Questions tagged [scale-invariance]
The scale-invariance tag has no usage guidance.
180 questions
5
votes
1
answer
174
views
Difficulty with a scaling argument
I am trying to make sense of an argument in this paper, "Fracture strength: Stress concentration, extreme value statistics and the fate of the Weibull distribution".
The paper deals with how ...
9
votes
4
answers
695
views
Is entropy scale-invariant?
The most common definition I’ve heard of entropy in physics is the number of micro-states for a given macro-state. Most examples use the atomic scale as the micro-setting and some kind of simple, ...
0
votes
0
answers
52
views
Four-point function in CFT, two constraints are missing
I am deriving the four-point functions, using translation and Lorentz invariance I start with the following form:
$$
\langle \phi_1(x_1)\phi_2(x_2)\phi_3(x_3)\phi_4(x_4)\rangle=C_{1234}x_{12}^ax_{13}^...
5
votes
1
answer
188
views
Ambiguity for boundary conditions after conformal transformation
Abbreviations
EOM = Equations of motion
BCs = Boundary conditions
CT = conformal transformation
Intro
I was playing around a bit with EOMs, action principle, CTs and BCs. There, I met a problem. ...
1
vote
1
answer
100
views
Scaling symmetry of Klein-Gordon Equation
Why does the scaling transformation \begin{align}
x^\mu &\rightarrow \lambda x^\mu \\
\end{align}
of a scalar field under the Klein-Gordon equation, \begin{equation}
\Box \phi(x) = 0
\end{equation}...
0
votes
0
answers
48
views
Why is Dilatation subgroup $SO(1,1)$?
Why is dilatation subgroup of the conformal group $SO(1,1)$? I am not sure if this is a result for 2D theories or even higher dimensional ones too. (Parameter counting suggests there is one parameter ...
3
votes
1
answer
164
views
Cyclic Universe Problems
In Penroses's hypothesis, at the end of each iteration the universe undergoes a conformal transformation, meaning distances are rescaled. If I am right, it implies that a planet from the previous ...
3
votes
2
answers
131
views
Could a universe be expanding if its physics were scale invariant?
Imagine a universe where every field is massless and has scale-invariance. Would the expansion/contraction of the universe still be happening there? would it be detectable? Would it affect the ...
0
votes
0
answers
71
views
What is the definition of a conformal symmetry? [duplicate]
I have been very confused by this after some recent reading. So as far as I know, a conformal transformation (according to the definition in di Francesco et. al.'s book on CFT) is an active coordinate ...
1
vote
0
answers
70
views
Reference request scale anomaly
Can anyone recommend some books, notes and review-oriented papers on scale anomaly, with a view towards its relation to renormalization? Such as an anomaly perspective on RG, Callan-Symanzik equations ...
0
votes
1
answer
256
views
Field transformation under conformal transformation
In 1 (see references below), I'm trying to derive how a spinless field transforms under a conformal transformation, specifically eq. (2.41). CFT references/lectures are the most confusing I've seen ...
0
votes
1
answer
36
views
Confusion regarding scale symmetry for certain charge configurations
I had a question on symmetry operations that exactly resembles this post. The selected answer there mentions the required symmetry operation to be scale symmetry, and says:
An infinite plate looks ...
4
votes
1
answer
119
views
State-Operator Correspondence and symmetry in CFT in general dimension
Let us assume to have a QFT ($\mathcal{L}$) with translational, Lorentz, scale and conformal invariance.
I ask because we can, for example the free scalar free theory, canonically quantize the system ...
3
votes
1
answer
137
views
How is the dimensionful renornalization scale $\mu$ related to break of scale invariance in String Theory?
In the $7.1.1$ of David Tong's String Theory notes it is said the following about regularization of Polyakov action in a curved target manifold:
$$\tag{7.3} S= \frac{1}{4\pi \alpha'} \int d^2\sigma \ ...
0
votes
0
answers
55
views
Scale transformation of scalars in curved backgrounds
I am puzzled by the concept of scalar fields that arise in conformal field theory in curved backgrounds. In general relativity, so far as I understand it, a scalar field is basically a function ...
0
votes
0
answers
98
views
What is the difference between scale invariance and scale free?
A question several years ago asked, "What is the difference between scale invariance and self-similarity. It appears that a new term has become popular in recent years, which is "scale ...
6
votes
5
answers
422
views
Does pure Yang-Mills have a scale?
Consider pure Yang-Mills (YM) in 4 dimensions. The YM mass gap problem (as described in https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf) tells us that this is supposed to have a mass-...
2
votes
1
answer
169
views
Scale transformation of the scalar field and gauge field
I am reading this paper: "Magnetic monopoles in gauge field theories", by Goddard and Olive. I don't understand some scale transformations that appear in Page 1427.
Start from the energy ...
1
vote
0
answers
26
views
Inflation in new aeons (cyclic cosmology)
I have a question pertaining to Penrose's ideas about cyclic cosmology. As predicted therein, the end of each cycle comes about when massive particles are extinct and time is no longer measured. What ...
1
vote
0
answers
61
views
A sense in which inverse square law forces are scale invariant?
For a fixed separation, the gravitational acceleration between two uniform spheres of density $\rho$ is proportional to their radius $r$. But since angular sizes and distances across the celestial ...
4
votes
1
answer
160
views
Ising model rescaling
Consider the 2D classical Ising model. It's understood that there is a critical temperature $T_c$, and that the correlation length $\xi(T)$ defined by:
$$\langle \sigma_i \sigma_j \rangle_\mathrm{...
1
vote
1
answer
146
views
Conserved current of quartic interaction QFT ($φ⁴$-Theory)
The Lagrangian of the real massless $φ⁴$-theory is
\begin{align}
L=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\lambda\phi^4
\end{align}
Therefore the action integral has the global symmetry
\begin{...
2
votes
1
answer
199
views
Do conserved currents have to be primary?
In many texts about CFT it is proven that spin-1 conserved currents have the dimension $d-1$. In the proof it is used that, sometimes only implicitly, the current $J^\mu$ is a primary operator. ...
0
votes
1
answer
225
views
Metric in dilatation transformation of massless scalar field
The lagrangian density of the massless real scalar field is
\begin{align}
L = \frac{1}{2}\eta^{\mu\nu}\partial_\mu\Phi\partial_\nu\Phi = \frac{1}{2}\partial_\mu\Phi\partial^\mu\Phi.
\end{align}
I want ...
0
votes
0
answers
229
views
Conformal invariance and tracelessness of the energy-momentum tensor: contradictory statements
Before starting my question, let me define a couple terms to avoid the confusion that usually accompanies this topic:
I define the $c$-number valued energy-momentum tensor as $T^{\mu\nu} = \frac{2}{\...
1
vote
0
answers
102
views
Momentum as ladder operator in CFT
I'm following these lectures: https://arxiv.org/abs/1601.05000.
In eq, 3.19 the author writes
$$
[D,\mathcal{O}(0)] =-i\Delta \mathcal{O}(0),\tag{3.19}
$$
where $D$ is the generator of dilations and $\...
1
vote
1
answer
355
views
Intuitive interpretation of the scaling dimension of an operator?
I am reading Field Theories of Condensed Matter Physics by Fradkin and in equation (4.10) it shows that an operator transforms irreducibly under scalings as
$$\phi_n(xb^{-1}) = b^{\Delta_n}\phi_x(x)$$
...
0
votes
0
answers
121
views
What does the non-commuting nature of the translation and dilation generators mean for the scaling dimension of a field?
I am reading about CFTs from the book by Di Francesco, Mathieu and Senechal and in page 98 was introduced to the conformal group and the algebra of the generators. In particular, we have the dilation/...
1
vote
2
answers
142
views
Does the stress energy tensor scale with the metric tensor?
Question
I had some thoughts from a previous question of mine.
If I have a metric $g^{\mu \nu}$
$$g^{\mu \nu} \to \lambda g^{\mu \nu}$$
Then does it automatically follow for the stress energy tensor $...
11
votes
2
answers
820
views
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 ...
0
votes
0
answers
106
views
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 ...
0
votes
0
answers
122
views
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 ...
2
votes
0
answers
214
views
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 ...
0
votes
1
answer
239
views
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 ...
4
votes
1
answer
334
views
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 ...
0
votes
0
answers
90
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
1
answer
152
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
1
answer
434
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
1
answer
424
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 ...
1
vote
0
answers
72
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 ...
5
votes
1
answer
564
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
1
answer
127
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'...
2
votes
2
answers
141
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 Joshua D. Qualls https://arxiv.org/abs/1511.04074. ...
1
vote
1
answer
106
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 ...
4
votes
2
answers
2k
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
0
answers
35
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 ...
8
votes
2
answers
1k
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
vote
1
answer
91
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
1
answer
1k
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
votes
0
answers
406
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, ...