Questions tagged [scale-invariance]

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What does the non-commuting nature of the translation and dilation generators mean for the scaling dimension of a field?

I am reading about CFTs from the book by Di Francesco, Mathieu and Senechal and in page 98 was introduced to the conformal group and the algebra of the generators. In particular, we have the dilation/...
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Does the stress energy tensor scale with the metric tensor?

Question I had some thoughts from a previous question of mine. If I have a metric $g^{\mu \nu}$ $$g^{\mu \nu} \to \lambda g^{\mu \nu}$$ Then does it automatically follow for the stress energy tensor $...
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Why are CFTs not usually studied in momentum space?

Conformal symmetry in QFT has been extremely useful for physics. However, while most of QFT is usually done in momentum space, CFTs are usually studied in position space or in terms of Mellin ...
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How to show that in 2D CFT the marginal operator must have $(h,\bar h)=(1,1)$?

A related post might be What are marginal fields in CFT? where Qmechanic♦ pointed to Ginsparg secion 8.6. However, I heard about two argument. Claim 1:In a $D$ dimension CFT, the marginal operator ...
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Space-Time Symmetries and Scaling

Typically, when a course examines symmetries of space-time and their consequences (e.g. symmetries of Lagrangians and conserved quantities), either the Lorentz group or the Poincaré group are ...
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What is the physical significance of conformal invariance of conformally invariant field theories?

Edited Question The absence of certain terms can make a field theory conformally invariant. For example, the absence of a mass term in the Maxwell action makes it conformally invariant. Here is a nice ...
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Scale invariance in curved spacetime?

Question What does it mean for the metric to be scale invariant in curved spacetime (in the sense when I say a property is scale invariant in thermodynamics)? I'm confused as to how to define this. It ...
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What does the pole in the running of the QED coupling represent?

In the case of QCD, the $\Lambda_{QCD}$ introduces a scale in the theory that can be also modified in presence of strongly interacting fermions. This mass-scale breaks the classical 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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1 answer
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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 ...
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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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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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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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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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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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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 Joshua D. Qualls https://arxiv.org/abs/1511.04074. ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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8 votes
1 answer
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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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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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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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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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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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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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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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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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1 vote
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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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27 votes
11 answers
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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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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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8 votes
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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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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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2 votes
1 answer
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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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6 votes
1 answer
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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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