Symmetries
in Quantum Physics| Symmetries
in Quantum Physics |
In General > s.a. hilbert
space; particle statistics; quantum
systems;
symplectic manifolds and structures.
* Idea: Have to give the action 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: TDLee 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 ht/01-in
[particles]; Kempf PRD(01)
[symmetric operators and unitary transformations]; Khruschov a0807 [symmetry
algebras from observables]; > 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 a0801 [in
quantum field theory]; > s.a. loop
quantum
gravity.
Types of Symmetries > s.a. conformal
symmetry [including scale
symmetry];
CPT; Permutations.
* Types: 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: They
are stronger, more restrictive versions of internal symmetries; 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, associated with
spin-1 (vector) fields,
which are massless if the symmetry is exact.
* Chiral: Associated
with spin-1/2 particles, which are massless if the symmetry is exact.
@ Gauge: Bergmann & Flaherty JMP(78); Ying et al ht/99 [integral
view]; Pons SHPMP(05)phy/04
[Dirac's
analysis].
@ Chiral: Brown & Rho PRP(02)
[in nuclear physics].
@ In quantum field theory, other: Lehto et al PLB(89)
[time translation]; 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 
– prediction
on the basis of symmetry classification scheme]; > s.a. kg
fields [symmetry
reduction].
Deformations, Violations, Quantum-Motivated Generalizations >
s.a. anomalies; deformations; symmetry
breaking.
@ Generalizations: Taylor JMP(80) [generalized groups]; Mezey & Maruani
MolP(90) [fuzzy, "syntopy"]; Oeckl JGP(02)mp/01 [ito
quantum groups]; Lukierski ht/04-in
[deformations]; > 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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Send feedback and suggestions to bombelli at olemiss.edu – Modified
19 jul 2008