Questions tagged [scale-invariance]
The scale-invariance tag has no usage guidance.
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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 ...
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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 ...
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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:
$$F(s\vec{x})=ms\vec{\ddot{x}} \Longleftrightarrow \frac{F(s\vec{x})}{s}=m\...
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Noether's Theorem and scale invariance
Noether's theorem usually considers coordinate/field transformations which leave the Lagrangian invariant up to a divergence term, i.e.
$$\mathcal{L} \rightarrow \mathcal{L} + \partial_{\mu}f^{\mu}$$
...
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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 ...
user2389
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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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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: &...
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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 ...
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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 ...
user2523
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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:
...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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, ...
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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 ...
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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 ...
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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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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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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 ...
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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)...
user87745
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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-...
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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 ...
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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 $\...
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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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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 ...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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?
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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 ...
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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 ...
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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.
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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 ...
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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 ...
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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 ...
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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 ...
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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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Is scale invariance an axiom in physics?
Is scale invariance axiomatic within physics, and if so, how does it get around the transition from the microscopic, quantum world, to the macroscopic, classical world?
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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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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 $$ \beta(\mu) \Big |_{\mu^*} = 0 \; \; \Longrightarrow \; \; T^{\mu}...
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Scale invariance in QFT?
About scale invariance in "beyond the standard model".
At the base of the analysis is the principle of scale invariance. So what is being said: what if there were another sector of the theory that ...
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Primary field in CFT and path integral
I should feel ashamed to ask such a naive question, but anyway let me start with the $\phi^4$ theory in the Minkowski spacetime, which has a Lagrangian of the form
$$\frac{1}{2}(\partial\phi)^2-\frac{...
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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{...
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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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Is a universe without massive particles scale-invariant?
In a popular talk by Roger Penrose about spacetime geometry, when introducing his conformal cyclic cosmology starting at 17:15 I think he says that as soon as there are no massive particles left in ...
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