SU(2)
Group| SU(2)
Group |
In General > s.a. lie group;
mathematical physics; 10j Symbol.
* Idea: The 3-dimensional
group of 2 × 2 unitary matrices of
unit determinant (this is the defining representation).
* Alternatively: Can
be represented by the unit quaternions |p|
= 1, acting on q 
H
by q 
pq;
Hamilton represented it by "turns'', equivalence classes of directed
great circle arcs on the unit sphere S2.
* Casimir invariant: S2 = Sx2 + Sy2 +
Sz2, with eigenvalues 
2j(j+1).
* Finite subgroups: The cyclic groups Zn, the binary dihedral
groups D4n*, the binary
tetrahedral group T*, the binary
octahedral group
O*, and the binary icosahedral group I*.
Generators
* And group parametrization: There are three Si, in terms of which
the group can be parametrized by
where S+/–:= 2–1/2 (S1 
i
S2), and N is a normalization
factor.
* Commutation relations: The Si are
chosen to satisfy [Si,
Sj] =
i 
ijk Sk
(or [Ji, Jj]
= i ijk Jk).
* Pauli matrices: The
basis 
i
= (i/2) 
i,
with i =
1, 2, 3, for the Lie algebra of SU(2) in the fundamental representation, or
for
the
set of quaternions; The matrices are given by
they satisfy the identity ij = 
ij +
i ijkk,
and their commutation relations are
[i,
j]
= ijk k .
@ References: Pittenger & Rubin PRA(00)qp [higher-dimensional generalization].
Representations and Other Related Topics > s.a. group
reps and lie
group reps; Elliptic
Space.
@ References: Zalka qp/04 [high-dimensional,
on quantum computer]; Kibler qp/05-in
[non-standard scheme].
> And physics: see classical
particles [on group manifold]; connection
formulation
of general relativity; coherent states.
6j-Symbol > s.a. angular
momentum; clebsch-gordan coefficients.
* Idea: A real number which can be associated to a labelling of the
six
edges of a tetrahedron by irreducible representations of SU(2).
* Racah formula: The formula for 6j symbols
@ References: Carter et al 95; Watson JPA(99) [asymptotics]; Roberts G&T(99)mp/98 [Ponzano-Regge and geometry]; Alisauskas mp/02 [SO(n)]; Freidel & Louapre CQG(03)ht/02 [asymptotics]; Coquereaux JGP(07)ht/05 [and Ocneanu cells]; Freidel et al JMP(07) [duality relations]; Kwee & Lebed JPA(08) [identity].
Other Similar Symbols
@ 9j: Jang JMP(68) [identities]; Rosengren JMP(99)
[triple sum formula]; Alisauskas JMP(00)m.QA/99 [SU(2)
and uq(2)
sum formulas].
@ 12j: Alisauskas JMP(02)
[SU(2) and uq(2) sum formulas];
> s.a. spin foam.
@ 3nj: Wei & Dalgarno mp/03 [factorization
+ calculation]; Lorente mp/04-in
[and Ponzano-Regge model].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
20 jun 2008