Groups (Subscript[SO, 3]) and (Subscript[SU, 2]) and their representations

Introduction

The rotations group in 3 dimensional space,  (Subscript[SO, 3]) , and its covering group (Subscript[SU, 2])are defined and their elements programmed with the commonly used parametrizations. Relations between parametrizations are defined and programmed. The Lie–algebras of these groups, (Subscript[so, 3]), respectively (Subscript[su, 2]), are defined and programmed. Finally, the unitary irreducible representations of (Subscript[SU, 2])are defined and programmed. 
The main definitions and properties are given in text preceding the inputs. For more details or proves, refer to the books:
"Irreducible Tensorial Sets" by U. Fano and G. Racah, 
ACADEMIC PRESS  INC., NEW YORK 1959.
"Angular Momentum in Quantum Mechanics" by A.E. Edmonds,
Princeton University Press, 1957.
The package SU2_SO3.m offers the same functionality as this Notebook, but with few text.

The groups and their Lie-algebras

Definitions

\(The\ group\ \(SU\_2\) 
      is\ the\ set\ af\ all\ unitary\ 2×2\ matrices\ of\ determinant\ 1. \ \)
Any element may be written

u (x)=(({{(Subscript[x, 0]) + i (Subscript[x, 3]), (Subscript[x, 2]) + i (Subscript[x, 1])}, {-(Subscript[x, 2]) + i (Subscript[x, 1]), (Subscript[x, 0]) - i (Subscript[x, 3])}}))

with a redondant parameter set x = ((Subscript[x, 0]), (Subscript[x, 1]), (Subscript[x, 2]), (Subscript[x, 3]))of real numbers satisfying 
the condition :

x · x  ≡  
        det (u (x))  = 1 .

\(The\ group\ SO\_3\ is\ the\ set\ of\ all\ orthogonal\ 3×3\ real\ matrices\ of\ determinant\ 1. \ \)
Any element may be written

R (p, q, r)=(({{(Subscript[p, 1]), (Subscript[p, 2]), (Subscript[p, 3])}, {(Subscript[q, 1]), (Subscript[q, 2]), (Subscript[q, 3])}, {(Subscript[r, 1]), (Subscript[r, 2]), (Subscript[r, 3])}}))

where the nine redondant real parameters have to satisfy the six conditions

p·p  =  q· q  =  r· r  =  
          1 ,     p· 
        q  =  q· r  =  r· p  =  
          0

plus the discrete one:

det (R)  ≡  p · q ⋀ r  =  
      1

\(The\ Lie\ algebra\ su\_2\ of\ \(SU\_2\) 
        is\ the\ vector\ space\ of\ 2×2\ traceless\ skew–hermitian\) 
matrices :  (g^†) = -g , Tr (g) = 
        0.  This set is closed for the Lie product :  if g,h∈ (Subscript[su, 2]), then [g, h] ≡ gh - hg  ∈  
          (Subscript[su, 2]) .  Every element of (Subscript[SU, 2]) may be written  as u = 
        (e^g) , g ∈  (Subscript[su, 2]).

\(The\ Lie\ algebra\ so\_3\ of\ SO\_3\ is\ the\ vector\ space\ of\ 3×3\ real\ skew–symmetric\ \(matrices : \ \(Ω\^~\)\) = \(-Ω\  . \ 

            This\)\ set\ is\ closed\ for\ the\ Lie\ \(product : \ 
          if\ Ω\), Ω'\  ∈ \ so\_3, \ 
    then\ [Ω, Ω'] ∈ \ 
        so\_3 . \ 
Every\ element\ of\ SO\_3\ may\ be\ written\ as\ R = 
      e\^Ω\ , \ Ω\  ∈ \ \(\(so\_3\)\(.\)\)\)

Parametrization of the Lie algebras

\(Both\ Lie\ algebras\ are\ 3  –\ dimensional\ vector\ spaces\ on\ ℝ . \ \ The\ standard\ basis\ of\ su\_2\ is\ \)
\(the\ set\ of\ generators\ γ\_k = \(i\/2\) σ\_k\ , \ 
    k = 1, 2, 
    3, \ \(\(proportional\)\(\ \)\(to\)\(\ \)\(the\)\(\ \)\(Pauli\)\(\ \)\(matrices\)\(\ \)\)\)

(Subscript[σ, 1]) = (({{0, 1}, {1, 0}}))     (Subscript[σ, 2]) = (({{0, -i}, {i, 0}}))     (Subscript[σ, 3]) = (({{1, 0}, {0, -1}}))

\(The\ generators\ fullfil\ the\ relations\ [γ\_1, γ\_2] = \(-γ\_3\)\ and\ circular\ permutations\ of\ the\ \(\(subscripts\)\(.\)\)\)

\(The\ real\ corresponding\ basis\ of\ so\_3\ is\ the\ set\ of\ generators\ I\_k\ , \ k = 1, 2, 3, \ 
    fullfiling\ the\ same\ commutation\ \(rules : \ \([I\_1, 
            I\_2]\)\) = \(-I\_3\)\ and\ circular\ \(\(permutations\)\(.\)\(\ \)\)\)

(Subscript[I, 1])=(({{0, 0, 0}, {0, 0, 1}, {0, -1, 0}}))     (Subscript[I, 2])=(({{0, 0, -1}, {0, 0, 0}, {1, 0, 0}}))     (Subscript[I, 3])=(({{0, 1, 0}, {-1, 0, 0}, {0, 0, 0}}))

A fundamental relation connects the two basis:

(Subscript[(((Subscript[I, k]))), mn]) = 
      Tr ((Subscript[σ, m]) (Subscript[γ, k]) (Subscript[σ, n]))

\(In\ summary, \ 
    any\ g ∈ \(\(su\_2\)\(\ \)\(may\)\(\ \)\(be\)\(\ \)\(written\)\(\ \)\)\)
\(g = \(∑\+\(i = 1\)\%3\( ω\_i\) γ\_i\ \ and\ \ any\ \ Ω\  ∈ \ 
          SO\_3\ \ correspondingly\ \ Ω = ∑\+\(i = 1\)\%3\( ω\_i\) \(\(I\_i\)\(\ \)\(.\)\)\)\)

Relation between (Subscript[SU, 2]) and (Subscript[SO, 3])

The group (Subscript[SU, 2]) is simply connected,  
    homeomorphic to the sphere (𝒮^3),  
    as is evident from 
\(the\ x–parametrization\ mentioned\ above . \ 
            It\ is\ the\ universal\ \ covering\ group\ of\ SO\_3 . \ 

            The\ group\ homomorphism\ \ SU\_2 ∋ u\  -> \ 
      R \((u)\) ∈ \(\(SO\_\(\(3\)\(\ \)\)\) \(is\)\(\ \)\(explicitely\)\(\ \)\(given\)\(\ \)\(by\)\(\ \)\)\)

((Subscript[R, ik])) = (1/2) (Tr (u^†) (Subscript[σ, i]) 
            u (Subscript[σ, k]))

\(which\ follows\ from\ the\ relations\ \(Trσ\_i\) σ\_k = δ\_ik\ 1, \ and\)

(u^†) (Subscript[σ, k]) 
        u = (∑_(i = 1)^3) 
              (Subscript[R, ki]) (Subscript[σ, i]) .

Parametrization of the groups

Parametrizations

\(The\ Taylor\ expansion\ defining\ the\ exponentials\ of\ generators\ can\ be\ summed\ up\ giving\ rise\ \)
to matrices whose elements are functions of the parameters .  
        These matrices are linear 
combinations of the identity matrix idM[
        2] and of elements of the triplet sigma of 
\(σ–matrices\ for\ SU\_2 . \ For\ \(SO\_3\) 
      the\ basis\ is\ the\ indentity\ idM[3], \ 
    the\ triplet\ of\ generators\ genR, \ 

    and\ a\ \ projector\ proj[
        ax]\ onto\ a\ rotation\ axis\ \(\(ax\)\(.\)\)\)

idM[n_] := IdentityMatrix[n]

sigma = {{{0, 1}, {1, 0}}, {{0, -I}, {I, 0}}, {{1, 
              0}, {0, -1}}};

genR = {{{0, 0, 0}, {0, 0, 1}, {0, -1, 0}}, {{0, 0, -1}, {0, 
              0, 0}, {1, 0, 0}}, {{0, 1, 0}, {-1, 0, 0}, {0, 0, 
              0}}};

proj[ax_] := 
      Table[ax[[i]] ax[[k]]/ax . ax, {i, 3}, {k, 3}]

Axial parametrization

\(The\ axial\ parametrization\ is\ given\ by\ a\ 3  –vector\ ω, \ whose\ direction\ gives\ the\ \)
rotation axis and its length  the rotation angle :  
          uAx[ω] = u (ω) = 
        (e^(ω·γ)),  
    rAx[ω] = R (ω) = (e^(ω·
                I)).

\(uAx[ω_] := 
      Module[{α, n}, α = 
          Sqrt[ω . ω]; 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ If[α == 0, idM[2], 
          n = ω/α\ ; 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Cos[α/2] idM[2] + 
            I*Sin[α/2]*n . sigma]\ ]\)

\(rAx[ω_] := 
      Module[{α, n}, α = 
          Sqrt[ω . ω]; 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ If[α == 0, idM[3], 
          n = ω/α\ ; 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Cos[α]\ idM[3] + \((1 - Cos[α])\)\ proj[
                n] + 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Sin[α] n . genR\ \ ]\ ]\)

\(To\ cover\ the\ group, 
    the\ vector\ ω\ must\ run\ through\ the\ bowl\ \ of\ radius\ π\ for\ SO\_3\ \)
and radius 2  π for (Subscript[SU, 2]).

Euler parametrization

The Euler parametrization is given by a triplet of angles ϵ=(ψ,θ,φ) of rotations: first a rotation of angle φ  around the coordinate axis 3, then a rotation of angle θ around the new axis 2, finally a rotation ψ around the new axis 3. (Mathematitians use sometimes the sequence 3-1-3). In this parametrization: u(ϵ)=uEu[ϵ], R(ϵ)=rEu[ϵ].

\(uEu[{ψ_, θ_, φ_}] := 
\ \ \ {{Exp[
              I*\((ψ + φ)\)/2] Cos[θ/2], 
          Exp[I*\((ψ - φ)\)/2] 
            Sin[θ/2]}, {\(-Exp[\(-I\)*\((ψ - φ)\)/2]\) 
            Sin[θ/2], 
          Exp[\(-I\)*\((ψ + φ)\)/2] 
            Cos[θ/2]}}\)

rEu[{ψ_, θ_, φ_}] := {{Cos[ψ], Sin[ψ], 
            0}, {-Sin[ψ], Cos[ψ], 0}, {0, 0, 
            1}} . {{Cos[θ], 0, -Sin[θ]}, {0, 1, 
            0}, {Sin[θ], 0, Cos[θ]}} . {{Cos[φ], 
            Sin[φ], 0}, {-Sin[φ], Cos[φ], 
            0}, {0, 0, 1}}

\(To\ cover\ the\ groups\ the\ angles\ must\ be\ in\ the\ ranges\ 0 ≤ θ ≤ π, \ 
    0 ≤ φ < 
      2  π, \ 
and\ \ 0 ≤ ψ < 
      2  π\ \ for\ \ SO\_3\ \ but\ \ 0 ≤ ψ < 
      4  π\ \ for\ \ \(\(SU\_2\)\(.\)\)\)

X - parametrization for (Subscript[SU, 2])

This parametrization has been evocated above. It simple and usefull sometimes: u(x)=uX[x_].

uX[x_] := 
      Module[{y}, 
        y = x/Sqrt[
              x . x]; 
                  {{y[[1]] + I*y[[4]], 
            y[[3]] + I*y[[2]]}, {-y[[3]] + I*y[[2]]}, 
          y[[1]] - I*y[[4]]}  ]

Cayley transform parametrization

It is simple and may be usefull for algebraic computations, b is any real 3–vector.

uCa[b_] := (idM[2] - I*b . sigma) . 
        Inverse[idM[2] + I*b . sigma]

rCa[b_] :=  (idM[3] - b . genR) . 
        Inverse[idM[3] + b . genR]

The b–parametrization covers only half of the groups.

Relations between parameters

\(These\ relations\ are\ easily\ obtained\ by\ equating\ elements\ of\ SU\_2\ differently\ parametrized . \ 

        They\ are\ valid\ for\ SO\_3\ and\ for\ the\ forthcomming\ \(\(representations\)\(.\)\)\)

Relation x⇔ ω

The function xOmega[ω] gives x versus ω, conversally omegaX[x] gives ω versus x.
(ω is any 3–vector, and x any 4–vector which is automatically normalized)

xOmega[ω_] := 
      Module[{lo, so}, lo = Sqrt[ω . ω]; 
        so = Sin[lo/2]/lo; 
        {Cos[lo/2], 
          so*ω[[1]], so*ω[[2]], 
          so*ω[[3]]} ]

omegaX[x_] := 
      Module[{lx, lo, no}, 
        lx = Sqrt[x . x]; 
         lo = 
          2*ArcCos[x[[1]]/lx]; no = lo/(lx*Sin[lo/2]);  
        no*{x[[2]], x[[3]], x[[4]]} ]

Relation x⇔ϵ

The function xEuler[ϵ] gives x versus ϵ=(ψ,θ,φ), conversally eulerX[x] gives ϵ versus x  (any 4–vector ,the program adjusts its length to 1).

xEuler[ϵ_] := 
      Module[{a, b, c}, a = ϵ[[1]]/2; b = ϵ[[2]]/2; 
        c = ϵ[[3]]/2;  {Cos[b] Cos[a + c], Sin[b] Sin[a - c], 
          Sin[b] Cos[a - c], Cos[b] Sin[a + c]} ]

eulerX[x_] := 
      Module[{y, p, t, f}, 
        y = x/Sqrt[
              x . x]; 
                           p = 
          Arg[y[[1]] + 
              I*y[[4]]]; 
                         t = 
          Sqrt[y[[1]]^2 + 
              y[[4]]^2]; 
                         f = 
          Arg[y[[3]] + 
              I*y[[2]]]; 
                        {p + f, 2*ArcCos[t], p - f}  ]

Relation ϵ⇔ω

The function omegaEu[ϵ] gives ω versus ϵ={ψ,θ,φ}, and eulerOm[ω] gives ϵ versus ω.

omegaEu[ϵ_] := omegaX[xEuler[ϵ]]

eulerOm[ω_] := eulerX[xOmega[ω]]

Unitary irreducible representations of SU2

\(A\ unitary\ representation\ of\ SU\_2\ is\ given\ by\ a\ map\ SU\_2 ∋ u\  -> \ U \((u)\)\ \ into\ a\ group\ of\ \)
\(unitary\ matrices, \ 
    which\ respects\ the\ product\ \(law : \ 
          U \((u)\) U \((u')\)\) = \(U \((uu')\) . \ 

            The\ representation\ is\ said\ irreductible\ if\ the\ relation\ [
            U \((u)\), M] = 0\ \(∀ \ u\)\)\ \ \)
\(implies\ that\ M\ must\ be\ a\ multiple\ of\ the\ identity . 
\ 
            This\ map\ induces\ a\ linear\ map\ su\_2 ∋ \ 
        g\  -> G \((g)\)\ \ between\ generators\ \ \)
\(g\ of\ u\ and\ G\ of\ U \((u)\), \ 
    which\ respects\ the\ Lie\ \(product : \ \([g, g']\)\) -> \ \([G, 
            G']\) . \ \ 

          This\ defines\ a\ representation\ of\ \(\(su\_2\)\(.\)\(\ \)\)\)

\(Up\ to\ an\ equivalence\ exists\ a\ single\ representation\ of\ SU\_2\ in\ each\ dimension\ d = 2  j + 1, \ j = 0, 1/2, 1, 3/2, 2,  ... , 
    infinity . \ 
          Physicists\ use\ to\ write\ the\ basis\ generators\ G\_k\%\((j)\) = 
      iJ\_k\%\((j)\)\ \)
\(because\ J\_k\%\((j)\)\ is\ hermitian\ and\ is\ interpreted\ as\ a\ component\ of\ the\ angular\ momentum\ \(\(operator\)\(.\)\)\)

Irreducible representations of (Subscript[su, 2])

\(The\ standard\ form\ of\ the\ basis\ elements\ in\ the\ j–representation\ \)
is given by the triplet mJ[j] :  
        (Subscript[J, k]^((j))) = mJ[j, [[k]]].

mJ[j_] := 
      Module[{j3, jp}, 
           j3 = 
          DiagonalMatrix[Table[j - k + 1, {k, 2*j + 1}]]; 
        jp = Table[
            Table[If[k2 == k1 + 1, Sqrt[k1 (2*j - k1 + 1)], 
                0], {k2, 2*j + 1}], {k1, 
              2*j + 1}]; 
{(jp + Transpose[jp])/
            2, (jp - Transpose[jp])/(2*I), j3} ]

Unitary irreducible representations of (Subscript[SU, 2])

\(In\ a\ j–representation\ the\ image\ of\ u \((ω)\) = \(e\^\(ω·γ\)\ is\ U \((ω)\) = 
        e\^\(iω·J\)\ defined\ \(\(by\)\(:\)\)\)\)

mU[j_, ω_] := MatrixExp[I*ω . mJ[j]]

\(But\ the\ computation\ is\ very\ time\ consuming\ as\ soon\ as\ j\ is\ greater\ as\ a\ few\ units . \ The\ best\ \)
\(way\ is\ to\ work\ in\ Euler\ parametrization, \ \(\(remembering\)\(\ \)\(that\)\(\ \)\(the\)\(\ \)\(relations\)\(\ \)\)\)
\(x ↔ ω ↔ \(\(ϵ\)\(\ \ \)\(given\)\(\ \)\(above\)\(\ \)\(are\)\(\ \)\(valid\)\(\ \)\(for\)\(\ \)\(any\)\(\ \)\(representation . \ 
          To\)\(\ \)\(u\) \((ψ, θ, φ)\)\(\ \)\(corresponds\)\(\ \)\(a\)\(\ \)\)\)
\(standard\ matrix\ \(D\^\((j)\)\) \((ψ, θ, φ)\)\ given\ by\ a\ product\ whose\ factors\ are\ analytical\ expressions\ \)
\(corresponding\ to\ rotations\ around\ axis\ 3, 2, 
    3\ \((mathematicians\ sometimes\ use\ 3, 1, 3)\) . \ 

        The\ D\^\((j)\)\ matrices\ are\ given\ by\ mD[
        j, {ψ, θ, φ}]\ in\ the\ integrated\ version, \ 
or\ by\ mDprod[]\ \ when\ expressed\ as\ a\ product\ of\ three\ \(\(matrices\)\(.\)\)\)

mDprod[j_, {ψ_, θ_, φ_}] := 
      d3[j, ψ] . d2[j, θ] . d3[j, φ]

d3[j_, a_] := 
      DiagonalMatrix[Table[Exp[I*(j - k)*a], {k, 0, 2*j}]]

d2[j_, b_] := 
      Module[{tb}, 
tb = 
          Table[Table[
              Sqrt[Binomial[2*j, k - 1]/Binomial[2*j, m - 1]] 
                Cos[b/2]^(2*j + 2 - k - m) Sin[b/2]^(k - m) 
                JacobiP[m - 1, k - m, 2*j + 2 - k - m, Cos[b] ], {k, m, 
                2*j + 2 - m}], {m, Floor[j + 1]}]; 
Table[
          Table[
Which[
              1 ≤ q ≤ Floor[j + 1]  &&  
                q ≤ p ≤ 
                  2*j + 2 - 
                    q, 
             tb[[q]][[p - q + 
                    1]], 
          2 ≤ q ≤ 2*j  &&  
                1 ≤ p < 
                  j + 1 - 
                    Abs[j + 1 - 
                        q], 
           tb[[p]][[q - p + 1]]*(-1)^(q - 
                      p), 
          2 ≤ q ≤ 2*j  &&  
                j + 1 + Abs[j - q + 1] < p ≤ 
                  2*j + 1, 
           tb[[2*
                        j + 2 - p]][[p - q + 
                    1]], 
          j + 1 < 
                  q ≤ 2*j + 1  &&  
                2*j + 2 - q ≤ p ≤ 
                  q, 
           tb[[2*j + 
                        2 - q]][[q - p + 1]]*(-1)^(q - 
                      p) 
          ], {p, 
              2*j + 1}], {q, 
            2*j + 1}]
                                        ]

To obtain the "mathematician" sequence 3–1–3 of rotations, one just need to replace d2[j,b] by d1[j,b]=d3[j,π/2]d2[j,b]d3[j,-π/2]≡mD[π/2,b,-π/2]:

d1[j_, b_] := mD[j, π/2, b, -π/2]

For large angular momenta j an integrated form of mD may be less time consuming by avoiding the product of three large matrices.

mD[j_, {ψ_, θ_, φ_}] := 
      Module[{tb}, 
tb = 
          Table[Table[
              Sqrt[Binomial[2*j, k - 1]/Binomial[2*j, m - 1]] 
                Cos[θ/2]^(2*j + 2 - k - m) 
                Sin[θ/2]^(k - m) 
                JacobiP[m - 1, k - m, 2*j + 2 - k - m, Cos[θ] ], {k, 
                m, 2*j + 2 - m}], {m, 
              Floor[j + 1]}]; 
Table[
          Table[Exp[I*(j - q + 1) ψ] 
              Exp[I*(j - p + 1) φ]
Which[
                1 ≤ q ≤ Floor[j + 1]  &&  
                  q ≤ p ≤ 
                    2*j + 2 - 
                      q, 
             tb[[q]][[p - q + 
                      1]], 
          2 ≤ q ≤ 2*j  &&  
                  1 ≤ p < 
                    j + 1 - 
                      Abs[j + 1 - 
                          q], 
           tb[[p]][[q - p + 1]]*(-1)^(q - 
                        p), 
          2 ≤ q ≤ 2*j  &&  
                  j + 1 + Abs[j - q + 1] < p ≤ 
                    2*j + 1, 
           tb[[2*j + 2 - p]][[p - q + 
                      1]], 
          j + 1 < 
                    q ≤ 2*j + 1  &&  
                  2*j + 2 - q ≤ p ≤ 
                    q, 
           tb[[2*j + 
                          2 - q]][[q - p + 1]]*(-1)^(q - 
                        p) 
          ], {p, 
              2*j + 1}], {q, 
            2*j + 1}]
                                        ]