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)) Φb − θ 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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