Free particle on SU(2) group manifold

The dynamics of a free particle on SU(2) group manifold is described by the Lagrangian where g ∈ SU(2) and 〈 〉 denotes the normalized trace which defines a scalar product in su(2) algebra. This Lagrangian gives rise to equations of motion 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 where h₁, h₂ ∈ SU(2)

According to the Noether's theorem these symmetries lead to the matrix valued conserved quantities and To construct integrals of motion out of C and S let us introduce the basis of su(2) algebra — three matrices T₁, T₂, T₃ with T₁ being 2×2 diagonal matrix while T₂ and T₂ are 2×2 off-diagonal matrices The elements of su(2) are traceless anti-hermitian matrices, and any A ∈ su(2) can be parameterized in the following way Scalar product ensures that Now we can introduce six functions which are integrals of motion.

Conservation of C and S leads to general solution of Euler-Lagrange equations 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 in sense that variation of C provides 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 where function plays the role of Hamiltonian and one-form 〈Cg^{− 1}dg〉 is a symplectic potential θ. External differential of θ is the symplectic form that determines Poisson brackets, the form of Hamilton's equation and provides isomorphism between vector fields and one-forms For any smooth SU(2) valued smooth function f ∈ SU(2) one can define Hamiltonian vector field X_f by where i_Xω denotes the contraction of X with ω. According to its definition Poisson bracket of two functions is where L_Xfg denotes Lie derivative of g with respect to vector filed X_f. The skew symmetry of ω provides skew symmetry of Poisson bracket.

Hamiltonian vector fields that correspond to C_n, S_m and g functions are and give rise to the following commutation relations 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

Let's introduce operators They act on the square integrable functions (see Appendix A) on SU(2) and satisfy quantum commutation relations The Hamiltonian is defined as and the complete set of observables that commute with each other is 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, Ĉ_a and Ŝ_b have the form where j takes positive integer and half integer values with c and s taking values in the following range Further we construct the corresponding eigenfunctions ψ_jsc. The first step of this construction is to note that the function 〈Tg〉 where T = (1 + iT_a)(1 + iT_b) is an eigenfunction of Ĥ, Ĉ_a and Ŝ_b with eigenvalues ¾, ½, ½ respectively. Proof of this proposition is straightforward. Using 〈Tg〉 one can construct the complete set of eigenfunctions of Ĥ, Ĉ_a and Ŝ_b operators 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 Where h(t) ∈ U(1) ⊂ SU(2) is an element of U(1). Without loss of generality we can take Since T₃ is antihermitian h(t) ∈ U(1) and since h(t) depends on t Lagrangian is not invariant under (38) local gauge transformations.

To make (40) gauge invariant we should replace time derivative with covariant derivative where B can be represented as follows with transformation rule or in terms of b variable The new Lagrangian 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 put it back in (45) and rewrite last obtained Lagrangian in terms of gauge invariant variables. It's obvious that the following element of su(2) algebra is gauge invariant. Since Z ∈ su(2) it can be parameterized as follows where z^a are real functions on SU(2)

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 otherwise

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 L_G takes the form This Lagrangian describes free particle on the sphere. Indeed, since Z = z^aT_a it's easy to show that 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 variation of u provides Rewriting Λ_G in terms of C and g leads to Due to the gauge invariance of Λ_G we obtain constrained Hamiltonian system, where 〈Cg^{− 1}dg〉 is symplectic potential, function plays the role of Hamiltonian and b is a Lagrange multiple leading to the first class constrain 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 Hilbert space of the initial system, that is linear span of wave functions, reduces to the linear span of wave functions. Indeed, Ŝ₃ψ_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 ψ_jcs rewriten in terms of gauge invariant variables up to a constant multiple should coincide with well known spherical harmonics One can check the following This is an example of using large initial model in quantization of coset model.

Scalar product in Hilbert space is defined as follows It's easy to prove that under this scalar product operators Ĉ_n and Ŝ_m are hermitian. Indeed Where integration by part has been used and the additional term coming from measure vanished since For more transparency one can introduce the following parameterization of SU(2). For any g ∈ SU(2). Then the symplectic potential takes the form and scalar product becomes that coincides with (65) because of

Without loss of generality we can take Ĥ, Ŝ₃ and Ĉ₃ as a complete set of observables. Assuming that operators Ĥ, Ŝ₃ and Ĉ₃ have at least one common eigenfunction it is easy to show that eigenvalues of Ĥ are non-negative E ≥ 0 and conditions are satisfied. Indeed, operators Ĉ and Ŝ are selfadjoint so To prove (74) we shall consider Ĉ₁² + Ĉ₂² and Ŝ₁² + Ŝ₂² operators and thus E − c² ≥ 0.

Now let's introduce new operators These operators are not selfadjoint, but (Ĉ₋)^† = Ĉ₊ and (Ŝ₋)^† = Ŝ₊ and they fulfill the following commutation relations where • takes values +, −, 3 using these commutation relations it is easy to show that if ψ_λcs is eigenfunction of Ĥ, Ŝ₃ and Ĉ₃ with corresponding eigenvalues : then Ĉ_±ψ_λcs and Ŝ_±ψ_λcs are the eigenfunctions with corresponding eigenvalues λ, s ± 1, c and λ , s, c ± 1. Consequently using Ĉ_±, Ŝ_± operators one can construct a family of eigenfunctions with eigenvalues but conditions (74) give restrictions on a possible range of eigenvalues. Namely we must have In other words, in order to interrupt (84) sequences we must assume for some j and k, therefore s and c could take only the following values The number of values is 2j + 1 and 2k + 1 respectively. Since number of values should be integer, j and k should take integer or half integer values Now using commutation relations we can rewrite Ĥ in terms of Ĉ_±, Ĉ₃ operators and it is clear that j = k and λ = j(j + 1) = k(k + 1)