Skip to main content

Questions tagged [scale-invariance]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
71 votes
5 answers
23k views

Conformal transformation/ Weyl scaling are they two different things? Confused!

I see that the weyl transformation is $g_{ab} \to \Omega(x)g_{ab}$ under which Ricci scalar is not invariant. I am a bit puzzled when conformal transformation is defined as those coordinate ...
vishmay's user avatar
  • 1,108
30 votes
2 answers
5k views

What is the difference between scale invariance and self-similarity?

I always thought that these two terms are some kind of synonyms, meaning that if you have a self-similar or scale invariant system, you can zoom in or out as you like and you will always see the same ...
Dilaton's user avatar
  • 9,581
28 votes
11 answers
6k 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\...
Godzilla's user avatar
  • 791
28 votes
1 answer
8k views

Noether's Theorem and scale invariance

Noether's theorem usually considers coordinate/field transformations which leave the Lagrangian invariant up to a divergence term, i.e. $$\mathcal{L} \rightarrow \mathcal{L} + \partial_{\mu}f^{\mu}$$ ...
user2640461's user avatar
28 votes
3 answers
4k views

Does dilation/scale invariance imply conformal invariance?

Why does a quantum field theory invariant under dilations almost always also have to be invariant under proper conformal transformations? To show your favorite dilatation invariant theory is also ...
user avatar
17 votes
5 answers
4k 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 ...
Deschele Schilder's user avatar
16 votes
2 answers
2k views

CFT and the Coleman-Mandula Theorem

The Coleman-Mandula theorem states that under certain seemingly-mild assumptions on the properties of the S-matrix (roughly: one particle states are left invariant and the amplitudes are analytic in ...
Morrissey87's user avatar
16 votes
1 answer
2k views

Is Weyl invariance absolutely necessary for string worldsheets?

The Polyakov action for a string worldsheet has Weyl invariance. In the conformal gauge augmented with Weyl gauge-fixing, we can always impose a flat worldsheet metric in Minkowski coordinates. The ...
user avatar
16 votes
3 answers
948 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: &...
P. C. Spaniel's user avatar
14 votes
3 answers
5k views

Why correlation length diverges at critical point?

I want to ask about the behavior near critical point. Let me take an example of ferromagnet. At $T < T_c$, all spins are aligned to the same direction thus it is in the ordered state, scale ...
john's user avatar
  • 327
13 votes
1 answer
3k views

Why Weyl invariance is important for consistent string theory?

This post is related to this link. I know there is a Weyl invariance for the Polyakov action at least in classical level. My question arises from obtaining effective action in string theory, such as ...
user26143's user avatar
  • 6,401
12 votes
1 answer
1k 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: ...
knzhou's user avatar
  • 103k
12 votes
1 answer
1k 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 ...
apt45's user avatar
  • 2,197
11 votes
2 answers
3k views

Simple conceptual question conformal field theory

I come up with this conclusion after reading some books and review articles on conformal field theory (CFT). CFT is a subset of FT such that the action is invariant under conformal transformation ...
user260822's user avatar
11 votes
1 answer
3k views

Is John Nash's "Interesting Equation" really interesting?

As recently mentioned in the news, before his passing, John Nash worked on general relativity. According to the linked article John Nash's work is available online from his webpage. His work is ...
asmaier's user avatar
  • 9,890
11 votes
2 answers
957 views

Scale invariance plus unitarity implies conformal invariance?

What has the reaction been towards the recent paper claiming to have a proof that scale invariance plus unitarity implies conformal invariance in 4d?
Prahar's user avatar
  • 26.5k
11 votes
1 answer
528 views

From which dimensionful constants does proton mass arise?

It is well known that the most of the proton (or any other hadron with light quarks) mass is not made up from quark masses, but it is dynamically generated by QCD mess inside. I've also heard that, ...
Varin Esan's user avatar
10 votes
2 answers
762 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 ...
Ari's user avatar
  • 2,849
10 votes
1 answer
711 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 ...
Bruce Lee's user avatar
  • 5,257
9 votes
5 answers
3k views

Are the physical laws scale-dependent?

If you read the article "More Is Different", by P.W. Anderson (Science, 4 August 1972), you will find a deep question: are the physical laws dependent of the size of the system under study? As an ...
asanlua's user avatar
  • 560
9 votes
2 answers
5k 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 \...
Bernoulli's user avatar
  • 279
9 votes
2 answers
2k views

Is $\phi^4$ theory in 4d conformally invariant at the classial level?

I used to believe the three following statements to be true (at the classical level only): From scale invariance full conformal invariance follows. Scale invariance is present if there are no ...
Weather Report's user avatar
9 votes
1 answer
527 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)...
user avatar
8 votes
2 answers
1k 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-...
gj255's user avatar
  • 6,425
8 votes
1 answer
3k views

Why does Weyl invariance imply a traceless energy-momentum tensor?

I've begun to self-study String Theory from Polchinski and Becker, Becker and Schwarz. I don't see why the fact that the Polyakov action is invariant under Weyl transformations is related to the ...
Ryan Unger's user avatar
  • 8,853
8 votes
1 answer
593 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 $\...
Shuheng Zheng's user avatar
7 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 ...
Solidification's user avatar
7 votes
2 answers
3k views

How to determine correlation length when the correlation function decays as a power law?

I am studying a system for which I observe a power-law decay in the correlation function: $\left\langle s\!\left(0\right)\cdot s\!\left(r\right) \right\rangle \propto r^{-\alpha}$ I am interested in ...
okj's user avatar
  • 819
7 votes
1 answer
4k views

CFT conformal weight vs. scaling dimension

I was wondering if anybody could clarify what the difference between the conformal scaling dimension $\Delta$ and the conformal weight $h$ is? Is it correctly understood that $\Delta$ is related to ...
Rexbye's user avatar
  • 194
7 votes
2 answers
4k views

Relation of conformal symmetry and traceless energy momentum tensor

In usual string theory, or conformal field theory textbook, they states traceless energy momentum tensor $T_{a}^{\phantom{a}a}=0$ implies (Here energy momentum tensor is usual one which is symmetric ...
phy_math's user avatar
  • 3,622
6 votes
1 answer
350 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 ...
mr.no's user avatar
  • 356
6 votes
1 answer
1k 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 ...
Janosh's user avatar
  • 1,264
6 votes
5 answers
364 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-...
dennis's user avatar
  • 742
6 votes
2 answers
452 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 ...
SRS's user avatar
  • 26.7k
6 votes
1 answer
927 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 ...
Wein Eld's user avatar
  • 3,691
6 votes
2 answers
375 views

Scale relativity vs. scale invariance

What is the difference between Nottale's "scale relativity", and the ordinary concept of scale invariance e.g. that appears in conformal field theory?
Mitchell Porter's user avatar
6 votes
2 answers
226 views

Why does renormalizability mean that "ultimately locality will have to be abandoned"?

This is stated by Zinn-Justin in his paper Quantum Field Theory: renormalization and the renormalization group: Low energy physics does not depend on all the details of the microscopic model ...
jak's user avatar
  • 10.1k
5 votes
2 answers
3k views

What are marginal fields in CFT?

In this article they call weight $(h,\bar{h})=(1,1)$ fields marginal. Why are these fields called marginal? Why are they to be distinguished.
Nom de plume's user avatar
5 votes
1 answer
1k views

Universality classes

I would like to ask about the universality classes. I know that these are the statistical models which describes different phase transitions with different critical exponents. But I would like to know ...
Alíz's user avatar
  • 187
5 votes
1 answer
550 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 ...
Aditya Vijaykumar's user avatar
5 votes
0 answers
988 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{\...
vanger's user avatar
  • 262
5 votes
1 answer
386 views

Conformally invariant theory. Relationship between conformal transformations and conformal rescaling (Weyl scaling)

So, I'm learning about Twistors, and in every book I've read they say the same: "If a flat theory is Poincaré-invariant and it is invariant under conformal rescaling (Weyl scaling), it is then ...
raul's user avatar
  • 428
5 votes
0 answers
225 views

RG flow from a UV scale invariant field theory to a gapped phase in the IR

On the section 3 of http://arxiv.org/abs/1309.2921 the authors consider the RG flow from a scale invariant field theory in the UV to a gapped theory in the IR. The theory is couple to a background ...
blima's user avatar
  • 86
4 votes
3 answers
843 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 ...
Matthew Christopher Bartsh's user avatar
4 votes
1 answer
509 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-...
physics_fan_123's user avatar
4 votes
1 answer
121 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}...
PPIP's user avatar
  • 141
4 votes
1 answer
2k views

Scale invariance in QFT?

About scale invariance in "beyond the standard model". At the base of the analysis is the principle of scale invariance. So what is being said: what if there were another sector of the theory that ...
Beyond-formulas's user avatar
4 votes
1 answer
498 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{...
Wenzhe's user avatar
  • 143
4 votes
1 answer
302 views

The definition of the transformed field in CFT

I am a little puzzled by what people call "the transformed field" in CFT. The usual definition of the scale-invariant function is \begin{equation} \phi(\lambda z) = \lambda^\Delta \phi(z) \end{...
mavzolej's user avatar
  • 2,921
4 votes
1 answer
144 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{...
QCD_IS_GOOD's user avatar
  • 6,896