Symmetries in Quantum Physics  

In General > s.a. hilbert space; particle statistics; symplectic manifolds and structures.
* Idea: One has to specify the action of a transformation under consideration as an operator on the Hilbert space, and check what it does to observables.
* Applications: A symmetry gives many properties of a system, due to linearity, using group representations; The parameters that classify particles (mass and the various quantum numbers), are just a characterization of how they transform under all symmetry groups of the laws of nature; A particle "is" how it transforms under these symmetries.
* Appoaches: In the Dirac approach, gauge symmetries are generated by first-class constraints; In the Faddeev-Jackiw approach, gauge, reparametrization and other symmetries are generated by the null eigenvectors of the sympletic matrix.
@ Simple: Lee 88; Gross PT(95)dec.
@ General references: Wigner PAPS(49); Sakurai 64; Houtappel et al RMP(65); Greiner & Müller 94; Wotzasek AP(95)ht [Faddeev-Jackiw approach]; Fano & Rau 96; Chaichian & Hagedorn 97; Wilczek NPPS(98)ht/97 [examples]; Larsson mp/01 [deepest]; Kim NPPS(01)ht [particles]; Kempf PRD(01) [symmetric operators and unitary transformations]; Khruschov a0807 [symmetry algebras from observables]; Ziaeepour JPCS(15)-a1502 [symmetry as a foundational concept]; Lombardi & Fortin EJTP-a1602 [role of symmetry in the interpretation of quantum mechanics]; > s.a. Wigner's Theorem.
@ Invariant actions: Michel CRAS(71); Gaeta LMP(93); Hurth & Skenderis NPB(99) [construction of quantum field theories].
@ Symmetry reduction: D'Avanzo et al IJGMP(05)mp [algebras of differential operators]; Wu PRD(09)-a0801 [in quantum field theory]; Torre JMP(09)-a0901 [and quasi-free states]; Bates et al RVMP(09) [singularly reduced systems]; Hochs & Mathai AiM(15)-a1309 [result on commutation of quantization with reduction]; Kumar & Sarovar JPA(14)-a1412; > s.a. loop quantum gravity.
> Related topics: see quantum systems; open quantum systems.

Types of Symmetries > s.a. conformal symmetry [including scale symmetry]; CPT symmetry; Permutations.
* In general: Permutations of particles (exact); Continuous spacetime transformations (exact); Discrete transformations, C, P, T, G-parity, etc (only CPT exact); Gauge transformations (only some are exact).
* Gauge symmetries: They are stronger, more restrictive versions of internal symmetries; They can be related to constraints in the classical theory, but a more general approach uses null vectors of the symplectic structure; In quantum field theory, they are associated with spin-1 (vector) fields, which are massless if the symmetry is exact.
* Chiral symmetry: Associated with spin-1/2 particles, which are massless if the symmetry is exact.
@ Particle physics, in general: Kibble CP(65), reprint CP(09).
@ Time translation: Lehto et al PLB(89) [in quantum field theory]; Nagata & Nakamura IJTP(09) [as a constraint on non-local realism]; > s.a. crystals [breaking].
@ Gauge symmetries: Bergmann & Flaherty JMP(78); Ying et al ht/99 [integral view]; Pons SHPMP(05)phy/04 [Dirac's analysis].
@ Chiral symmetry: Brown & Rho PRP(02) [in nuclear physics].
@ In quantum field theory, other: Łopuszański 90 [lecture notes]; Connes & Marcolli JGP(06)ht/05 [universal symmetry as motivic Galois group]; Jaffe & Ritter a0704 [representations of spacetime symmetries]; Bangu SHPMP(08) [Gell-Mann–Ne'eman Ω prediction on the basis of symmetry classification scheme]; Wallace SHPMP(09) [vs classical fields]; Lowdon a1509 [non-manifest symmetries, and their anomalies]; Trautner a1608-PhD [outer automorphisms of symmetries, "symmetries of symmetries"]; > s.a. klein-gordon fields [symmetry reduction].

Deformations, Violations, Quantum-Motivated Generalizations > s.a. anomalies; deformations; symmetry breaking.
* Idea: The fact that a classical theory has a certain symmetry does not imply that the corresponding quantum theory has that symmetry, because of possible anomalies.
@ Generalizations: Taylor JMP(80) [generalized groups]; Mezey & Maruani MolP(90) [fuzzy, "syntopy"]; Oeckl JGP(02)mp/01 [in terms of quantum groups]; Lukierski in(05)ht/04 [deformations]; Kunzinger & Popovych a0903-proc [re non-classical symmetries]; Schreiber & Škoda a1004-ln [categorified symmetries in quantum field theory]; > s.a. quantum group.
@ Related topics: Balachandran et al NPB(87) [central extensions of symmetry groups upon quantization]; Bohr & Ulfbeck RMP(95) [and origin of quantum mechanics].

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