# What is Conformal invariance: Definition and 13 Discussions

In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetry has 15 degrees of freedom: ten for the Poincaré group, four for special conformal transformations, and one for a dilation.
Harry Bateman and Ebenezer Cunningham were the first to study the conformal symmetry of Maxwell's equations. They called a generic expression of conformal symmetry a spherical wave transformation. General relativity in two spacetime dimensions also enjoys conformal symmetry.

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1. ### I Why is Scalar Massless Wave Equation Conformally Invariant?

It can be shown mathematically that the scalar massless wave equation is conformally invariant. However, doing so is rather tedious and muted in terms of physical understanding. As such, is there a physically intuitive explanation as to why the scalar massless wave equation is conformally invariant?
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. ### I Restrictions of 1st Order Perturbation Theory

Hello guys, I'm wondering if there are some important restrctions on the 'applicability' of first order perturbation theory. I know there's a way to deduce Schwarzschild's solution to Einstein's field equations that assummes one can decompose the 4D metric ##g_{\mu\nu}## as Minkowski...
4. ### I Has Large Hadron Collider shown Higgs conformal invariance

the higgs naturalness problems has several solutions 1- natural susy 2- technicolor 3- extra dimensions 4- conformal invariance Large Hadron Collider has to date strongly disfavored susy, technicolor, extra dimensions. it is highly unlikely susy is the answer to the higgs hierarchy problem...
5. ### I How to show that Electrodynamics is conformally invariant?

[Moderator's note: changed thread title to be more descriptive of the actual question.] Consider Maxwell's action ##S=\int L## over Minkovski space, where the Lagrangian density is ##L = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}##, and the Electromagnetic tensor is given by ##F^{\mu\nu} = \partial^\mu...
6. ### I Conformal transformation of the line element

Let us see how the line element transforms under conformal transformations. Consider the Minkovski metric gij, a line element ds2=dxigijdxj, and a conformal transformation δk(x)=ak + λ xk + Λklxl + x2sk - 2xkx⋅s We have δ(dxk)=dδ(x)k=λ dxk + Λkldxl + 2 x⋅dx sk - 2dxkx⋅s - 2xkdx⋅s And so the...
7. ### Questions about Neveu-Schwarz B-field

In string theory, the Neveu-Schwarz B-field appears in the action: S_{NS}=\frac{1}{4\pi\alpha^\prime}\int d^2\xi\;\epsilon^{\mu\nu}B_{ij}\partial_\mu X^i\partial_\nu X^j. In Polchinski's text, the antisymmetric tensor appears in the form of \frac{1}{4\pi\alpha^\prime}\int...
8. ### Conformal Invariance Klein Gordon Action

Wald Appendix D talks on why g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi is not conformally invariant when n is not equal to 2. I want to prove that the Klein Gordon Action (V=0) is not conformally invariant. However the term that I have in the action is just...
9. ### Conformal invariance of gluon amplitudes

Hi, I'm very ashamed to not understand how even the simplest gluon amplitudes are conformally invariant. See eg http://arxiv.org/abs/hep-th/0312171 pages 11-12. M(1^-,2^-,3^+)=\delta(\sum_i \lambda_i\tilde{\lambda}_i)\frac{\langle12\rangle^4}{\langle12\rangle \langle...
10. ### Conformal invariance of null geodesics

Hi, folks. I hope this is the right forum for this question. I'm not actually taking any classes, but I am doing self-study using D'Inverno's Introducing Einstein's Relativity. I have a solution, and I want someone to check it for me. Homework Statement Prove that the null geodesics of two...
11. ### Conformal invariance / reparametrization invariance

Hi! I have little questions about symmetries. I begin in the field, so... First about conformal symmetry. As I studied, in 2-d, a transformation (\tau, \sigma) \to (\tau', \sigma') changing the metric by a multiplicative factor implies that the transformation (\tau, \sigma) \to (\tau'...
12. ### Conformal group, Conformal algebra and Conformal invariance in field theory

I have noticed that questions about this subject get either ignored or receive some confusing answers. So I decided to write a "brief" but self-contained introduction to the subject. I'm sure you will find it useful. It is going to take about 13 or 14 post to complete the work. Be patient with...
13. ### Gauge Groups, Riemann Tensors & Conformal Invariance in GR & QG

In trying to get my head round GR and quantum gravity, I'm puzzled about the following questions: Is the gauge group for gravity defined as the group of all possible Weyl tensors on a general 4D Riemann manifold? How is this group defined in matrix algebra? Is it a subgroup of GL(4). How do...