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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