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 Δxa = Xab δωb, Δφ = Φa δωa of the Lagrangian for a field theory there corresponds a conserved current J ab with ∂a J ab = 0, and a conserved quantity, the charge Qb:

J ab:= (∂\(\cal L\)/∂(∂aφ)) Φbθ ac X cb ,   Qb:= Σ dSa J ab.

@ Books, reviews: Neuenschwander 11; Bañados & Reyes IJMPD(16)-a1601 [pedagogical, and boundary terms]; Leone a1804 [intro, and Routh reduction].
@ General references: Noether NKGG(18) [translation TTSP(71)phy/05]; Govinder & Leach PLA(95) [integrals]; Fatibene et al a1001 [and covariant conservation laws, rev]; Tsamparlis & Paliathanasis GRG(11) [geometric nature]; Francaviglia et al a1309-conf [epistemological implications]; Neuenschwander AJP(14)mar [in the undergraduate curriculum]; Silagadze EJP(15)-a1507 [invariance of the Noether charge]; Deser a1905, Brown a2010 [the converse result].
@ History: Byers phy/98; Quigg a1902 [colloquium]; Kosmann-Schwarzbach a2004-in.
@ 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]; Navarro & Sancho IJGMP(14)-a1312-conf [on any natural bundle].
@ And Killing vectors: Bokhari & Kara GRG(07); Hussain GRG(10).
@ Hamiltonian / canonical version: García & Pons IJMPA(01)ht/00; Struckmeier JPCS(12)-a1206; Herman a1409-MS [and the Legendre transform]; Sardanashvily a1510 [all conserved quantities as symmetries].
@ Quantum version: Brown & Holland AJP(04)jan [first theorem, and electromagnetism]; Albeverio et al JMP(06); Lima et al AP(12)-a0912 [for gauge theories with anomalies].
@ Related topics: Sanyal & Modak CQG(01)gq [and field couplings]; Butterfield phy/05-fs; Bokhari et al IJTP(06) [and spacetime isometries]; Bering a0911-proc [proof, for a fixed integration region]; Dallen & Neuenschwander AJP(11)mar [in a rotating frame]; Pons JMP(11) [energy-momentum tensors and conformal symmetry]; Fiorani et al a1505 [Lie algebras of conservation laws]; Baez a2006 [algebraic approach].

In Specific Theories > s.a. energy-momentum tensor; quantum theory in curved spaces.
@ Classical mechanics: Desloge & Karch AJP(77)apr; Sardanashvily mp/03; Marinho EJP(07), comment Rejmer EJP(09); > s.a. classical particles.
@ 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]; Avery & Schwab JHEP(16)-a1512 [second theorem and Ward identities for gauge symmetries].
@ Gravity: Sorkin PRS(91) [Noether operator, and electromagnetism]; Majhi & Padmanabhan PRD(12), Majhi AHEP(13)-a1210 [Noether charge from Einstein-Hilbert action, and Bekenstein-Hawking entropy]; Petrov & Lompay GRG(13)-a1211 [metric theories]; > s.a. energy-momentum; multipole moments.
@ In cosmological models: Vakili PLB(08)-a0804; Paliathanasis et al PRD(14) [scalar-tensor cosmology]; > s.a. minisuperspace models.
@ Other applications: García & Pons IJMPA(00)ht/99 [constrained systems]; Hanc et al AJP(04)apr [examples and teaching]; Fan PRD(18)-a1801 [and equations of motion, holographic transport]; Cîrstoiu et al PRX(20) [open quantum systems].

Generalizations > s.a. symmetries [and conservation laws].
@ General references: Rosen AP(72), AP(74), AP(74); Torres m.OC/03-conf [non-smooth solutions]; Fassò & Sansonetto IJGMP(09) [non-holonomic]; Hydon & Mansfield PRS(11)-a1103 [simple local proof and extension to finite-difference systems]; Marvian & Spekkens nComm(14)-a1404 [quantifying the asymmetry of quantum states]; Fiorani & Spiro JGP(15)-a1411 [Lie algebras of conservation laws]; Finster & Kleiner a1506 [for causal variational principles]; Halder et al a1812.
@ More general types of symmetries: Lunev TMP(90) [non-local symmetries]; Govinder et al PLA(98) [approximate symmetries]; Paal in(09)mp/06, CzJP(06)mp-conf [from Moufang transformations]; Agostini et al MPLA(07)ht/06, Arzano & Marcianò PRD(07)ht, Amelino-Camelia et al PTPS(07)-a0710-conf [for Hopf-algebra spacetime symmetries]; Cicogna & Gaeta JPA(07) [for μ-symmetries]; Alamino a1305 [symmetry on average, and Noether's theorem with dissipative currents]; Webb & Mace JPP(15)-a1403 [fluid relabelling symmetries]; Zhang et al a1903 [scaling symmetry]; Bravetti & Garcia-Chung a2009 [geometric approach].
@ Higher-order Lagrangians: Gràcia & Pons JPA(95); Townsend a1605.
@ More general types of theories: Cariñena & Rañada LMP(88) [singular Lagrangians]; Magro et al AP(02)ht/01 [superfields]; Holman a1009 [for field theories formulated in Minkowski spacetime]; Baez & Fong JMP(13)-a1203 [for Markov Processes]; Sardanashvily a1411 [reducible degenerate Grassmann-graded Lagrangian theories]; Kegeles & Oriti JPA-a1506, Krivoruchenko & Tursunov a1602 [non-local theories]; Anco a1605-in [non-variational partial differential equations]; Peng a1607 [differential-difference equations]; D'Ambrosio a1902 [discrete covariant mechanics]; > s.a. higher-order lagrangians [non-local].
@ Non-Lagrangian theories: Kaparulin et al JMP(10)-a1001; Delphenich a1109 [based on the virtual work functional].
@ More general settings: Agostini IJMPA(09)-a0711 [in κ-Minkowski]; Muslih a1003 [for fractional classical fields]; González & Cabo FP(18)-a1709 [stochastic version].

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