In physics, a charge may refer to one of many
different quantities, such as the electric
charge in electromagnetism or the color charge
in quantum chromodynamics. Charges correspond
to the time-invariant generators of a symmetry
group, and specifically, to the generators
that commute with the Hamiltonian. Charges
are often denoted by the letter Q, and so
the invariance of the charge corresponds to
the vanishing commutator
[
Q
,
H
]
=
0
{\displaystyle [Q,H]=0}
, where H is the Hamiltonian. Thus, charges
are associated with conserved quantum numbers;
these are the eigenvalues q of the generator
Q.
== 
Abstract definition ==
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. The
thing that "flows" in the current is the "charge",
the charge is the generator of the (local)
symmetry group. This charge is sometimes called
the Noether charge.
Thus, for example, the electric charge is
the generator of the U(1) symmetry of electromagnetism.
The conserved current is the electric current.
In the case of local, dynamical symmetries,
associated with every charge is a gauge field;
when quantized, the gauge field becomes a
gauge boson. The charges of the theory "radiate"
the gauge field. Thus, for example, the gauge
field of electromagnetism is the electromagnetic
field; and the gauge boson is the photon.
The word "charge" is often used as a synonym
for both the generator of a symmetry, and
the conserved quantum number (eigenvalue)
of the generator. Thus, letting the upper-case
letter Q refer to the generator, one has that
the generator commutes with the Hamiltonian
[Q, H] = 0. Commutation implies that the eigenvalues
(lower-case) q 
are time-invariant: dq/dt = 0.
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 simple roots are used, as
all the other roots can be obtained as linear
combinations of these. The general roots are
often called raising and lowering operators,
or ladder operators.
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.
== Examples ==
Various charge quantum numbers have been introduced
by theories of particle physics. These include
the charges of the Standard Model:
The color charge of quarks. The color charge
generates the SU(3) color symmetry of quantum
chromodynamics.
The weak isospin quantum numbers of the electroweak
interaction. It generates the SU(2) part of
the electroweak SU(2) × U(1) symmetry. Weak
isospin is a local symmetry, whose gauge bosons
are the W and Z bosons.
The electric charge for electromagnetic interactions.
In mathematics texts, this is sometimes referred
to as the
u
1
{\displaystyle u_{1}}
-charge of a Lie algebra module.Charges of
approximate symmetries:
The strong isospin charges. The symmetry groups
is SU(2) flavor symmetry; the gauge bosons
are the pions. The pions are not elementary
particles, and the symmetry is only approximate.
It is a special case of flavor symmetry.
Other quark-flavor charges, such as strangeness
or charm. Together with the u–d isospin
mentioned above, these generate the global
SU(6) flavor symmetry of the fundamental particles;
this symmetry is badly broken by the masses
of the heavy quarks. Charges include the hypercharge,
the X-charge and the weak hypercharge.Hypothetical
charges of extensions to the Standard Model:
The hypothetical magnetic charge is another
charge in the theory of electromagnetism.
Magnetic charges are not seen experimentally
in laboratory experiments, but would be present
for theories including magnetic monopoles.In
supersymmetry:
The supercharge refers to the generator that
rotates the fermions into bosons, and vice
versa, in the supersymmetry.In conformal field
theory:
The central charge of the Virasoro algebra,
sometimes referred to as the conformal central
charge or the conformal anomaly. Here, the
term 'central' is used in the sense of the
center in group theory: it is an operator
that commutes with all the other operators
in the algebra. The central charge is the
eigenvalue of the central generator of the
algebra; here, it is the energy-momentum tensor
of the two-dimensional conformal field theory.In
gravitation:
Eigenvalues of the energy-momentum tensor
correspond to physical mass.
== 
Charge conjugation ==
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. It is usually the case that
the two charge-conjugate representations are
complex conjugate fundamental representations
of the Lie group. Their product then forms
the adjoint representation of the group.
Thus, a common example is that the product
of two charge-conjugate fundamental representations
of SL(2,C) (the spinors) forms the adjoint
rep of the Lorentz group SO(3,1); abstractly,
one writes
2
⊗
2
¯
=
3
⊕
1.
{\displaystyle 2\otimes {\overline {2}}=3\oplus
1.\ }
That is, the product of two (Lorentz) spinors
is a (Lorentz) vector and a (Lorentz) scalar.
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. The decomposition is given
by the Clebsch-Gordan coefficients.
A similar phenomenon occurs in the compact
group SU(3), where there are two charge-conjugate
but inequivalent fundamental representations,
dubbed
3
{\displaystyle 3}
and
3
¯
{\displaystyle {\overline {3}}}
, the number 3 denoting the dimension of the
representation, and with the quarks transforming
under
3
{\displaystyle 3}
and the antiquarks transforming under
3
¯
{\displaystyle {\overline {3}}}
. The Kronecker product of the two gives
3
⊗
3
¯
=
8
⊕
1.
{\displaystyle 3\otimes {\overline {3}}=8\oplus
1.\ }
That is, an eight-dimensional representation,
the octet of the eight-fold way, and a singlet.
The decomposition of such products of representations
into direct sums of irreducible representations
can in general be written as
Λ
⊗
Λ
′
=
⨁
i
L
i
Λ
i
{\displaystyle \Lambda \otimes \Lambda '=\bigoplus
_{i}{\mathcal {L}}_{i}\Lambda _{i}}
for representations
Λ
{\displaystyle \Lambda }
. The dimensions of the representations obey
the "dimension sum rule":
d
Λ
⋅
d
Λ
′
=
∑
i
L
i
d
Λ
i
.
{\displaystyle d_{\Lambda }\cdot d_{\Lambda
'}=\sum _{i}{\mathcal {L}}_{i}d_{\Lambda _{i}}.}
Here,
d
Λ
{\displaystyle d_{\Lambda }}
is the dimension of the representation
Λ
{\displaystyle \Lambda }
, and 
the 
integers
L
{\displaystyle {\mathcal {L}}}
being the Littlewood-Richardson coefficients.
The decomposition of the representations is
again given by the Clebsch-Gordan coefficients,
this time in the general Lie-algebra setting.
== See also ==
Casimir operator
