I need to know if conformal symmetry can be localized in the same manner that global symmetries like $SU(2)$ is localized and gauge bosons pop up?(I assume the trace anomaly doesn't violate the scale ...
2
votes
1answer
114 views
Is nonlocality consistent with scale invariance?
For sure I'm excluding gravity at first step, the question is that if nonlocality is compatible with scale invariance. At the classical and quantum levels for field theory in Minkowski spacetime.
Then ...
2
votes
1answer
67 views
Intuition for the trace-free energy-momentum tensor condition in CFTs
It is a textbook exercise to show that
\begin{equation}T^{\mu}_{\,\,\,\mu}=0
\end{equation}
is a sufficient condition for there to be a conserved current associated with a dilation symmetry. This ...
0
votes
1answer
73 views
What is the meaning of power laws in physics?
I know the basic meaning of a power law. Scale invariance. Also that with negative exponents, bigger events occur more rarely, like in say earthquakes. But I can't seem to find more physical ...
2
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0answers
41 views
How to check the conformal prefactor in a correlation function?
In CFT it is usual practice to extract the so-called conformal prefactor from the correlators in order to isolate a function which depends only on the cross-ratios. For example the $4$-point function ...
4
votes
1answer
179 views
Intuition behind power-law scale invariance
I have seen this notion of a scale-invariant power law curve exhibiting the property that $f(cx) = a(cx)^{-k} = c^{-k}f(x)$, and I am confused about how I should be thinking of this as "scale-...
0
votes
1answer
90 views
How to amend General Relativity to include a position-dependent gravitational 'constant' $G$?
What is the best way to amend General Relativity to include a variable gravitational 'constant' $G$, that depends on the positions of all other masses?
That is, if the amount of 'bending of space-time'...
1
vote
0answers
33 views
Representation of dilations
I am having some trouble getting some signs right on the representation of the dilation operator on a field. Let us follow the conventions of https://arxiv.org/abs/1511.04074. According to equation (2....
1
vote
1answer
46 views
Scale invariance beyond the critical point
Using Anderson localization as an example, I understand how scale invariance comes into play at a critical point - at a critical point, the localization length $\xi$ (the average "radius" of ...
2
votes
1answer
91 views
Proving the energy-momentum tensor for a conformal field theory is traceless
A trick to derive Noether currents that is frequently used in conformal field theory literature is the following: suppose we have an action $S[\phi]$ which has the infinitesimal symmetry $\phi(x) \...
2
votes
0answers
31 views
Boundary condition for $\Box\vec{E}(t,\vec{x})=0$ that preserves scale-invariance
In short, this is a question about the symmetry of a differential equation preserved by its boundary condition.
In free space, the vector wave equation satisfied by the electric and the magnetic field ...
7
votes
2answers
541 views
What does scale invariance or non-invariance of electromagnetism physically imply?
According to Wikipedia,
classical electromagnetism is scale-invariant.
I understand what it means mathematically as explained in Wikipedia. But what does it really imply physically?
Next, here it ...
1
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1answer
30 views
What dictates the efficiency of a semiconductor?
Semiconductors can be used for a heat exchange but are less efficient than a Freon air-conditioning system. What dictates this efficiency?
2
votes
1answer
133 views
Proof that free scalar field is conformally invariant
So, under conformal transformations $$x\mapsto x'\\
\phi\mapsto\phi'(x')=\Omega^{(2-D)/2}\phi(x),$$ where $$\eta_{\mu\nu}\frac{\partial x^\mu}{\partial x^{'\alpha}}\frac{\partial x^\nu}{\partial x^{'\...
0
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0answers
70 views
Are infinitesimal dilatation transformations local?
In quantum field theories, a local transformation of a scalar field $\phi(x)$ is a transformation that involves the field and its derivatives at same point. See for instance Weinberg's QFT textbok, ...
0
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0answers
18 views
Scaling dimension of non-relativistic field
Let us consider a field theory determined by an action $S$ which depends on a single field $\phi$. In usual relativistic field theories, in natural units, the action is dimensionless, which fixes (via ...
2
votes
0answers
188 views
Is there a valid solution to Einstein's equations for this cosmological model?
The cosmological model below has been developed in order to explain the flatness problem.
At first it's from Newtonian considerations, then a solution of the Friedman equations is looked for
$$\left(\...
1
vote
1answer
113 views
Another static solution of the Friedmann equations - interpretation of $p=-\rho c^2$
Looking for solutions of the Friedmann equations
$$(\frac{\dot a}{a})^2+\frac{kc^2}{a^2} = \frac{8 \pi G \rho+\Lambda c^2}{3}, \tag{1}$$
$$\frac{\ddot a}{a} = \frac{-4 \pi G}{3} (\rho + \frac{3p}{c^2})...
4
votes
3answers
238 views
What determines the magnitude of the atmospheric scale height of a planet?
What determines the magnitude of the atmospheric scale height of a planet?
https://en.wikipedia.org/wiki/Scale_height says that:
"Approximate atmospheric scale heights for selected Solar System ...
1
vote
0answers
73 views
Experimental Test for the cyclic $G_{earth}$ prediction of a Cosmological Model
Can anyone suggest a way to measure or rule out a tiny cyclic variation in the earth’s gravitational constant $G_{earth}$, predicted by an alternative cosmological model? It’s an annual cyclic ...
16
votes
5answers
3k views
Does a slowed down version of small stone falling in water look the same as a big rock falling in real time?
I was wondering: If you let a small stone drop on a body of water, record it on film, and replay the scene in slow motion, will it be possible to see the difference with a huge rock that falls, in ...
1
vote
0answers
49 views
Monte Carlo approach to determining mean free path of Lévy dust
Problem statement
I am trying to determine the mean free path $\lambda$ of a so-called Lévy dust, i.e. $M$ points in a square $L\times L$ environment with distances between subsequent points ...
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0answers
98 views
Can scale invariance symmetry in a conformal field theory (a theory with beta function=0) be localized?
Imagine one has a truly Scale-invariant theory (I mean not classical scale invariance but quantum mechanical, beta function vanishing one), can this symmetry become localized?
If yes, what can be the ...
1
vote
1answer
74 views
Definition of conformal flatness
I was trying to prove that the Ricci scalar $R$ is not invariant under conformal transformations and when we talk about conformal transformations we have the relation:
$$ \hat g_{\mu\nu} = \Omega(x) \...
3
votes
3answers
178 views
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 ...
0
votes
1answer
58 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}...
8
votes
1answer
158 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)...
0
votes
1answer
37 views
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 $k^2>0$
$$\frac{\partial ^2y}{\partial x^2}+k^...
2
votes
1answer
105 views
Interacting CFT fixed point of an RG flow
Suppose we have a gauge theory defined in the UV and it flows to an interacting CFT in the IR, i.e. the beta function vanishes for some finite value of the coupling. I am confused about the meaning of ...
1
vote
1answer
199 views
Why does the Ising model at the critical point have scale invariance?
If my current understanding of phase transitions and the renormalization group (RG) method is true, RG is a kind of 'zooming out' process, since this procedure makes a block of neighboring spins and ...
0
votes
0answers
42 views
Scaling invariance of fluid energy
I have read that the incompressible Navier Stokes equation is preserved by the scaling
$$x',y',z'=\lambda x, \lambda y, \lambda z$$
$$t'=\lambda^2 t$$
$$u'=(1/\lambda) u$$
As I understand it, fluid ...
0
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0answers
75 views
Scaling of correlation functions
In book Renormalization Group and Fixed Points there is formula for scaling of correlation function:
But this formula contradict my understanding of scaling of correlation functions.
For free theory ...
2
votes
1answer
92 views
Importance of Tracelessness of Tensor?
What makes the trace-free tensor (or part of it) so important?
As in trace-free Ricci tensor or Weyl tensor.
3
votes
1answer
74 views
RG fixed points and $T_{\mu\nu}$
It is common to refer to fixed points of the renormalization group as scale invariant theories. This statement can be formulated as $$ \beta(\mu) \Big |_{\mu^*} = 0 \; \; \Longrightarrow \; \; T^{\mu}...
0
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0answers
29 views
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 ...
1
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0answers
87 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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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:
$$F(s\vec{x})=ms\vec{\ddot{x}} \Longleftrightarrow \frac{F(s\vec{x})}{s}=m\...
2
votes
0answers
141 views
Scale invariance of lagrangians and transformation properties of fields under dilations
Consider a field theory, and a rescaling transformation of the coordinates
\begin{equation}
T_\epsilon[\phi(x)]=\phi((1+\epsilon)x).
\end{equation}
From what I understand, one usually requires that, ...
1
vote
0answers
36 views
Is a function of conformal ratios invariant under conformal transformations?
If I have a function $f:=f(r,s)$ a function of the conformal ratios $r$ and $s$ only, i.e. for example:
$$r := \frac{(x1-x2)^2(x3-x4)^2}{(x1-x3)^2(x2-x4)^2} \qquad, \qquad s := \frac{(x1-x4)^2(x2-x3)^...
7
votes
2answers
2k 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
$$
x^\mu \...
1
vote
1answer
134 views
Scaling transformations, definitions and all that's not mentioned
If we transform the massless scalar field Lagrangian $$\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
135 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 ...
3
votes
1answer
193 views
Do all classical-statistical 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
vote
1answer
66 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,
$$\...
2
votes
2answers
2k views
A traceless stress energy tensor?
I'm trying to solve this exercise:
Suppose an arbitrary theory (Flat space-time?) with a single field (Is a scalar field?) invariant under dilations, i.e.
$x\mapsto b x$ and $\phi \mapsto \phi$. ...
2
votes
0answers
88 views
How does it make sense to talk about the size of a string if the string action is conformally invariant?
From what I understand the Polyakov action in string theory is essentially something like
$$S(\xi, g, G)=\kappa \int_{\Sigma} d \mu_{g} \operatorname{Tr}_{g} \xi^{*} G$$
where $\Sigma$ is a given ...
1
vote
0answers
101 views
Conformal transformation
I am reading some lecture notes on Conformal Field Theory by Joshua D. Qualls (https://arxiv.org/abs/1511.04074).
At the end of page 5 of these notes, it is stated that the four momentum transforms ...
0
votes
0answers
54 views
Derivation of the Ising free energy close to a critical point
In "Statistical physics of fields" Mehran Kardar states that the Ising free energy scales with,
$$
f(t,h)\sim t^\alpha g_f\left(\frac{h}{t^\Delta}\right),
$$
wherein $t=\vert T-T_c\vert/T_c$ ...
4
votes
0answers
501 views
Finite conformal transformations of fields from infinitesimal
I know that in conformal field theories conformal group acts not by pushforwards but (e.g. for scalar field $\phi$ with conformal dimension $\Delta$)
$$
\phi(x) \mapsto \phi'(x') = \left| \frac{\...
8
votes
1answer
423 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 $\...
