Conformal transformations Definition and 14 Threads

I'm studying CFT, and I find the lecture notes and books really confusing and devoid of explanations (more details). In a scale transformation ##x' = \lambda x##, the field ##\phi(x)## should also be affected by the scale transformation, i.e., ##\phi'(x') = \phi'(\lambda x) = \lambda^{-\Delta}...

Conformal field theory is way over my head at the moment, but I decided to "dip my toes into it," and I watched a little video talking about conformal transformations. Now, I know that in a conformal transformation, $$x^\mu \to x'^\mu ,$$ the metric must satisfy $$\Lambda (x) g_{\mu \nu} =...

Homework Statement The exercise needs us to first show that ##P^2## (with ##P_\mu=i\partial_\mu##) is not a Casimir invariant of the Conformal group. From this, it wants us to deduce that only massless theories could be conformally invariant. Homework Equations The Attempt at a Solution I...

I want to clarify the relations between a few different sets of operators in a conformal field theory, namely primaries, descendants and operators that transform with an overall Jacobian factor under a conformal transformation. So let us consider the the following four sets of...

I am confused about conformal transformations on Riemannian manifolds. Here's what I have so far. 1. Under a conformal transformation the metric changes by: g' -> Ω2g 2. Under a Weyl transformation the metric changes by: g' -> exp(-2f)g 3. Any 2D Riemann manifold is locally conformally...

Homework Statement As the title says, I need to show this. A conformal transformation is made by changing the metric: ##g_{\mu\nu}\mapsto\omega(x)^{2}g_{\mu\nu}=\tilde{g}_{\mu\nu}## Homework Equations The Weyl tensor is given in four dimensions as: ##...

It is well known that from a two-dimensional solution of Laplace equation for a particular geometry, other solutions for other geometries can be obtained by making conformal transformations. Now, I have a function defined on a disc centered at the origin and is given by f(r) = a r where a is...

Homework Statement Find the infinitesimal dilation and conformal transformations and thereby show they are generated by ##D = ix^{\nu}\partial_{\nu}## and ##K_{\mu} = i(2x_{\mu}x^{\nu}\partial_{\nu} - x^2\partial_{\mu})## The conformal algebra is generated via commutation relations of elements...

Hi all, my question is rather a simple one and regards conformal transformations. On "Applied CFT" by P.Ginsparg, http://arxiv.org/pdf/hep-th/9108028.pdf , on page 10, gives the transformation rule of a quasi primary field and relates the exponent of 1.12 to the one of 1.10. My first question...

Conformal transformations as far as I knew are defined as g_{mn}\rightarrow g'_{mn}=\Omega g_{mn}. Now I come across a new definition, such that a smooth mapping \phi:U\rightarrow V is called a conformal transformation if there exist a smooth function \Omega:U\rightarrow R_{+} such that...

A conformal transformation is a coordinate transformation that leaves the metric invariant up to a scale change g_{\mu\nu}(x) \to g'_{\mu\mu}(x)=\Omega(x)g_{\mu\nu}(x). This means that the length of vectors is not preserved: g_{\mu\nu}x'^{\mu}x'^{\nu}\not=g_{\mu\nu}x^{\mu}x^{\nu} But is...

Hello, I read somewhere that in 2D, the Möbius transformations do not represent all the possible conformal transformations, while according to Liouville's theorem, in spaces of dimension greater than 2 all the conformal transformation can be expressed as combinations of...

While it's pretty easy to derive the infinitesimal version of the special conformal transformation of the coordinates: x'^{\mu}=x^{\mu}+c_{\nu}(x^{\mu} x^{\nu}-g^{\mu \nu} x^2) with c infinitesimal, how does one integrate it to obtain the finite version transformation...

Hello, In conformal geometry there is a 15-parameter symmetry group. I have an rough conceptual understanding of the 3 spatial translations, the 1 temporal translation, the 3 rotations, the 3 Lorentz "boosts", and the 1 dilation transformation. I am having trouble conceptualizing the...