Department of Theoretical Physics,A. Razmadze Institute of Mathematics,1 Aleksidze Street, Tbilisi 0193, Georgia

The dynamics of a free particle on $SU(2)$ group manifold is described by the LagrangianL = 〈g^{− 1}ġg^{− 1}ġ〉 where $g\; \in \; SU(2)$ and $\langle \rangle $ denotes the normalized trace〈 · 〉 = − ½Tr( · ) which defines a scalar product in $su(2)$ algebra. This Lagrangian gives rise to equations of motiond dt g^{− 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 transformationsg → h_{1}gg → gh_{2} where $h$_{1}, h_{2} ∈ SU(2)

According to the Noether's theorem these symmetries lead to the matrix valued conserved quantitiesC = g^{− 1}ġ d dt C = 0 andS = ġg^{− 1} d dt S = 0 To construct integrals of motion out of $C$ and $S$ let us introduce the basis of$su(2)$ algebra — three matrices:T_{1} =i | 0 |

0 | − i |

T_{2} =0 | − 1 |

1 | 0 |

T_{3} =0 | i |

i | 0 |

The elements of $su(2)$ are traceless anti-hermitian matrices, and any$A\; \in \; su(2)$ can be parameterized in the following wayA = A^{n}T_{n} n = 1, 2, 3 Scalar product AB = 〈AB〉 = − ½Tr(AB) ensures thatA^{n} = 〈AT_{n}〉 (〈T_{n}T_{m}〉 = δ_{nm}) Now we can introduce six functionsC_{n} = 〈T_{n}C〉 n = 1, 2, 3 C = C^{n}T_{n}S_{n} = 〈T_{n}S〉 n = 1, 2, 3 S = S^{n}T_{n} which are integrals of motion.

Conservation of $C$ and $S$ leads to general solution of Euler-Lagrange equationsd dt g^{− 1}ġ = 0 ⇒ g^{− 1}ġ = constg = e^{Ct}g(0) These are well known geodesics on Lie group.

Working in a first order Hamiltonian formalism we can construct new Lagrangianwhich is equivalent to the initial oneΛ = 〈C(g^{− 1}ġ − v)〉 + ½〈v^{2}〉 in sense that variation of C providesg^{− 1}ġ = v and $\Lambda $ reduces to $L$.Variation of $v$ gives $C\; =\; v$ and therefore we can rewriteequivalent Lagrangian $\Lambda $ in terms of C and g variablesΛ = 〈Cg^{− 1}ġ〉 − ½ 〈C^{2}〉 where functionH = ½〈C^{2}〉 plays the role of Hamiltonian andone-form $\langle Cg-\; 1dg\rangle $ is a symplectic potential $\theta $.External differential of $\theta $ is the symplectic formω = dθ = − 〈g^{− 1}dg ∧ dC〉 − 〈Cg^{− 1} dg ∧ g^{− 1}dg〉 that determines Poisson brackets, the form of Hamilton's equationand provides isomorphism between vector fields and one-formsX → i_{X}ω For any smooth $SU(2)$ valued smooth function$f\; \in \; SU(2)$ one can define Hamiltonian vector field $X$_{f} byi_{Xf}ω = − df where $i$_{X}ω denotes the contraction of $X$ with $\omega $.According to its definition Poisson bracket of two functions is{f , g} = L_{Xf}g = i_{Xf}dg = ω(X_{f} , X_{g}) where $L$_{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.

Hamiltonian vector fields that correspond to $C$_{n}, S_{m} and $g$ functions areX_{n} = X_{Cn} = ([C ,T_{n}] , gT_{n})Y_{m} = X_{Sm} = ([C , gT_{m}g^{− 1}] , T_{m}g ) and give rise to the following commutation relations{S_{n} , S_{m}} = − 2ε_{nm}^{k} S_{k}{C_{n} , C_{m}} = 2ε_{nm}^{k} C_{k}{C_{n} , S_{m}} = 0{C_{n} , g} = gT_{n}{S_{m} , g} = T_{m}g 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 equationsġ = {H , g} = gR Ċ = {H , C} = 0

Let's introduce operatorsĈ_{n} = i 2 L_{Xn} Ŝ_{m} = − i 2 L_{Ym} They act on the square integrable functions (see Appendix A) on $SU(2)$ and satisfy quantumcommutation relations[Ŝ_{n} , Ŝ_{m}] = iε_{nm}^{k} Ŝ_{k} [Ĉ_{n} , Ĉ_{m}] = iε_{nm}^{k} Ĉ_{k} [Ĉ_{n} , Ŝ_{m}] = 0 The Hamiltonian is defined asĤ = Ĉ^{2} = Ŝ^{2} and the complete set of observables that commute with each other isĤ, Ĉ_{a}, Ŝ_{b} 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$\u0124,\; \u0108$_{a} and $\u015c$_{b} have the formĤψ_{jsc} = j(j + 1)ψ_{jsc} where $j$ takes positive integer and half integer valuesj = 0, 1 2 , 1, 3 2 , 2 ... Ĉ_{a}ψ_{jsc} = cψ_{jsc} Ŝ_{b}ψ_{jsc} = sψ_{jsc} with $c$ and $s$ taking values in the following range− j, − j + 1, ... , j − 1, j Further we construct the corresponding eigenfunctions$\psi $_{jsc}. The first step of this construction is to note thatthe 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} operatorsψ_{jsc} =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 transformationsg → h(t)g Where $h(t)\; \in \; U(1)\; \subset \; SU(2)$ is an element of $U(1)$. Without loss of generality we can takeh = e^{β(t)T3} Since $T$_{3} is antihermitian $h(t)\; \in \; U(1)$ and since $h(t)$ depends on $t$ LagrangianL = 〈g^{− 1}ġg^{− 1}ġ〉 is not invariant under (38) local gauge transformations.

To make (40) gauge invariant we should replace time derivativewith covariant derivatived dt g → ∇g = (d dt + B)g where $B$ can be represented as followsB = bT_{3} ∈ su(2) with transformation ruleB → hBh^{− 1} − dh dt h^{− 1} or in terms of $b$ variableb → b − dβ dt The new LagrangianL_{G} = 〈g^{− 1}∇gg^{− 1}∇g〉 is invariant under (38) local gauge transformations. But thisLagrangian as well as every gauge invariant Lagrangian is singular.It contains additional non-physical degrees of freedom. Toeliminate them we should eliminate $B$ using Lagrange equations∂L_{G} ∂B → b = − 〈ġg^{− 1}T_{3}〉 put it back in (45) and rewrite last obtained Lagrangian in terms of gauge invariant variables.L_{G} = 〈(g^{− 1}ġ − S_{3}T_{3})^{2}〉 It's obvious that the followingZ = g^{− 1}T_{3}g ∈ su(2) element of $su(2)$ algebra is gauge invariant. Since $Z\; \in \; su(2)$ it can be parameterized as followsZ = z^{a}T_{a} where $za$ are real functions on $SU(2)$z_{a} = 〈ZT_{a}〉

So we have three gauge invariant variables $za(a\; =\; 1,\; 2,\; 3)$ but it's easy tocheck that only two of them are independent. Indeed〈Z^{2}〉 = 〈g^{− 1}T_{3}gg^{− 1}T_{3}g〉 = 〈T_{3}^{2}〉 = 1 otherwise〈Z^{2}〉 = 〈z^{a}T_{a}z^{b}T_{b}〉 = z^{a}z_{a}

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 formL_{G} = ¼〈Z^{− 1}ŻZ^{− 1}Ż〉 This Lagrangian describes free particle on the sphere. Indeed,since $Z\; =\; zaT$_{a} it's easy to show thatL_{G} = ¼〈Z^{− 1}ŻZ^{− 1}Ż〉 =¼〈ZŻZŻ〉 = ½ż^{a}ż_{a} So $SU(2)/U(1)$ coset model describes free particle on $S2$ manifold.

Working in a first order Hamiltonian formalism one can introduce equivalent LagrangianΛ_{G} = 〈C(g^{− 1}ġ − u)〉 + ½ 〈(u + g^{− 1}Bg)^{2}〉 variation of $u$ providesC = u + g^{− 1}Bg u = C − g^{− 1}Bg Rewriting $\Lambda $_{G} in terms of $C$ and $g$ leads toΛ_{G} = 〈Cg^{− 1}ġ〉 − ½ 〈C^{2}〉 − 〈BgCg^{− 1}〉 = 〈Cg^{− 1}ġ〉 − ½ 〈C^{2}〉 − b〈gCg^{− 1}T_{3}〉 = 〈Cg^{− 1}ġ〉 − ½ 〈C^{2}〉 − bS_{3} Due to the gauge invariance of $\Lambda $_{G} we obtain constrained Hamiltonian system,where $\langle Cg-\; 1dg\rangle $ is symplectic potential, functionH =½〈C^{2}〉 plays the role of Hamiltonian and$b$ is a Lagrange multiple leading to the first class constrainφ = 〈gCg^{− 1}T_{3}〉 = 〈ST_{3}〉 = S_{3} = 0 So coset model is equivalent to the initial one with (59) constrain.Using technique of the constrained quantization, instead ofquantizing coset model we can subject quantum model that corresponds to the free particle on $SU(2)$,to the following operator constrainŜ_{3}|ψ〉 = 0 Hilbert space of the initial system, that is linear span ofψ_{jcs} j = 0, 1 2 , 1, 3 2 , 2, ... wave functions, reduces tothe linear span ofψ_{jc0} j = 0, 1, 2, 3, ... wave functions. Indeed,$\u015c$_{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 invariantvariables up to a constant multiple should coincide with well knownspherical harmonicsψ_{jc0} ∼ J_{jc} One can check the followingψ_{jc0} ∼ Ŝ_{−}^{j}Ĉ_{−}^{j − c} 〈Tg〉^{2j} ∼ Ĉ_{−}^{j − c} 〈T_{+}g^{− 1}T_{3}g〉^{j} ∼ Ĉ_{−}^{j − c}sin^{j}θe^{ijθ} ∼ Ĉ_{−}^{j − c}J_{jj} ∼J_{jc} This is an example of using large initial model in quantization ofcoset model.

Scalar product in Hilbert space is defined as follows〈ψ_{1}|ψ_{2}〉 =∫ SU(2) 3 ∏ a = 1 〈g^{− 1}dgT_{a}〉(ψ_{1})^{†}ψ_{2} It's easy to prove that under this scalar product operators$\u0108$_{n} and $\u015c$_{m} are hermitian.Indeed〈ψ_{1}|Ĉ_{n}ψ_{2}〉 =∫ SU(2) 3 ∏ a = 1 〈g^{− 1}dgT_{a}〉(ψ_{1})^{†}(i 2 L_{Xn}ψ_{2}) = ∫ SU(2) 3 ∏ a = 1 〈g^{− 1}dgT_{a}〉(i 2 L_{Xn}ψ_{1})^{†}ψ_{2} Where integration by part has been used and the additional term coming from measure3 ∏ a = 1 〈g^{− 1}dgT_{a}〉 vanished sinceL_{Xn}〈g^{− 1}dgT_{a}〉 = 0 For more transparency one can introduce the following parameterization of$SU(2)$. For any $g\; \in \; SU(2)$.g = e^{qaTa} Then the symplectic potential takes the form〈Cg^{− 1}dg〉 = C_{a}dq^{a} and scalar product becomes〈ψ_{1}|ψ_{2}〉 =2π ∫ 0 2π ∫ 0 2π ∫ 0 d^{3}q(ψ_{1})^{†}ψ_{2} that coincides with (65) because ofdq_{a} = 〈g^{− 1}dg T_{a}〉

Without loss of generality we can take$\u0124,\; \u015c$_{3} and$\u0108$_{3} as a complete set of observables.Assuming that operators $\u0124,\; \u015c$_{3} and $\u0108$_{3}have at least one common eigenfunctionĤψ = EψĈ_{3}ψ = cψŜ_{3}ψ = sψ it is easy to show that eigenvalues of $\u0124$ are non-negative $E\; \ge \; 0$and conditionsE − c^{2} ≥ 0E − s^{2} ≥ 0 are satisfied. Indeed, operators $\u0108$ and $\u015c$ are selfadjoint so〈ψ|Ĥ|ψ〉 = 〈ψ|Ĉ^{2}|ψ〉 = 〈ψ|Ĉ_{a}Ĉ^{a}|ψ〉 =〈ψ|(Ĉ_{a})^{†}Ĉ^{a}|ψ〉 =〈Ĉ_{a}ψ|Ĉ^{a}ψ〉 = ∥Ĉ_{a}ψ∥ ≥ 0 To prove (74) we shall consider$\u0108$_{1}^{2} + Ĉ_{2}^{2} and$\u015c$_{1}^{2} + Ŝ_{2}^{2} operators〈ψ|Ĉ_{1}^{2} + Ĉ_{2}^{2}|ψ〉 =∥Ĉ_{1} ψ∥ + ∥Ĉ_{2} ψ∥ ≥ 0 and〈ψ|Ĉ_{1}^{2} + Ĉ_{2}^{2}|ψ〉 =〈ψ|Ĥ − Ĉ_{3}^{2}|ψ〉 = (E − c^{2})〈ψ|ψ〉 thus $E\; -\; c2\ge \; 0$.

Now let's introduce new operatorsĈ_{+} = iĈ_{1} + Ĉ_{2} Ĉ_{−} =iĈ_{1} − Ĉ_{2} Ŝ_{+} = iŜ_{1} + Ŝ_{2} Ŝ_{−} =iŜ_{1} − Ŝ_{2} These operators are not selfadjoint, but $(\u0108$_{−})^{†} = Ĉ_{+} and$(\u015c$_{−})^{†} = Ŝ_{+}and they fulfill the following commutation relations[Ĉ_{±} , Ĉ_{3}] = ± Ĉ_{±} [Ŝ_{±} , Ŝ_{3}] = ± Ŝ_{±} [Ĉ_{+} , Ĉ_{−}] = 2Ĉ_{3} [Ŝ_{+} , Ŝ_{−}] = 2Ŝ_{3} [Ĉ_{•} , Ŝ_{•}] = 0 where $\u2022$ takes values $+,\; -,\; 3$ using these commutation relations it is easy to showthat if $\psi $_{λcs} is eigenfunction of$\u0124,\; \u015c$_{3} and$\u0108$_{3} with corresponding eigenvalues :Ĥψ_{λcs} = λψ_{λcs}Ŝ_{3}ψ_{λcs} = sψ_{λcs}Ĉ_{3}ψ_{λcs} = cψ_{λcs} then $\u0108$_{±}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 constructa family of eigenfunctions with eigenvaluesc, c ± 1, c ± 2, c ± 3, ...s, s ± 1, s ± 2, s ± 3, ... but conditions (74) give restrictions on a possible range of eigenvalues.Namely we must haveλ − c^{2} ≥ 0λ − s^{2} ≥ 0 In other words, in order to interrupt (84) sequences we must assumeŜ_{+} ψ_{λcj} = 0 Ŝ_{−}ψ_{λc, − j} = 0Ĉ_{+}ψ_{λks} = 0 Ĉ_{−}ψ_{λ, − ks} = 0 for some $j$ and $k$, therefore $s$ and $c$ could take only the following values− j, − j + 1, ... , j − 1, j− k, − k + 1, ... , k − 1, k The number of values is $2j\; +\; 1$ and $2k\; +\; 1$ respectively. Since number of valuesshould be integer, $j$ and $k$ should take integer or half integer valuesj = 0, 1 2 , 1, 3 2 , 2, ...k = 0, 1 2 , 1, 3 2 , 2, ... Now using commutation relations we can rewrite $\u0124$ in terms of$\u0108$_{±}, Ĉ_{3} operatorsĤ = Ĉ_{+} Ĉ_{−} + Ĉ_{3}^{2} + Ĉ_{3} and it is clear that $j\; =\; k$ and $\lambda \; =\; j(j\; +\; 1)\; =\; k(k\; +\; 1)$

- V. I. Arnold Mathematical methods of classical mechanics Springer-Verlag, Berlin 1978
- A. Bohm Quantum mechanics: foundations and applications Springer-Verlag 1986
- G. Jorjadze, L. O'Raifeartaigh, I. Tsitsui Quantization of a free relativistic particle on the SL(2,R) manifold based on Hamiltonian reduction Physics Letters B 336, 388-394 1994
- N. M. J. Woodhouse Geometric Quantization Claredon, Oxford 1992