The dynamics of a free particle on SU(2) group manifold is described by the Lagrangian L = 〈g^{− 1}ġg^{− 1}ġ〉 where g ∈ SU(2) and 〈 〉 denotes the normalized trace 〈 · 〉 = − ½Tr( · ) which defines a scalar product in su(2) algebra. This Lagrangian gives rise to equations of motion ddtg^{− 1}ġ = 0 that describe dynamics of particle on group manifold. Also, one can notice that it has SU(2) "right" and SU(2) "left" symmetry. It means that it is invariant under the following transformations g       →      h1g g       →    gh2 where h1, h2 ∈ SU(2)

According to the Noether's theorem these symmetries lead to the matrix valued conserved quantities C = g^{− 1}ġ           ddtC = 0 and S = ġg^{− 1}           ddtS = 0 To construct integrals of motion out of C and S let us introduce the basis of su(2) algebra — three matrices: T1 = i 0 0 − i       T2 = 0 − 1 1 0       T3 = 0 i i 0 The elements of su(2) are traceless anti-hermitian matrices, and any A ∈ su(2) can be parameterized in the following way A = AⁿTn            n = 1, 2, 3 Scalar product AB = 〈AB〉 = − ½Tr(AB) ensures that Aⁿ = 〈ATn〉            (〈TnTm〉 = δnm) Now we can introduce six functions Cn = 〈TnC〉          n = 1, 2, 3           C = CⁿTn Sn = 〈TnS〉           n = 1, 2, 3            S = SⁿTn which are integrals of motion.

Conservation of C and S leads to general solution of Euler-Lagrange equations ddtg^{− 1}ġ = 0           ⇒           g^{− 1}ġ = const g = e^Ctg(0) These are well known geodesics on Lie group.

Working in a first order Hamiltonian formalism we can construct new Lagrangian which is equivalent to the initial one Λ = 〈C(g^{− 1}ġ − v)〉 + ½〈v²〉 in sense that variation of C provides g^{− 1}ġ = v and Λ reduces to L. Variation of v gives C = v and therefore we can rewrite equivalent Lagrangian Λ in terms of C and g variables Λ = 〈Cg^{− 1}ġ〉 − ½ 〈C²〉 where function H = ½〈C²〉 plays the role of Hamiltonian and one-form 〈Cg^{− 1}dg〉 is a symplectic potential θ. External differential of θ is the symplectic form ω = dθ = − 〈g^{− 1}dg ∧ dC〉 − 〈Cg^{− 1} dg ∧ g^{− 1}dg〉 that determines Poisson brackets, the form of Hamilton's equation and provides isomorphism between vector fields and one-forms X      →      iXω For any smooth SU(2) valued smooth function f ∈ SU(2) one can define Hamiltonian vector field Xf by iXfω = − df where iXω denotes the contraction of X with ω. According to its definition Poisson bracket of two functions is {f , g} = LXfg = iXfdg = ω(Xf , Xg) where LXfg denotes Lie derivative of g with respect to vector filed Xf. The skew symmetry of ω provides skew symmetry of Poisson bracket.

Hamiltonian vector fields that correspond to Cn, Sm and g functions are Xn = XCn = ([C ,Tn] , gTn) Ym = XSm = ([C , gTmg^{− 1}] , Tmg ) and give rise to the following commutation relations {Sn , Sm} = − 2εnm^k Sk {Cn , Cm} = 2εnm^k Ck {Cn , Sm} = 0 {Cn , g} = gTn {Sm , g} = Tmg The results are natural. C and S that correspond respectively to the "right" and "left" symmetry commute with each other and independently form su(2) algebras. Now knowing Poisson bracket structure one can write down Hamilton's equations ġ = {H , g} = gR Ċ = {H , C} = 0

Let's introduce operators Ĉn = i2LXn Ŝm = − i2LYm They act on the square integrable functions (see Appendix A) on SU(2) and satisfy quantum commutation relations [Ŝn , Ŝm] = iεnm^k Ŝk [Ĉn , Ĉm] = iεnm^k Ĉk [Ĉn , Ŝm] = 0 The Hamiltonian is defined as Ĥ = Ĉ² = Ŝ² and the complete set of observables that commute with each other is Ĥ,           Ĉa,            Ŝb with some fixed a and b. Using a simple generalization of a well known algebraic construction (see Appendix B) one can check that the eigenvalues of the quantum observables Ĥ, Ĉa and Ŝb have the form Ĥψjsc = j(j + 1)ψjsc where j takes positive integer and half integer values j = 0, 12, 1, 32, 2 ... Ĉaψjsc = cψjsc Ŝbψjsc = sψjsc with c and s taking values in the following range − j, − j + 1, ... , j − 1, j Further we construct the corresponding eigenfunctions ψjsc. The first step of this construction is to note that the function 〈Tg〉 where T = (1 + iTa)(1 + iTb) is an eigenfunction of Ĥ, Ĉa and Ŝb with eigenvalues ¾, ½, ½ respectively. Proof of this proposition is straightforward. Using 〈Tg〉 one can construct the complete set of eigenfunctions of Ĥ, Ĉa and Ŝb operators ψjsc = Ŝ−^{j − s}Ĉ−^{j − c}〈Tg〉^2j in the manner described in Appendix B.

Free particle on 2D sphere can be obtained from our model by gauging U(1) symmetry. In other words let's consider the following local gauge transformations g      →      h(t)g Where h(t) ∈ U(1) ⊂ SU(2) is an element of U(1). Without loss of generality we can take h = e^β(t)T3 Since T3 is antihermitian h(t) ∈ U(1) and since h(t) depends on t Lagrangian L = 〈g^{− 1}ġg^{− 1}ġ〉 is not invariant under (38) local gauge transformations.

To make (40) gauge invariant we should replace time derivative with covariant derivative ddtg     →    ∇g = ddt + Bg where B can be represented as follows B = bT3 ∈ su(2) with transformation rule B      →     hBh^{− 1} − dhdth^{− 1} or in terms of b variable b    →     b − dβdt The new Lagrangian LG = 〈g^{− 1}∇gg^{− 1}∇g〉 is invariant under (38) local gauge transformations. But this Lagrangian as well as every gauge invariant Lagrangian is singular. It contains additional non-physical degrees of freedom. To eliminate them we should eliminate B using Lagrange equations ∂LG∂B      →      b = − 〈ġg^{− 1}T3〉 put it back in (45) and rewrite last obtained Lagrangian in terms of gauge invariant variables. LG = 〈(g^{− 1}ġ − S3T3)²〉 It's obvious that the following Z = g^{− 1}T3g ∈ su(2) element of su(2) algebra is gauge invariant. Since Z ∈ su(2) it can be parameterized as follows Z = z^aTa where z^a are real functions on SU(2) za = 〈ZTa〉

So we have three gauge invariant variables z^a (a = 1, 2, 3) but it's easy to check that only two of them are independent. Indeed 〈Z²〉 = 〈g^{− 1}T3gg^{− 1}T3g〉 = 〈T3²〉 = 1 otherwise 〈Z²〉 = 〈z^aTaz^bTb〉 = z^aza

So configuration space of SU(2)/U(1) coset model is sphere. By direct calculations one can check that after being rewritten in terms of gauge invariant variables LG takes the form LG = ¼〈Z^{− 1}ŻZ^{− 1}Ż〉 This Lagrangian describes free particle on the sphere. Indeed, since Z = z^aTa it's easy to show that LG = ¼〈Z^{− 1}ŻZ^{− 1}Ż〉 = ¼〈ZŻZŻ〉 = ½ż^aża So SU(2)/U(1) coset model describes free particle on S² manifold.

Working in a first order Hamiltonian formalism one can introduce equivalent Lagrangian ΛG = 〈C(g^{− 1}ġ − u)〉 + ½ 〈(u + g^{− 1}Bg)²〉 variation of u provides C = u + g^{− 1}Bg u = C − g^{− 1}Bg Rewriting ΛG in terms of C and g leads to ΛG = 〈Cg^{− 1}ġ〉 − ½ 〈C²〉 − 〈BgCg^{− 1}〉 = 〈Cg^{− 1}ġ〉 − ½ 〈C²〉 − b〈gCg^{− 1}T3〉 = 〈Cg^{− 1}ġ〉 − ½ 〈C²〉 − bS3 Due to the gauge invariance of ΛG we obtain constrained Hamiltonian system, where 〈Cg^{− 1}dg〉 is symplectic potential, function H = ½〈C²〉 plays the role of Hamiltonian and b is a Lagrange multiple leading to the first class constrain φ = 〈gCg^{− 1}T3〉 = 〈ST3〉 = S3 = 0 So coset model is equivalent to the initial one with (59) constrain. Using technique of the constrained quantization, instead of quantizing coset model we can subject quantum model that corresponds to the free particle on SU(2), to the following operator constrain Ŝ3|ψ〉 = 0 Hilbert space of the initial system, that is linear span of ψjcs           j = 0, 12, 1, 32, 2, ... wave functions, reduces to the linear span of ψjc0            j = 0, 1, 2, 3, ... wave functions. Indeed, Ŝ3ψjcs = 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 ψjcs rewriten in terms of gauge invariant variables up to a constant multiple should coincide with well known spherical harmonics ψjc0 ∼ Jjc One can check the following ψjc0 ∼ Ŝ−^jĈ−^{j − c} 〈Tg〉^2j ∼ Ĉ−^{j − c} 〈T+g^{− 1}T3g〉^j ∼ Ĉ−^{j − c}sin^jθe^ijθ ∼ Ĉ−^{j − c}Jjj ∼Jjc This is an example of using large initial model in quantization of coset model.