Symmetries in Physical Theories  

In General > s.a. Central Charge; crystals.
* 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; classification of crystals.
> In mathematics: see differential equations (ordinary and partial); group theory; integral equations; killing fields.

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.
@ Internal vs external: Wisnivesky IJMPA(00) [unified]; Kantorovich SHPMP(03) [internal]; Aldaya & Sánchez-Sastre JPA(06)mp; > s.a. types of gauge theories.
@ Special types: Krauss GRG(90) [discrete local]; Wotzasek AP(95)ht [hidden]; Anco & Bluman JMP(96) [non-local, and conservation laws]; Mostafazadeh & Samani MPLA(00) [topological]; Varlamov IJTP(01)mp/00 [discrete]; > s.a. wave equations.

Spacetime Symmetries > s.a. axisymmetry; conformal invariance; Helical; killing fields; Translations.
* Types: A vector field X may generate different kinds of symmetries, Isometry, X gab = 0; Conformal isometry, X gab = gab, with a function; Affine collineation, X abc = 0; Projective collineation, X abc = 2 a[b f, c], with f a function; Curvature collineation, X Rabcd = 0.
@ General references: Katzin & Levine JMP(81); Hall CQG(89), GRG(98); Duggal & Sharma 99; Hall 02; Harte CQG(08)-a0805 [approximate, and conservation laws]; Roberts BJPS(08) [dynamical symmetries vs empirical symmetries]; Saifullah NCB(07)-a0902 [classification].
@ And field theory: Halliwell PRD(91) [parametrization of hypersurface embedding].
> And gravity: see bianchi models; general relativity; initial-value formulation; minisuperspace; solutions with symmetries; supergravity.

And Dynamical Theories > s.a. conservation laws; formulations of classical mechanics; lagrangian dynamics; 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 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 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]; Lange SHPMP(07) [and conservation laws]; Ferrario & Passerini EJP(07) [Lagrangian vs dynamical].
@ And constraints: Lee & Wald JMP(90); Giulini MPLA(95)gq/94; Chitaia et al PRD(97), PRD(97); > s.a. 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.

References > s.a. lie algebra; physics teaching.
@ Books, I: Lederman & Hill 04 [and physics]; Zee 07 [and physics, r JPA(08)#1].
@ Books: Weyl 52; Takens 77; Elliot & Dawber 79; Rosen 83; Froggatt & Nielsen 91; Doncel et al 87; Yaglom 88; Bunch 89; Rosen 95; Stewart & Golubitsky 93 [I]; Singer 01 [II/III]; Brading & Castellani ed-03 [philosophical]; Prakash 03 [including super]; Debs & Redhead 07 [objectivity, invariance, and convention; r SHPMP(09)].
@ General articles: Feynman TPT(66); Rosen FP(90); Sudarshan FP(95) [classifying systems in terms of symmetry groups]; Gières ht/97-in; Kosso BJPS(00) [observation]; Suppes FP(00) [invariance and covariance]; Brading & Castellani qp/03-in; Esposito & Marmo mp/05-in [rev]; Hon & Goldstein PhSc(06)oct [types of symmetry arguments]; Giulini a0802 [types, in Pauli's work].
@ Formalisms: Rosen AJP(81)apr-RL [group theory]; Guay & Hepburn PhSc(09)apr [groups vs grupoids].
@ History: Brading SHPMP(02) [Noether & Weyl]; Katzir HSPBS(04) [origin]; Hon & Goldstein SHPSA(05) [evolution].
@ And teaching: Mangelsdorf & Heald AJP(90)feb; Bloembergen AJP(90)feb; Rosen AJP(90)aug; Hill & Lederman phy/00.

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