Conformal symmetry
Encyclopedia
In theoretical physics, conformal symmetry is a symmetry
under dilatation (scale invariance
) and under the special conformal transformations. Together with the Poincaré group
these generate the conformal symmetry group.
The conformal group has the following representation
:
Symmetry
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 perfection...
under dilatation (scale invariance
Scale invariance
In physics and mathematics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor...
) and under the special conformal transformations. Together with the Poincaré group
Poincaré group
In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...
these generate the conformal symmetry group.
The conformal group has the following representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
:
-
where
are the LorentzLorentz groupIn physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
generatorsGenerating set of a groupIn abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...
,
generates translationTranslation (physics)In physics, translation is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker:...
s,
generates scaling transformations (also known as dilatations or dilations) and 
generates the special conformal transformations.
The commutation relations are as follows:
-
other commutators vanish.
Additionally,
is a scalar and 
is a covariant vector under the Lorentz transformationLorentz transformationIn physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
s.
The special conformal transformations are given by
where aμ is a parameter describing the transformation. This special conformal transformation can also be written as
, where
which shows that it consists of an inversion, followed by a translation, followed by a second inversion.
In two dimensional spacetimeSpacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
, the transformations of the conformal group are the conformal transformationsConformal geometryIn mathematics, conformal geometry is the study of the set of angle-preserving transformations on a space. In two real dimensions, conformal geometry is precisely the geometry of Riemann surfaces...
.
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 groupSymmetry groupThe symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
of a non-supersymmetricSupersymmetryIn particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
interactingInteractionInteraction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way causal effect...
field theoryQuantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
is a direct productDirect product of groupsIn the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
of the conformal group with an internal group. Such theories are known as conformal field theoriesConformal field theoryA conformal field theory is a quantum field theory that is invariant under conformal transformations...
.
One particular application is to critical phenomenaCritical phenomenaIn physics, critical phenomena is the collective name associated with thephysics of critical points. Most of them stem from the divergence of thecorrelation length, but also the dynamics slows down...
(phase transitions of the second order) in systems with local interactions. 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 theoriesConformal field theoryA conformal field theory is a quantum field theory that is invariant under conformal transformations...
. Conformal invariance is also discovered in two-dimensional turbulence at high Reynolds number.
Several spaces and theories in high-energy physics admit the conformal symmetry:- N = 4 supersymmetricSupersymmetryIn particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
Yang-Mills. - The theory over the worldsheetWorldsheetIn string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind around 1967 as a direct generalization of the world line concept for a point particle in special and general relativity.The type of string,...
in string theoryString theoryString theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
.
See also
- Coleman–Mandula theorem
- Renormalization groupRenormalization groupIn theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
- Scale invarianceScale invarianceIn physics and mathematics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor...
- Superconformal algebraSuperconformal algebraIn theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. It generates the superconformal group in some cases .In two dimensions, the superconformal algebra is infinite-dimensional...
- Harry BatemanHarry BatemanHarry Bateman FRS was an English mathematician.-Life and work:Harry Bateman first grew to love mathematics at Manchester Grammar School, and in his final year, won a scholarship to Trinity College, Cambridge. There he distinguished himself in 1903 as Senior Wrangler and by winning the Smith's Prize...
- Ebenezer CunninghamEbenezer CunninghamEbenezer Cunningham was a British mathematician who is remembered for his research and exposition at the dawn of special relativity....
- N = 4 supersymmetric
-