Questions tagged [scale-invariance]
182 questions
72
votes
5 answers
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 transformations that effect the above metric transformation…
vishmay
- 1,156
32
votes
2 answers
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 picture (physics).
But now I have just read in…
Dilaton
- 9,771
29
votes
11 answers
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:
$$F(s\vec{x})=ms\vec{\ddot{x}} \Longleftrightarrow \frac{F(s\vec{x})}{s}=m\vec{\ddot{x}}$$
Which is clearly just $F(\vec{x})$!…
Godzilla
- 821
29
votes
1 answer
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.
$$\mathcal{L} \rightarrow \mathcal{L} + \partial_{\mu}f^{\mu}$$
However there is a more general class of…
user2640461
- 821
28
votes
3 answers
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 invariant under proper conformal transformations is…
user2389
16
votes
2 answers
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 external momenta) the largest possible Lie algebra…
Morrissey87
- 161
16
votes
1 answer
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 residual gauge symmetries take on the form of…
user2523
16
votes
5 answers
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 real-time, in a body of water?
Let's take for the…
Deschele Schilder
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- 105
16
votes
4 answers
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: "Why does the correlation length become…
P. C. Spaniel
- 4,476
14
votes
3 answers
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 invariant, its correlation length is effectively infinite.…
john
- 327
14
votes
2 answers
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 of the fields and coordinate but leave the metric…
user260822
- 271
13
votes
1 answer
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 section 7.3 in this lecture note
A consistent…
user26143
- 6,571
12
votes
6 answers
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 example, we can ask ourselves, are the description of…
asanlua
- 600
12
votes
1 answer
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:
The action stays the same under a step of RG…
knzhou
- 107,105
12
votes
1 answer
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 then we will set the metric to be flat.
The…
apt45
- 2,247