Noether Symmetries / Theorem  

In General > s.a. [hamiltonian and lagrangian symmetries]; symmetries.
* Idea: Exploit a symmetry of a theory so as to reduce the number of variables needed to treat a problem.
* History: Soon after Hilbert's discovery of the variational principle for general relativity, people including Hilbert, Klein, and Einstein were concerned about the failure of local energy conservation in the theory; Noether's theorems solved the problem.
$ Def: To every continuous symmetry xa = Xab b, = a a of the Lagrangian for a field theory there corresponds a conserved current Jab with a Jab = 0, and a conserved quantity, the charge Qb:

Jab:= (/(a)) b – ac Xcb ,   Qb:= Sigma dSa Jab.

@ General references: Noether NKGG(18) [translation TTSP(71)phy/05]; Govinder & Leach PLA(95) [integrals]; Byers phy/98 [historical].
@ Second theorem: Gogilidze & Surovtsev ht/96 [and constraints]; Bashkirov et al JPA(05)m.DG/04 [generalized setting], JMP(05)mp/04 [BRST symmetries]; Cariñena et al m.DG/05 [gauge symmetries in classical mechanics].
@ Related topics: García & Pons IJMPA(01)ht/00 [canonical realization]; Sanyal & Modak CQG(01)gq [and field couplings]; Brown & Holland AJP(04)jan [first theorem, in quantum mechanics and electromagnetism]; Butterfield phy/05-in; Albeverio et al JMP(06) [quantum]; Bokhari et al IJTP(06) [and spacetime isometries]; Bokhari & Kara GRG(07) [vs Killing vectors]; Bering a0911 [proof, for a fixed integration region].

In Specific Theories > s.a. energy-momentum tensor.
@ Classical mechanics: Desloge & Karch AJP(77)apr; Sardanashvily mp/03; Marinho EJP(07), comment Rejmer EJP(09).
@ Gauge theories / quantum field theories: Buchholz et al AP(86); Karatas & Kowalski AJP(90)feb; Danos FP(97)ht; Fatibene et al JMP(97); Julia & Silva CQG(98)gq; Gràcia & Pons JMP(00)mp; Bashkirov JPA(05) [reducible gauge symmetries]; Darvas a0811 [new conserved current].
@ In gravitation: Sorkin PRS(91) [Noether operator, and electromagnetism];
@ In cosmological models: Vakili PLB(08)-a0804.
@ Other applications: García & Pons IJMPA(00)ht/99 [constrained systems]; Hanc et al AJP(04)apr [examples and teaching].

Generalizations
@ References: Rosen AP(72), AP(74), AP(74); Cariñena & Rañada LMP(88) [singular Lagrangians]; Lunev TMP(90) [non-local symmetries]; Gràcia & Pons JPA(95) [higher-order Lagrangians]; Govinder et al PLA(98) [approximate symmetries]; Magro et al AP(02)ht/01 [superfields]; Torres m.OC/03-in [non-smooth solutions]; Paal mp/06, CzJP(06)mp-in [from Moufang transformations]; Agostini et al MPLA(07)ht/06, Arzano & Marcianò PRD(07)ht, Amelino-Camelia et al a0710-in [for Hopf-algebra spacetime symmetries]; Cicogna & Gaeta JPA(07) [for -symmetries]; Agostini IJMPA(09)-a0711 [in -Minkowski].

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