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# CHARGE

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 are associated with conserved quantum numbers.

## Formal definition

More 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.

Sometimes, the word "charge" is used as a synonym for "generator" in referring to the generator of the symmetry. More precisely, when the symmetry group is a Lie group, then the charges are understood to correspond to the root system of the Lie group; the discreteness of the root system accounting for the quantization of the charge.

## 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.

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 fundamental particles, and the symmetry is only approximate. It is a special case of flavor symmetry.
• Particle flavor charges, such as strangeness or charm. These generate the global SU(6) flavor symmetry of the fundamental particles; this symmetry is badly broken by the masses of the heavy quarks.

Hypothetical charges of extensions to the Standard Model:

• The magnetic charge, 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 the formalism of particle theories charge-like quantum numbers can sometimes be inverted by means of a charge conjugation operator called C. Chiral fermions often cannot. 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 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\otimes\overline{2}=3\oplus 1.\$

# CHARGE

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 are associated with conserved quantum numbers.

## Formal definition

More 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.

Sometimes, the word "charge" is used as a synonym for "generator" in referring to the generator of the symmetry. More precisely, when the symmetry group is a Lie group, then the charges are understood to correspond to the root system of the Lie group; the discreteness of the root system accounting for the quantization of the charge.

## 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.

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 fundamental particles, and the symmetry is only approximate. It is a special case of flavor symmetry.
• Particle flavor charges, such as strangeness or charm. These generate the global SU(6) flavor symmetry of the fundamental particles; this symmetry is badly broken by the masses of the heavy quarks.

Hypothetical charges of extensions to the Standard Model:

• The magnetic charge, 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 the formalism of particle theories charge-like quantum numbers can sometimes be inverted by means of a charge conjugation operator called C. Chiral fermions often cannot. 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 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 $2\otimes\overline{2}=3\oplus 1$.

# CHARGE

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