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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
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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
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914:
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is another charge in the theory of electromagnetism. Magnetic charges are not seen experimentally in laboratory experiments, but would be present for theories including
438:
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.
1262:
1117:
529:
1125:
234:
934:
887:
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254:
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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
1292:. The decomposition of the representations is again given by the Clebsch–Gordan coefficients, this time in the general Lie-algebra setting.
840:
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
711:
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
771:
1025:. The decomposition of such products of representations into direct sums of irreducible representations can in general be written as
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100:
969:
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72:
748:
operator called C. Charge conjugation simply means that a given symmetry group occurs in two inequivalent (but still
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79:
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342:. The charges of the theory "radiate" the gauge field. Thus, for example, the gauge field of electromagnetism is the
119:
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This article is about charge in the most general physics sense. For charge as relates to electrical phenomena, see
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362:
172:
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859:
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57:
280:. A "charge" can also refer to a point-shaped object with an electric charge and a position, such as in the
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In the formalism of particle theories, charge-like quantum numbers can sometimes be inverted by means of a
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939:
892:
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160:
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916:, the number 3 denoting the dimension of the representation, and with the quarks transforming under
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refers to the generator that rotates the fermions into bosons, and vice versa, in the supersymmetry.
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1018:
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part of the electroweak SU(2) Ă— U(1) symmetry. Weak isospin is a local symmetry, whose
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24:
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153:
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708:
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for electromagnetic interactions. In mathematics texts, this is sometimes referred to as the
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343:
1207:{\displaystyle d_{\Lambda }\cdot d_{\Lambda '}=\sum _{i}{\mathcal {L}}_{i}d_{\Lambda _{i}}.}
869:, where there are two charge-conjugate but inequivalent fundamental representations, dubbed
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8:
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of the physical system under study. When a physical system has a symmetry of some sort,
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263:
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1089:{\displaystyle \Lambda \otimes \Lambda '=\bigoplus _{i}{\mathcal {L}}_{i}\Lambda _{i}}
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328:
324:
145:
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420:
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141:
20:
572:, and the symmetry is only approximate. It is a special case of flavor symmetry.
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164:
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In the case of local, dynamical symmetries, associated with every charge is a
1345:
1022:
841:
783:
668:
756:. It is usually the case that the two charge-conjugate representations are
478:
459:
304:. The thing that "flows" in the current is the "charge", the charge is the
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580:
542:
Note that these charge quantum numbers show up in the
Lagrangian via the
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339:
335:
305:
1119:. The dimensions of the representations obey the "dimension sum rule":
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358:
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Various charge quantum numbers have been introduced by theories of
195:, and so the invariance of the charge corresponds to the vanishing
406:, then the charge operators correspond to the simple roots of the
775:
731:
Eigenvalues of the energy–momentum tensor correspond to physical
426:
The charge quantum numbers then correspond to the weights of the
133:
631:
flavor symmetry of the fundamental particles; this symmetry is
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is the
Hamiltonian. Thus, charges are associated with conserved
1017:
That is, an eight-dimensional representation, the octet of the
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product of two charge-conjugate fundamental representations
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651:
Hypothetical charges of extensions to the
Standard Model:
434:
of the Lie algebra. So, for example, when a particle in a
707:. Here, the term 'central' is used in the sense of the
635:
by the masses of the heavy quarks. Charges include the
1270:
1250:
1223:
1128:
1105:
1034:
1007:{\displaystyle 3\otimes {\overline {3}}=8\oplus 1.\ }
972:
942:
922:
895:
875:
830:{\displaystyle 2\otimes {\overline {2}}=3\oplus 1.\ }
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627:isospin mentioned above, these generate the global
357:refer to the generator, one has that the generator
60:. Unsourced material may be challenged and removed.
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858:and a singlet. The decomposition is given by the
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865:A similar phenomenon occurs in the compact group
140:is any of many different quantities, such as the
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763:of the Lie group. Their product then forms the
719:of the two-dimensional conformal field theory.
402:So, for example, when the symmetry group is a
260:; these are the eigenvalues of the generator
338:; when quantized, the gauge field becomes a
292:Abstractly, a charge is any generator of a
167:, and specifically, to the generators that
16:Physics property associated with symmetries
963:. The Kronecker product of the two gives
372:implies that the eigenvalues (lower-case)
544:Gauge covariant derivative#Standard_Model
120:Learn how and when to remove this message
1244:is the dimension of the representation
306:generator of the (local) symmetry group
1344:
1323:Affine Lie Algebras and Quantum Groups
936:and the antiquarks transforming under
308:. This charge is sometimes called the
287:
1320:
739:
58:adding citations to reliable sources
29:
770:Thus, a common example is that the
564:symmetry; the gauge bosons are the
549:Charges of approximate symmetries:
450:. These include the charges of the
13:
1290:Littlewood–Richardson coefficients
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466:. The color charge generates the
786:SO(3,1); abstractly, one writes
715:of the algebra; here, it is the
557:charges. The symmetry groups is
175:. Charges are often denoted by
34:
23:. For other uses of charge, see
956:{\displaystyle {\overline {3}}}
909:{\displaystyle {\overline {3}}}
782:) forms the adjoint rep of the
699:, sometimes referred to as the
327:. The conserved current is the
45:needs additional citations for
1325:, Cambridge University Press,
1314:
1281:{\displaystyle {\mathcal {L}}}
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205:
1:
1307:
346:; and the gauge boson is the
1237:{\displaystyle d_{\Lambda }}
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156:. Charges correspond to the
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860:Clebsch–Gordan coefficients
761:fundamental representations
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300:implies the existence of a
10:
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18:
579:-flavor charges, such as
1257:{\displaystyle \Lambda }
1112:{\displaystyle \Lambda }
701:conformal central charge
319:is the generator of the
483:electroweak interaction
481:quantum numbers of the
315:Thus, for example, the
282:method of image charges
1357:Quantum chromodynamics
1321:Fuchs, Jurgen (1992),
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717:energy–momentum tensor
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428:highest-weight modules
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524:{\displaystyle u_{1}}
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69:"Charge" physics
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587:. Together with the
570:elementary particles
568:. The pions are not
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436:quantum field theory
376:are time-invariant:
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54:improve this article
1362:Physical quantities
1264:, and the integers
485:. It generates the
294:continuous symmetry
288:Abstract definition
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229:{\displaystyle =0}
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929:{\displaystyle 3}
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882:{\displaystyle 3}
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758:complex conjugate
713:central generator
705:conformal anomaly
655:The hypothetical
302:conserved current
298:Noether's theorem
273:{\displaystyle Q}
249:{\displaystyle H}
188:{\displaystyle Q}
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853:rotation group
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767:of the group.
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693:central charge
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555:strong isospin
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495:W and Z bosons
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452:Standard Model
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432:representation
310:Noether charge
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165:symmetry group
158:time-invariant
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531:-charge of a
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71: –
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65:Find sources:
59:
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49:
48:
43:This article
41:
37:
32:
31:
26:
22:
1322:
1316:
1216:
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1016:
864:
839:
769:
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682:
666:
650:
633:badly broken
548:
541:
491:gauge bosons
479:weak isospin
460:color charge
445:
425:
416:discreteness
401:
390:
382:
373:
354:
352:
333:
323:symmetry of
314:
309:
291:
169:commute with
150:color charge
137:
131:
116:
110:October 2015
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
725:gravitation
677:supercharge
637:hypercharge
581:strangeness
533:Lie algebra
430:of a given
412:Lie algebra
408:root system
370:Commutation
363:Hamiltonian
340:gauge boson
336:gauge field
173:Hamiltonian
1346:Categories
1308:References
1288:being the
750:isomorphic
197:commutator
161:generators
80:newspapers
1252:Λ
1230:Λ
1191:Λ
1162:∑
1149:Λ
1140:⋅
1135:Λ
1107:Λ
1078:Λ
1054:⨁
1043:Λ
1039:⊗
1036:Λ
996:⊕
985:¯
977:⊗
949:¯
902:¯
845:real form
819:⊕
808:¯
800:⊗
404:Lie group
361:with the
1296:See also
1152:′
1046:′
1021:, and a
643:and the
641:X-charge
493:are the
442:Examples
359:commutes
236:, where
1023:singlet
842:compact
780:spinors
776:SL(2,C)
703:or the
695:of the
410:of the
395:
379:
148:or the
134:physics
94:scholar
1329:
1217:Here,
1002:
825:
709:center
639:, the
575:Other
562:flavor
536:module
464:quarks
414:; the
348:photon
138:charge
96:
89:
82:
75:
67:
25:Charge
867:SU(3)
849:su(2)
778:(the
629:SU(6)
585:charm
577:quark
566:pions
559:SU(2)
487:SU(2)
468:SU(3)
163:of a
101:JSTOR
87:books
1327:ISBN
889:and
856:O(3)
733:mass
691:The
675:The
553:The
500:The
477:The
458:The
321:U(1)
171:the
136:, a
73:news
774:of
723:In
683:In
667:In
583:or
462:of
397:= 0
366:= 0
152:in
144:in
132:In
56:by
1348::
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454::
423:.
399:.
391:dt
383:dq
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350:.
331:.
312:.
284:.
1274:L
1226:d
1202:.
1195:i
1186:d
1180:i
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1166:i
1158:=
1144:d
1131:d
1082:i
1072:i
1066:L
1058:i
1050:=
993:8
990:=
982:3
974:3
946:3
924:3
899:3
877:3
816:3
813:=
805:2
797:2
735:.
663:.
647:.
617:d
607:–
597:u
538:.
517:1
513:u
497:.
474:.
387:/
374:q
355:Q
268:Q
244:H
224:0
221:=
218:]
215:H
212:,
209:Q
206:[
183:Q
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117:(
112:)
108:(
98:·
91:·
84:·
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50:.
27:.
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