Conformal symmetry - Citizendia

In theoretical physics, conformal symmetry is a symmetry under dilatation (scale invariance) and under the special conformal transformations. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or In Physics and Mathematics, scale invariance is a feature of objects or laws that do not change if length scales (or energy scales are multiplied by a common factor Together with the Poincaré group these generate the conformal symmetry group. In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime

The conformal group has the following representation in spacetime:

$M_{\mu\nu}\equiv-i(x_\mu\partial_\nu-x_\nu\partial_\mu),$

$P_\mu\equiv-i\partial_\mu,$

$D\equiv-x_\mu\partial^\mu,$

$K_\mu\equiv-{i\over2}(x^2\partial_\mu-2x_\mu x_\nu\partial^\nu),$

where $M μν$ are the Lorentz generators, $P μ$ generates translations, $D$ generates scaling transformations (also known as dilatations or dilations) and $K μ$ generates the special conformal transformations. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Physics (and mathematics the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the In Physics, translation is movement that changes the position of an object as opposed to Rotation.

The commutation relations, in addition to those of the Poincaré group, are as follows:

[D, D] = 0

[D, K μ] = - K μ,

[D, P μ] = P μ

[K μ, K ν] = 0,

[K μ, P ν] = η μν D - i M μν,

Additionally, $D$ is a scalar and $K μ$ is a covariant vector under the Lorentz transformations. In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to

Proper conformal transformations are given by

$x^\mu \to \frac{x^\mu+a^\mu x^2}{1+ 2a\cdot x + a^2 x^2}$

where a^μ is a parameter describing the transformation.

A coordinate grid prior to a proper conformal transformation

The same grid after a proper conformal transformation

In two dimensional spacetime, the transformations of the conformal group are the conformal transformations. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a Riemannian manifold or Pseudo-Riemannian

In more than two dimensions, Euclidean conformal transformations map circles to circles, and hyperspheres to hyperspheres with a straight line considered a degenerate circle and a hyperplane a degenerate hypercircle.

In more than two Lorentzian dimensions, conformal transformations map null rays to null rays and light cones to light cones with a null hyperplane being a degenerate light cone.

Uses

The largest possible symmetry group of a non-supersymmetric interacting field theory is a direct product of the conformal group with an internal group. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that Interaction is a kind of action that occurs as two or more objects have an Effect upon one another In quantum field theory (QFT the forces between particles are mediated by other particles In Mathematics, one can often define a direct product of objectsalready known giving a new one In Physics, a field is a Physical quantity associated to each point of Spacetime. Such theories are known as conformal field theories. A conformal field theory (CFT is a Quantum field theory (or Statistical mechanics model at the Critical point) that is Invariant under

One particular application is to critical phenomena (phase transitions of the second order) in systems with local interactions. In Physics, critical phenomena is the collective name associated with thephysics of critical points Most of them stem from the divergence of the Correlation length In Thermodynamics, phase transition or phase change is the transformation of a thermodynamic system from one phase to another The fluctuations in such systems are conformally invariant at the critical point. That allows for classification of universality classes of phase transitions in terms of conformal field theories. A conformal field theory (CFT is a Quantum field theory (or Statistical mechanics model at the Critical point) that is Invariant under Conformal invariance is also discovered in two-dimensional turbulence at high Reynolds number. In Fluid mechanics and Heat transfer, the Reynolds number \mathrm{Re} is a Dimensionless number that gives a measure of the Ratio

Several spaces and theories in high-energy physics admit the conformal symmetry:

N = 4 supersymmetric Yang-Mills. Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that Yang-Mills is a Gauge theory of Quantum field theory based on the SU(N group.
The theory over the worldsheet in string theory. In String theory, the worldsheet is a two-dimensional Manifold which describes the embedding of the string in Spacetime. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings

Uses

See also