Questions tagged [scale-invariance]

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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 ...
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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
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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 ...
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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 ...
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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 ...
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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-...
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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'...
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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....
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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 ...
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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) \...
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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 ...
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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 ...
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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?
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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^{'\...
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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, ...
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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 ...
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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(\...
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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})...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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) \...
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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 ...
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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}...
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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)...
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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^...
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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 ...
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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 ...
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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 ...
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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 ...
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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.
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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}...
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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 ...
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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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11answers
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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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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, ...
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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)^...
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2answers
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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
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,\...
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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 ...
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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 ...
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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, $$\...
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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 $\phi \mapsto \phi$. ...
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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 ...
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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 ...
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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 $t=\vert T-T_c\vert/T_c$ ...
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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{\...
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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 $\...