What is Conformal transformations: Definition and 13 Discussions

In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.
More formally, let

U

{\displaystyle U}
and

V

{\displaystyle V}
be open subsets of

R

n

{\displaystyle \mathbb {R} ^{n}}
. A function

f
:
U

V

{\displaystyle f:U\to V}
is called conformal (or angle-preserving) at a point

u

0

U

{\displaystyle u_{0}\in U}
if it preserves angles between directed curves through

u

0

{\displaystyle u_{0}}
, as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature.
The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix.For mappings in two dimensions, the (orientation-preserving) conformal mappings are precisely the locally invertible complex analytic functions. In three and higher dimensions, Liouville's theorem sharply limits the conformal mappings to a few types.
The notion of conformality generalizes in a natural way to maps between Riemannian or semi-Riemannian manifolds.

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1. I Metrics and conformal transformations

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} =...
2. Massless theories can be conformally invariant

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...
3. A Primaries, descendents and transformation properties in CFT

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...
4. A Confusion regarding conformal transformations

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...
5. Weyl Tensor invariant under conformal transformations

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: ##...
6. Conformal and non-conformal transformations

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...
7. Infinitesimal conformal transformations

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...
8. Conformal Field Theory: Questions & Answers

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...
9. Conformal Transformations: Exploring a New Definition

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...
10. Easy question on conformal transformations

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...
11. Conformal transformations and Möbius transformations

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...
12. Integrating infinitesimal conformal transformations

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...
13. Special Conformal Transformations

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...