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].

* And group parametrization: There are three generators Si, in terms of which the group can be parametrized in one of the following ways:

\[ \def\ee{{\rm e}}\def\ii{{\rm i}} U(\theta,\phi,\psi) = \ee^{-\ii\phi S_3}\,\ee^{-\ii\theta S_2}\,\ee^{-\ii\psi S_3}\,,\quad U(\alpha_{_1},\alpha_{_2},\alpha_{_3}) = \ee^{-\ii{\bf\alpha}\,\cdot\,{\bf S}}\,\quad U(z,\psi) = N\,\ee^{zS_-}\,\ee^{-z^*S_+}\,\ee^{-\ii\psi S_3}\,. \] \[{\rm or}\quad U(\alpha,\beta) = \left(\matrix{\alpha&\beta\cr-\beta^*&\alpha^*}\right),\quad {\rm with}\quad \alpha = \cos{\mu\over2}\,\ee^{-\ii\nu_1},\ \beta = \sin{\mu\over2}\,\ee^{-\ii\nu_2} \ \ (\mu,\nu_i\in{\mathbb R})\;. \]

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

\[ \tau_{_1} = \left(\matrix{0 & 1\cr1 & 0}\right),\qquad \tau_{_2} = \left(\matrix{0 & i\cr-i & 0}\right),\qquad \tau_{_3} = \left(\matrix{1 & 0\cr0 & -1}\right); \]

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

\[ \left\{\matrix{j_{_1} & j_{_2} & j_{_{12}}\cr j_{_3} & j & j_{_{23}}}\right\} = \Delta( j_{_1}, j_{_2}, j_{_{12}})\, \Delta( j_{_1},j, j_{_{23}})\, \Delta( j_{_3}, j_{_2}, j_{_{23}})\, \Delta( j_{_3},j,j_{_{12}})\;w\left\{\matrix{j_{_1} & j_{_2} & j_{_{12}}\cr j_{_3} & j & j_{_{23}}}\right\}, \]

\[ \Delta(a,b,c) = \sqrt{{(a+b-c)!\,(a-b+c)!\,(-a+b+c)!\over(a+b+c+1)}^{\vphantom2}},\qquad
w\left\{\matrix{j_{_1} & j_{_2} & j_{_{12}}\cr j_{_3} & j & j_{_{23}}}\right\} =\ ... \]

@ 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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