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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