Charge (physics)

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In

. Charges are often denoted by ${\displaystyle Q}$, and so the invariance of the charge corresponds to the vanishing

${\displaystyle [Q,H]=0}$

${\displaystyle H}$

${\displaystyle Q}$

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

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

Various charge quantum numbers have been introduced by theories of particle physics. These include the charges of the Standard Model:

• The
  SU(3) color symmetry of quantum chromodynamics
  .
• 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 ${\displaystyle u_{1}}$-charge of a Lie algebra module.

Note that these charge quantum numbers show up in the Lagrangian via the Gauge covariant derivative#Standard_Model.

Charges of approximate symmetries:

• The
  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
  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
  .

• 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
  energy–momentum tensor of the two-dimensional conformal field theory.^[1]

• 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

Thus, a common example is that the

SO(3,1); abstractly, one writes

${\displaystyle 2\otimes {\overline {2}}=3\oplus 1.\ }$

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

Clebsch–Gordan coefficients

.

A similar phenomenon occurs in the compact group

SU(3)

, where there are two charge-conjugate but inequivalent fundamental representations, dubbed ${\displaystyle 3}$ and ${\displaystyle {\overline {3}}}$, the number 3 denoting the dimension of the representation, and with the quarks transforming under ${\displaystyle 3}$ and the antiquarks transforming under ${\displaystyle {\overline {3}}}$. The Kronecker product of the two gives

${\displaystyle 3\otimes {\overline {3}}=8\oplus 1.\ }$

That is, an eight-dimensional representation, the octet of the