Thus, charges are associated with conserved quantum numbers; these are the eigenvalues of the generator
Abstractly, a charge is any generator of a continuous symmetry of the physical system under study.
When a physical system has a symmetry of some sort, Noether's theorem implies the existence of a conserved current.
Thus, for example, the electric charge is the generator of the U(1) symmetry of electromagnetism.
So, for example, when the symmetry group is a Lie group, then the charge operators correspond to the simple roots of the root system of the Lie algebra; the discreteness of the root system accounting for the quantization of the charge.
The charge quantum numbers then correspond to the weights of the highest-weight modules of a given representation of the Lie algebra.
So, for example, when a particle in a quantum field theory belongs to a symmetry, then it transforms according to a particular representation of that symmetry; the charge quantum number is then the weight of the representation.
Various charge quantum numbers have been introduced by theories of particle physics.
These include the charges of the Standard Model: Note that these charge quantum numbers show up in the Lagrangian via the Gauge covariant derivative#Standard_Model.
Charges of approximate symmetries: Hypothetical charges of extensions to the Standard Model: In supersymmetry: In conformal field theory: In gravitation: In the formalism of particle theories, charge-like quantum numbers can sometimes be inverted by means of a charge conjugation operator called C. Charge conjugation simply means that a given symmetry group occurs in two inequivalent (but still isomorphic) group representations.
Note that the complex Lie algebra sl(2,C) has a compact real form su(2) (in fact, all Lie algebras have a unique compact real form).
The same decomposition holds for the compact form as well: the product of two spinors in su(2) being a vector in the rotation group O(3) and a singlet.
A similar phenomenon occurs in the compact group SU(3), where there are two charge-conjugate but inequivalent fundamental representations, dubbed
, the number 3 denoting the dimension of the representation, and with the quarks transforming under
The Kronecker product of the two gives That is, an eight-dimensional representation, the octet of the eight-fold way, and a singlet.
The decomposition of the representations is again given by the Clebsch–Gordan coefficients, this time in the general Lie-algebra setting.