^{− 1}ġg

^{− 1}ġ〉

^{− 1}ġ = 0

_{1}g

_{2}

_{1}, h

_{2}∈ SU(2)

^{− 1}ġ

^{− 1}

_{1}=

_{2}=

_{3}=

^{n}T

_{n}n = 1, 2, 3

^{n}= 〈AT

_{n}〉 (〈T

_{n}T

_{m}〉 = δ

_{nm})

_{n}= 〈T

_{n}C〉 n = 1, 2, 3 C = C

^{n}T

_{n}

_{n}= 〈T

_{n}S〉 n = 1, 2, 3 S = S

^{n}T

_{n}

^{− 1}ġ = 0 ⇒ g

^{− 1}ġ = const

^{Ct}g(0)

^{− 1}ġ − v)〉 + ½〈v

^{2}〉

^{− 1}ġ = v

^{− 1}ġ〉 − ½ 〈C

^{2}〉

^{2}〉

^{− 1}dg ∧ dC〉 − 〈Cg

^{− 1}dg ∧ g

^{− 1}dg〉

_{X}ω

_{f}by

_{Xf}ω = − df

_{X}ω denotes the contraction of $X$ with $\omega $. According to its definition Poisson bracket of two functions is

_{Xf}g = i

_{Xf}dg = ω(X

_{f}, X

_{g})

_{Xf}g denotes Lie derivative of $g$ with respect to vector filed $X$

_{f}. The skew symmetry of $\omega $ provides skew symmetry of Poisson bracket.

_{n}, S

_{m}and $g$ functions are

_{n}= X

_{Cn}= ([C ,T

_{n}] , gT

_{n})

_{m}= X

_{Sm}= ([C , gT

_{m}g

^{− 1}] , T

_{m}g )

_{n}, S

_{m}} = − 2ε

_{nm}

^{k}S

_{k}

_{n}, C

_{m}} = 2ε

_{nm}

^{k}C

_{k}

_{n}, S

_{m}} = 0

_{n}, g} = gT

_{n}

_{m}, g} = T

_{m}g

_{n}=

_{Xn}

_{m}= −

_{Ym}

_{n}, Ŝ

_{m}] = iε

_{nm}

^{k}Ŝ

_{k}

_{n}, Ĉ

_{m}] = iε

_{nm}

^{k}Ĉ

_{k}

_{n}, Ŝ

_{m}] = 0

^{2}= Ŝ

^{2}

_{a}, Ŝ

_{b}

_{a}and $\u015c$

_{b}have the form

_{jsc}= j(j + 1)ψ

_{jsc}

_{a}ψ

_{jsc}= cψ

_{jsc}

_{b}ψ

_{jsc}= sψ

_{jsc}

_{jsc}. The first step of this construction is to note that the function $\langle Tg\rangle $ where $T\; =\; (1\; +\; iT$

_{a})(1 + iT

_{b}) is an eigenfunction of $\u0124,\; \u0108$

_{a}and $\u015c$

_{b}with eigenvalues $\xbe,\; \xbd,\; \xbd$ respectively. Proof of this proposition is straightforward. Using $\langle Tg\rangle $ one can construct the complete set of eigenfunctions of $\u0124,\; \u0108$

_{a}and $\u015c$

_{b}operators

_{jsc}= Ŝ

_{−}

^{j − s}Ĉ

_{−}

^{j − c}〈Tg〉

^{2j}

^{β(t)T3}

_{3}is antihermitian $h(t)\; \in \; U(1)$ and since $h(t)$ depends on $t$ Lagrangian

^{− 1}ġg

^{− 1}ġ〉

_{3}∈ su(2)

^{− 1}−

^{− 1}

_{G}= 〈g

^{− 1}∇gg

^{− 1}∇g〉

_{G}

^{− 1}T

_{3}〉

_{G}= 〈(g

^{− 1}ġ − S

_{3}T

_{3})

^{2}〉

^{− 1}T

_{3}g ∈ su(2)

^{a}T

_{a}

_{a}= 〈ZT

_{a}〉

^{2}〉 = 〈g

^{− 1}T

_{3}gg

^{− 1}T

_{3}g〉 = 〈T

_{3}

^{2}〉 = 1

^{2}〉 = 〈z

^{a}T

_{a}z

^{b}T

_{b}〉 = z

^{a}z

_{a}

_{G}takes the form

_{G}= ¼〈Z

^{− 1}ŻZ

^{− 1}Ż〉

_{a}it's easy to show that

_{G}= ¼〈Z

^{− 1}ŻZ

^{− 1}Ż〉 = ¼〈ZŻZŻ〉 = ½ż

^{a}ż

_{a}

_{G}= 〈C(g

^{− 1}ġ − u)〉 + ½ 〈(u + g

^{− 1}Bg)

^{2}〉

^{− 1}Bg

^{− 1}Bg

_{G}in terms of $C$ and $g$ leads to

_{G}= 〈Cg

^{− 1}ġ〉 − ½ 〈C

^{2}〉 − 〈BgCg

^{− 1}〉 = 〈Cg

^{− 1}ġ〉

^{2}〉 − b〈gCg

^{− 1}T

_{3}〉 = 〈Cg

^{− 1}ġ〉 − ½ 〈C

^{2}〉 − bS

_{3}

_{G}we obtain constrained Hamiltonian system, where $\langle Cg-\; 1dg\rangle $ is symplectic potential, function

^{2}〉

^{− 1}T

_{3}〉 = 〈ST

_{3}〉 = S

_{3}= 0

_{3}|ψ〉 = 0

_{jcs}j = 0,

_{jc0}j = 0, 1, 2, 3, ...

_{3}ψ

_{jcs}= 0 implies $s\; =\; 0$, and if $s\; =\; 0$ then $j$ is integer. Thus $c$ takes $-\; j,\; -\; j\; +\; 1,\; ...,\; j\; -\; 1,\; j$ integer values only. Wave functions $\psi $

_{jcs}rewriten in terms of gauge invariant variables up to a constant multiple should coincide with well known spherical harmonics

_{jc0}∼ J

_{jc}

_{jc0}∼ Ŝ

_{−}

^{j}Ĉ

_{−}

^{j − c}〈Tg〉

^{2j}∼ Ĉ

_{−}

^{j − c}〈T

_{+}g

^{− 1}T

_{3}g〉

^{j}

_{−}

^{j − c}sin

^{j}θe

^{ijθ}∼ Ĉ

_{−}

^{j − c}J

_{jj}∼J

_{jc}

_{1}|ψ

_{2}〉 =

^{− 1}dgT

_{a}〉(ψ

_{1})

^{†}ψ

_{2}

_{n}and $\u015c$

_{m}are hermitian. Indeed

_{1}|Ĉ

_{n}ψ

_{2}〉 =

^{− 1}dgT

_{a}〉(ψ

_{1})

^{†}(

_{Xn}ψ

_{2})

^{− 1}dgT

_{a}〉(

_{Xn}ψ

_{1})

^{†}ψ

_{2}

^{− 1}dgT

_{a}〉

_{Xn}〈g

^{− 1}dgT

_{a}〉 = 0

^{qaTa}

^{− 1}dg〉 = C

_{a}dq

^{a}

_{1}|ψ

_{2}〉 =

^{3}q(ψ

_{1})

^{†}ψ

_{2}

_{a}= 〈g

^{− 1}dg T

_{a}〉

_{3}and $\u0108$

_{3}as a complete set of observables. Assuming that operators $\u0124,\; \u015c$

_{3}and $\u0108$

_{3}have at least one common eigenfunction

_{3}ψ = cψ

_{3}ψ = sψ

^{2}≥ 0

^{2}≥ 0

^{2}|ψ〉 = 〈ψ|Ĉ

_{a}Ĉ

^{a}|ψ〉 = 〈ψ|(Ĉ

_{a})

^{†}Ĉ

^{a}|ψ〉 =

_{a}ψ|Ĉ

^{a}ψ〉 = ∥Ĉ

_{a}ψ∥ ≥ 0

_{1}

^{2}+ Ĉ

_{2}

^{2}and $\u015c$

_{1}

^{2}+ Ŝ

_{2}

^{2}operators

_{1}

^{2}+ Ĉ

_{2}

^{2}|ψ〉 = ∥Ĉ

_{1}ψ∥ + ∥Ĉ

_{2}ψ∥ ≥ 0

_{1}

^{2}+ Ĉ

_{2}

^{2}|ψ〉 = 〈ψ|Ĥ − Ĉ

_{3}

^{2}|ψ〉 = (E − c

^{2})〈ψ|ψ〉

_{+}= iĈ

_{1}+ Ĉ

_{2}Ĉ

_{−}= iĈ

_{1}− Ĉ

_{2}

_{+}= iŜ

_{1}+ Ŝ

_{2}Ŝ

_{−}= iŜ

_{1}− Ŝ

_{2}

_{−})

^{†}= Ĉ

_{+}and $(\u015c$

_{−})

^{†}= Ŝ

_{+}and they fulfill the following commutation relations

_{±}, Ĉ

_{3}] = ± Ĉ

_{±}[Ŝ

_{±}, Ŝ

_{3}] = ± Ŝ

_{±}

_{+}, Ĉ

_{−}] = 2Ĉ

_{3}[Ŝ

_{+}, Ŝ

_{−}] = 2Ŝ

_{3}

_{•}, Ŝ

_{•}] = 0

_{λcs}is eigenfunction of $\u0124,\; \u015c$

_{3}and $\u0108$

_{3}with corresponding eigenvalues :

_{λcs}= λψ

_{λcs}

_{3}ψ

_{λcs}= sψ

_{λcs}

_{3}ψ

_{λcs}= cψ

_{λcs}

_{±}ψ

_{λcs}and $\u015c$

_{±}ψ

_{λcs}are the eigenfunctions with corresponding eigenvalues $\lambda ,\; s\; \pm \; 1,\; c$ and $\lambda \; ,\; s,\; c\; \pm \; 1$. Consequently using $\u0108$

_{±}, Ŝ

_{±}operators one can construct a family of eigenfunctions with eigenvalues

^{2}≥ 0

^{2}≥ 0

_{+}ψ

_{λcj}= 0 Ŝ

_{−}ψ

_{λc, − j}= 0

_{+}ψ

_{λks}= 0 Ĉ

_{−}ψ

_{λ, − ks}= 0

_{±}, Ĉ

_{3}operators

_{+}Ĉ

_{−}+ Ĉ

_{3}

^{2}+ Ĉ

_{3}