Charges |

**In General** > s.a. G-parity;
Isospin; noether
charge; topology in physics [topological charges].

* __Idea__: Loosely speaking,
a charge is a conserved quantity other than energy-momentum.

* __And conserved currents__:
When a charge is conserved because it is associated with a conserved current,
it can be calculated as a surface integral (see examples below); > s.a.
conservation laws.

> __Online resources__:
see Wikipedia page.

**In a Gauge Theory** > s.a. electromagnetism; gauge theory.

* __Abelian electric charge__:
The electric charge of an isolated system is

*Q* = \(1\over2\)\(\displaystyle\oint_S\)_{}
*F*^{ab} d*s*_{ab} =
\(\displaystyle\oint_S\)_{} *F*^{ 0b}
d*s*_{b} = \(\displaystyle\oint_S\)_{}_{ }*E*_{a}
d*s*^{a} ,

where *S* is any closed (*N*–1)-surface in *N*+1
spacetime dimensions enclosing the system, possibly a surface at infinity, *S*_{∞}.

* __Non-abelian theory__:
The electric and magnetic charges *Q* and *F* have
internal indices; For a Yang-Mills theory they are, respectively,

*Q*^{ i}_{j}
= \(1\over4\pi\)\(\displaystyle\oint_S\)_{}_{} *E*_{a}^{i}_{j}
d*s*^{a} and *P* = \(1\over2\)\(\displaystyle\oint_S\)_{}_{} **F*^{ab} d*s*_{ab} =
\(\displaystyle\oint_S\)_{}* B*_{a} d*s*^{a} ;

Several electric and magnetic charges can actually be defined; 'Brane source
charge' is gauge invariant and localized but not conserved or quantized, 'Maxwell
charge' is gauge invariant and conserved but not localized or quantized, while
'Page charge' conserved, localized, and quantized but not gauge invariant
[@ Marolf ht/00-proc].

@ __Abelian, electric charge__: Segers et al AJP(09)jan
[Einstein's "little machine" to measure small charges];
Dzhunushaliev a1002 [non-perturbative, finite quantum field theory corrections].

@ __General references__: Sniatycky & Schwartz RPMP(94)
[no local non-abelian charge density]; Marolf
ht/00-proc;
Lyre IJMPD(00)gq [equivalence principle for charge];
Barnich CQG(03)ht [boundary charges];
McMullan Sigma(07)ht;
> s.a. topology in physics.

**Specific Systems and Particles ** > s.a. electron [decay];
particle types.

@ __In gravity__: Hollands et al PRD(05)ht/05 [counter-term subtraction definition];
Bousso et al a1709
[asymptotic charges cannot be observed in finite retarded time];
> s.a. anti-de sitter space; gravitational energy.

**References** > s.a. renormalization.

@ __General__: Klauder & Wheeler RMP(57) [re neutrinos];
Toussaint GRG(00)gq/99 [general relativity / gauge theory];
Hofer qp/00 [origin?].

@ __Quantization__: Rosen IJTP(79);
Staruszkiewicz AP(89);
Rañada & Trueba PLB(98)ht [topological];
Rañada ht/99;
Olive ht/01-conf;
Allen et al ht/02;
Pérez-Lorenzana & Pires MPLA(03) [higher dimensions];
Barone & Helayël-Neto qp/04-proc,
comment MacKenzie et al qp/05 [and Aharonov-Bohm effect];
Minguzzi et al JPA(06) [from weak gauge principle];
Buitrago a1404 [from 2-spinor approach to U(1) gauge theory];
Solha JGP(16)-a1409 [topological argument, without magnetic monopoles];
> s.a. experiments in physics; kaluza-klein theory;
monopoles [Dirac's argument].

@ __Fractional charges__: Wilczek cm/02-in;
Gies et al PRL(06) [unquantized 'millicharged fermions', search];
> s.a. Hall Effect [fractional]; Quarks.

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