Symmetries in Physical Theories  

In General > s.a. Central Charge; crystals [including generalized symmetries].
* History: They entered physics as properties of structures (from an earlier notion that applied to architecture), then motions, finally laws and actions; There, they led to a fruitful connection with conserved quantities, through Hamel's and Noether's work; Their importance was recognized especially after Wigner's work in the 1930s.
* Applications: They are at the basis of gauge theories and particle statistics; They help establish physical laws and control their validity by imposing restrictions; They are associated with conservation laws in dynamical theories; They lead to the classification of crystals.
* Origin: They can be postulated as fundamental (e.g., in the action), or they might emerge dynamically (e.g., in solutions of field equations); Symmetries and physical laws might also arise naturally from some essentially random dynamics (e.g., as proposed in Froggatt & Nielsen 91).

Kinds of Symmetries > s.a. gauge symmetries; hamiltonian dynamics; Supersymmetry; symplectic manifolds [reduction].
* According to what they act on: Acting on the variables of the action, leading to conserved quantities; On the equations of motion or field equations; On a solution of the equations of motion or field equations; On the underlying manifold (internal/external).
* Global vs local:
* According to what they leave invariant: Symmetries realized by a unitary operator, under which wave functions may pick up a phase; Gauge symmetries, which leave all the physics invariant – all the observables in classical mechanics, and all the amplitudes in quantum mechanics.
@ General references: Sudarshan FP(95) [classifying systems in terms of symmetry groups]; Hon & Goldstein PhSc(06)oct [types of symmetry arguments]; Giulini a0802 [types, in Pauli's work]; Healey BJPS(09) [empirical vs theoretical]; Mouchet EPJH(13)-a1111 [different meanings]; Strocchi a1502 [symmetry, symmetry breaking and gauge symmetries].
@ Internal vs external: Wisnivesky IJMPA(00) [unified]; Kantorovich SHPMP(03) [internal]; Aldaya & Sánchez-Sastre JPA(06)mp; Kim & Noz a1007-proc [internal]; Laszlo a1512 [unification]; > s.a. types of gauge theories.
@ Discrete: Krauss GRG(90) [local]; Varlamov IJTP(01)mp/00; Ishimori et al PTPS(10)-a1003, 12 [in particle physics]; Sozzi 12.
@ Special types: Wotzasek AP(95)ht [hidden]; Anco & Bluman JMP(96) [non-local, and conservation laws]; Mostafazadeh & Samani MPLA(00) [topological]; Cariglia RMP(15)-a1411 [hidden]; Czachor QSMF(14)-a1412 [relativity of arithmetics]; Gomes IJMPA(16)-a1510-ln [emergent]; > s.a. wave equations.

Spacetime Symmetries > s.a. killing fields; lorentzian geometry; Relativity.
* Types: A vector field X may generate different kinds of symmetries, Isometry, \(\cal L\)X gab = 0; Conformal isometry, \(\cal L\)X gab = α gab, with α a function; Affine collineation, \(\cal L\)X Γabc = 0; Projective collineation, \(\cal L\)X Γabc = 2 δa[b f, c], with f a function; Curvature collineation, \(\cal L\)X Rabcd = 0.
@ General references: Katzin & Levine JMP(81); Hall CQG(89), GRG(98); Duggal & Sharma 99; Hall 04; Harte CQG(08)-a0805 [approximate, and conservation laws]; Roberts BJPS(08) [dynamical symmetries vs empirical symmetries]; Saifullah NCB(07)-a0902 [classification]; Houri & Yasui CQG(15)-a1410 [test]; Ayón-Beato & Velázquez-Rodríguez a1511 [residual symmetries of a gravitational Ansatz].
@ And field theory: Halliwell PRD(91) [parametrization of hypersurface embedding]; Costa & Fogli 12; Smolić CQG(15)-a1501 [symmetry inheritance, scalar fields].
> And gravity: see bianchi models; general relativity; initial-value formulation; minisuperspace; solutions with symmetries; supergravity.
> Special types: see axisymmetry; Collineations; conformal invariance; Helical Symmetry; Translations.

And Dynamical Theories > s.a. conservation laws; formulations of classical mechanics; interaction; Lambda Symmetries; Mechanical Similarity.
* Idea: We say that a physical theory has a certain symmetry if, given a solution q(t) for the equations of motion, the transformation q \(\mapsto\) q':= f(q) yields another solution; These transformations usually have a group structure.
* Conservation laws: Can be obtained from Noether's theorem, Lutzky's theorem, bi-Hamiltonian formalism, or bidifferential calculi.
* Canonical framework: A transformation q \(\mapsto\) q' can be extended to a canonical one.
@ General references: Caratù et al draft-77; Barnich & Brandt NPB(02)ht/01 [field theory, covariant theory]; Gitman & Tyutin BJP(06)ht/05 [equivalent Lagrangian and Hamiltonian systems]; de León et al IJGMP(04)mp [field theory, multisymplectic]; Bogoyavlenskij CMP(05) [hidden structures]; Ferrario & Passerini EJP(07) [Lagrangian vs dynamical]; Wilczek MPLA(10) [symmetry transmutation]; Boozer EJP(12) [in classical mechanics]; Andersson et al CQG(14) + Bäckdahl CQG+ [conditions for existence of symmetry operators for field equations]; Henkel Symm(15)-a1509 [dynamical symmetries and causality]; Fang et al CTP(16)-a1601 [quantifying approximate symmetries of Hamiltonians or states, degree of symmetry].
@ And conservation laws: Lange SHPMP(07); Cicogna MMAS(13)-a1307 [and generalizations]; Peng a1403 [for difference systems]; Sharapov Sigma(16)-a1607 [based on variational tricomplex with a presymplectic structure].
@ And constraints: Lee & Wald JMP(90); Giulini MPLA(95)gq/94; Chitaia et al PRD(97), PRD(97); > s.a. constrained systems; symmetries in quantum physics.
@ Noether symmetries: Rosenhaus & Katzin JMP(94) [for differential equations]; Pons & García IJMPA(00)ht/99 [constrained systems]; Brading & Brown ht/00 [gauge symmetries]; García & Pons IJMPA(01)ht/00 [canonically realized in enlarged phase space].
@ Non-Noether: Chavchanidze GMJ(01)mp, mp/01, JGP(03)mp/02 [and bi-Hamiltonian systems], mp/02 [and conservation laws].
@ Non-linear realizations: Love MPLA(05)ht [supersymmetry and others, rev]; > s.a. sigma-models.
> Types of theories: see fields with higher spin; gauge theory; integrable system; lagrangian dynamics; noether theorem [non-local theories]; special relativity.

References > s.a. lie algebra; physics teaching.
@ Books, I: Lederman & Hill 04 [and physics]; Zee 07 [and physics].
@ Books: Weyl 52; Takens 77; Elliott & Dawber 79; Rosen 83; García Doncel et al 87; Yaglom 88; Bunch 89; Froggatt & Nielsen 91; Rosen 95; Stewart & Golubitsky 93 [I]; Singer 01 [II/III]; Brading & Castellani ed-03 [philosophical]; Prakash 03 [including super]; Mainzer 05 [and complexity]; Debs & Redhead 07 [objectivity, invariance, and convention]; Haywood 10 [group theory, r CP(120#2]; Goldberg 13; Sundermeyer 14 [in fundamental physics].
@ General articles: Feynman TPT(66); Rosen FP(90); Gières ht/97-proc; Kosso BJPS(00) [observation]; Suppes FP(00) [invariance and covariance]; Chester ISPS(02)-a1202; Brading & Castellani qp/03-ch; Esposito & Marmo in(04)mp/05 [rev]; Zuber a1307-conf [and Klein's Erlangen program]; Alamino a1305 [generalization to symmetry on average]; Lederer a1401 [philosophical approach]; Mouchet a1503 [discovered or invented?]; Das & Kunstatter a1609-JAFS [and unification].
@ And group theory: Rosen AJP(81)apr [RL]; Guay & Hepburn PhSc(09)apr [groups vs groupoids]; > s.a. group theory; Semigroups.
@ Other mathematics: Yanofsky & Zelcer FoS(16)-a1502; > s.a. differential equations (ordinary and partial); integral equations; killing fields.
@ History: Brading SHPMP(02) [Noether & Weyl]; Katzir HSPBS(04) [origin]; Hon & Goldstein SHPSA(05) [evolution]; Wilczek MPLA(10)-a1008-in [and BCS theory]; Maldacena a1410 [in particle physics, and the Higgs boson].
@ And teaching: Mangelsdorf & Heald AJP(90)feb; Bloembergen AJP(90)feb; Rosen AJP(90)aug; Hill & Lederman TPT(00)phy.

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