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75

While other answers are correct, they fail to address your specific issue. It looks like you are treating Newton's second law like it defines a single function, when it does not. For example, in algebra if I say a function is $f(x)=x^2 + 3$, then I can "plug into" this function something like $sx$ so that $f(sx)=(sx)^2+3$ by how we defined the function. ...

43

In order to deal with the kind of analysis you want to do, you have to be careful. It's a bit awkward to write $F(\vec{x})=m\ddot{\vec{x}}$ in the first place but you can write that as long as you understand what it means. It means that you are considering the force and acceleration both as fields because you're considering Newton's law at each point in ...

42

The Weyl transformation and the conformal transformation are completely different things (although they are often discussed in similar contexts). A Weyl transformation isn't a coordinate transformation on the space or spacetime at all. It is a physical change of the metric, $g_{\mu\nu}(x)\to g_{\mu\nu}(x)\cdot \Omega(x)$. It is a transformation that changes ...

15

A conformal transformation is a space-time transformation which leaves the metric invariant up to scale and thus preserves angles. A Weyl transformation actively scales the metric. More formally: Let $M, N$ be two manifolds with inner products $g, h$ and coordinates $x=(x^i), y=(y^j)$ respectively. A map $f:M\rightarrow N$ is called conformal if there ...

12

From the point of view of non-linear dynamics where self-similarity plays an important role if the attractor is a fractal I would say that the difference is one between continuous and discrete transformations. A self-similar transformation like the one producing the Cantor set or the Sierpinski triangle proceeds by discrete stages. The fractal which is the ...

11

First of all, let's see what Noether's Theorem says about your specific case (Klein-Gordon under global rescaling of the fields). Noether's theorem states that To every differentiable symmetry of the Action of a system, there corresponds a conserved current. The current in object is given by $$J^{\mu}=-T_{\nu}^{\mu}\ \delta x^{\nu}+\frac{\partial \... 10 The (Belinfante-Rosenfeld) stress energy momentum tensor is defined as T^{\mu\nu}\propto \frac{1}{\sqrt{-g}} \frac{\delta S}{\delta g_{\mu\nu}} where the worldsheet metric is g_{\mu\nu}. By definition of the functional derivative, for any variation \delta g_{\mu\nu} we have \delta S = \int \frac{\delta S}{\delta g_{\mu\nu}} \delta g_{\mu\nu}. ... 10 The proper definition of an (unbroken) conformal field theory is that all scalar n-point functions (or equivalently, their generating functional, a kind of partition function) remain unchanged when each field (including the metric, if it is a field) transforms according to a representation of the conformal group and the volume element (if the metric is not a ... 9 Scale invariance is not present in most of the realistic physical theories and thus can in no way be considered axiomatic. It is already broken in some classical theories, but even those which possess scale invariance on the classical level (like Yang-Mills theory, for example) acquire a nontrivial scale dependence through renormalization when quantized. ... 9 No, dimensionful couplings do not have to be all set to zero at an RG fixed point. An RG fixed point is one where all of the beta functions vanish, and beta functions generally have the form$$\beta(g_i) = (d_i - d) g_i + \hbar A_{ij} g_j + \ldots$$where d_i is the dimension of the corresponding operator. If one truncates the series at O(\hbar^0) then ... 9 Imagine you take the transformation you mentioned above:$$x^i \rightarrow x'^i = \alpha x^i,\\ t \rightarrow t' = \alpha t,$$where \alpha \in \mathbb{R}. Then assuming the Newton's law holds in the new coordinates, it will be of the form$$F^i = m \frac{d^2x'^i}{dt' ^2} = m \frac{d^2 (\alpha x^i)}{ dt^2} \left(\frac{dt}{dt'} \right)^2.$$As you can see,... 8 Note that under an infinitesimal change in the metric of the form g \to g + \delta g the action changes to$$ \delta S = \int T^{ab} \delta g_{ab} $$Now, under Weyl transformations we have$$ g_{ab} \to e^{2\omega} g_{ab} \qquad \implies \qquad \delta g_{ab} = 2 \omega g_{ab} $$For Weyl transformations \omega is completely arbitrary. If we consider a ... 8 Imposing that an action should be conformally invariant has a subtle but important difference with respect to imposing that it should be diffeomorphism invariant. Let me try to emphasize the differences between Diffeomorphisms, Conformal transformations and Weyl transformations. I will also clarify how to impose the invariance of the action, namely I'll ... 8 Is Nash's equation interesting? That is a matter of taste, but objectively I can say that his equations have (independently) interested physicists in the recent past. The equation of motion in your question originates from an action with higher-derivatives and without the usual Einstein-Hilbert action:$$ S = \int d^4 x \sqrt{-g}\left[2 R^{\mu\nu}R_{\mu\nu}...

8

What you found here is not a inconsistency of Newton's mechanics, but a symmetry of the harmonic oscillator. Consider for simplicity a point particle in $\mathbb{R}^n$. The force can be considered as a function $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$ taking the position of the particle as argument. Newton's law states that a physical trajectory $$\gamma:\... 7 Scale invariance can be thought of as 'self-similarity'. What this really means is that regardless of how much you zoom into or out of an object (be it a function, or a physical object, or the like) it looks exactly the same. Fractals are good examples of self-similarity. Shown below is an animation of the self-similarity of the Mandelbrot set: Another ... 7 The discussion p. 24 on renormalons by Rosten does not look to me related to the issue of scale invariance at all. As for the one on pp. 8-9, I think this is just a remark in passing about typical spin configurations at the critical point. Let me give precise definitions to explain what is going on in the language of probability theory. Take say the nearest ... 7 A universality class is an equivalence class of physical models – field theories, quantum field theories, or models of classical or quantum statistical physics – where the equivalence is defined by two or several models' having the same mathematical description of the behavior at very long time scales and distance scales. So if two models' behavior at very ... 7 "Arbitrary theory" probably means do not make a specific choice of metric (i.e. flat space time, your \eta_{\mu\nu}), do not make a specific choice of the symmetry operation (why did you define \phi \rightarrow e^{\omega \theta}\phi? If \theta, \omega \in \mathbb{R}, then it does not have magnitude 1. If one of them is an imaginary number, then you'... 6 Two dimensional CFTs separate into a left-moving sector and a right-moving sectors. The Virasoro generators L_n act on the left-moving sector and {\tilde L}_n act on the right-moving ones. Operators (or states due to the state-operator map) are labelled independently by representations of the left- and right-moving Virasoro. In particular, h and {\... 6 I cannot claim to speak for "the community" (whoever they might be), but so far I have only heard positive replies from knowledgeable people. Of course, people will need to read the paper in close detail, there will be discussions in seminars etc. so it'll take at least a couple of months before there will be a serious consensus. Let me give a few brief ... 6 I think its easiest to understand this if one has a minimal understanding of QFT. I'm not sure about your background knowledge but hopefully this isn't gibberish to you. The QCD Lagrangian for massless quarks is given by, {\cal L} = - g \sum_i \bar{\psi} _i A _\mu \gamma ^\mu \psi _i - \frac{1}{4} F _{ \mu \nu } F ^{ \mu \nu } \end{... 6 What you have shown is that Newton's law is not scale-invariant for a force F(x,\dot{x},t) that is scale-invariant, since you implicitly assumed that F transforms as a scalar under the dilation1. This is kind of a trivial statement: If the l.h.s. of an equation transforms non-trivially and you assume that the r.h.s. transforms trivially, the equation as ... 6 In the types of system in which second-order phase transitions are studied, forces are generally short range. This means that the other scales you mention will be finite in size. However, at the critical point the correlation length diverges, which means it becomes effectively infinite. If you look at the system on larger and larger scales, any finite scale ... 6 Classically a theory is invariant under a transformation if its action is invariant (up to boundary terms). In our case a conformal transformation is given by$$t'=\lambda t\\ Q'=\lambda ^{-\Delta}Q $$where \Delta is the scaling dimension of Q, which is just its energy dimension classically. For now let's assume a Lagrangian with only the kinetic term ... 6 The fact that all \beta-functions vanish in a theory is equivalent to the statement that the energy-momentum tensor is traceless by the operator identity$$ T^\mu_\mu(x) = \sum_\mathcal{O} \beta_\mathcal{O} \mathcal{O}(x) = 0. $$This is in turn sufficient to conclude that the theory is conformal: the charge$$ K^\mu \equiv \int d\Sigma_\nu \left( 2 x^\mu ...

6

For $\vec x$ scaled by some arbitrary constant $𝑠$, we obtain: $$F(s\vec{x})=ms\vec{\ddot{x}} \Longleftrightarrow \frac{F(s\vec{x})}{s}=m\vec{\ddot{x}}$$ This is not true. Remember that $\vec x$ and $\vec F(\vec x)$ represent something physical; $\vec x$ represents the position and $\vec F(\vec x)$ represents the net force as a function of position. ...

6

It looks to me like you attempted an illegal change of variables. You can't just substitute $s\vec{x}$ for $\vec{x}$. Remember that the equation $F(\vec{x}) = m \vec{\ddot{x}}$ isn't supposed to hold for all possible time-varying quantities $\vec{x}$. It's a particular assertion about $\vec{x}$, which is true for some time-varying quantities $\vec{x}$ and ...

5

The dilaton $\sigma$ is the Goldstone boson of scale invariance. Scale transformations $x\rightarrow x/\lambda$ are generated non linearly, e.g. $$\sigma(x)\rightarrow \sigma(\lambda x)+f \log\lambda\,,\qquad \lambda>0$$ where $f$ is the dilaton decay constant (see below). An effective field theory for this Goldstone boson can be easily written with ...

5

In the answer below I will only try to motivate why Weyl+diff invariance is (thought to be) necessary in (bosonic) string theory. Consider a (classical) string in a spacetime with coordinates $X^\mu$ and metric $G_{\mu\nu}$. As the string moves it defines a two dimensional surface $S$. Let $g$ denote the metric induced on the surface from the spacetime ...

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