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: It can be represented
by the unit quaternions |p| = 1, acting on q ∈ \(\mathbb H\)
by q \(\mapsto\) 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 \(\hbar\)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*.
@ References: Mukunda et al Pra(10)-a0904 [Hamilton's theory of turns].
Generators
* And group parametrization:
There are three generators Si,
in terms of which the group can be parametrized in one of the following ways:
where S±:=
2−1/2 (\(S_1 \pm
\ii S_2\)), and N is a normalization factor.
* Commutation relations:
The Si are chosen
to satisfy [Si,
Sj] =
i εijk
Sk
(or [Ji, Jj]
= i\(\hbar\) εijk
Jk).
* Pauli matrices:
The basis σi
= (i/2) τi,
with i = 1, 2, 3, for the Lie algebra of SU(2) in the fundamental
representation, or for the set of quaternions, where
They satisfy the identity σi σj = δij + i εijkσk, and their commutation relations are
[σi, σj] = εijk σk .
@ References: Pittenger & Rubin PRA(00)qp [higher-dimensional generalization]; Curtright et al Sigma(14)-a1402 [group elements as finite polynomials of Lie algebra generators].Representations and Other Related Topics > see regge calculus.
@ References:
Zalka qp/04 [high-dimensional, on quantum computer];
Kibler qp/05-conf [non-standard scheme];
Shurtleff a1001 [and spacetime rotations, boosts and translations];
Akhtarshenas a1003 [invariant vector fields and one-forms].
> Related topics: see Elliptic Space;
group representations and lie-group representations;
types of spinors [decompositions of products].
> And physics: see 2-spinors [compositions];
classical particles [on group manifold]; connection
formulation of general relativity; coherent states.
6j-Symbol > s.a. angular momentum;
clebsch-gordan coefficients; Racah Coefficients / Formula.
* 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
\[
\Delta(a,b,c) = \sqrt{{(a+b-c)!\,(a-b+c)!\,(-a+b+c)!\over(a+b+c+1)}^{\vphantom2}},\qquad
w\left\{\matrix{j_{_1} & j_{_2} & j_{_{12}}\cr j_{_3} & j & j_{_{23}}}\right\} =\ ...
\]
@ Asymptotics: Watson JPA(99);
Freidel & Louapre CQG(03)ht/02;
Gurău AHP(08)-a0808 [Ponzano-Regge asymptotic formula];
Littlejohn & Yu JPCA(09)-a0905 [uniform asymptotic approximation];
Dupuis & Livine CQG(10)-a0910;
Kamiński & Steinhaus JMP(13)-a1307 [using a coherent state approach];
Bartlett & Ranaivomanana a2012 [and Wigner derivative for spherical tetrahedra].
@ Related topics: Carter et al 95;
Roberts G&T(99)mp/98 [Ponzano-Regge and geometry];
Alisauskas JPA(02)mp [SO(n)];
Coquereaux JGP(07)ht/05 [and Ocneanu cells];
Freidel et al JMP(07) [duality relations];
Kwee & Lebed JPA(08) [identity];
Aquilanti et al JPA(12)-a1009 [semiclassical mechanics];
Bonzom & Livine AHP(11)-a1103 [recursion relations].
Other Similar Symbols > s.a. 3j 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];
Haggard & Littlejohn CQG(10)-a0912 [asymptotic formula];
Littlejohn & Yu PRA(11)-a1104 [limit of one small and eight large angular momenta].
@ 12j: Alisauskas JMP(02)
[SU(2) and uq(2) sum formulas];
Yu PRA(11)-a1104,
a1108 [some asymptotic limits];
> s.a. spin-foam models.
@ 15j:
Barrett et al IJMPA(10) [asymptotics].
@ 3nj: Wei & Dalgarno JPA(04)mp/03 [factorization + calculation];
Lorente mp/04-conf [and Ponzano-Regge model];
Bonzom & Fleury JPA(12)-a1108 [asymptotics];
Delfino a1110 [identities];
Speziale JMP(17)-a1609 [and boosts].
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