The dynamics of a free particle on *SU(2)* group manifold is described by the Lagrangian
*g ∈ SU(2)* and *〈 〉* denotes the normalized trace
*su(2)* algebra. This Lagrangian gives rise to equations of motion
*SU(2)* "right" and *SU(2)* "left" symmetry.
It means that it is invariant under the following transformations
*h*_{1}, h_{2} ∈ SU(2)

L = 〈g^{− 1}ġg^{− 1}ġ〉

where
〈 · 〉 = − ½Tr( · )

which defines a scalar product in
g^{− 1}ġ = 0

that describe dynamics of particle on group manifold.
Also, one can notice that it has
g → h_{1}g

g → gh_{2}

where g → gh

According to the Noether's theorem these symmetries lead to the matrix valued conserved quantities
*C* and *S* let us introduce the basis of
*su(2)* algebra — three matrices *T*_{1}, T_{2}, T_{3}
with *T*_{1} being *2×2* diagonal matrix
while *T*_{2} and *T*_{2} are *2×2* off-diagonal matrices
*su(2)* are traceless anti-hermitian matrices, and any
*A ∈ su(2)* can be parameterized in the following way

C = g^{− 1}ġ C = 0

and
S = ġg^{− 1} S = 0

To construct integrals of motion out of
T_{1} = diag(i , − i)

{T_{2}}_{21} = −{T_{2}}_{12} = 1

{T_{3}}_{12} = {T_{3}}_{21} = i

The elements of {T

{T

A = A^{n}T_{n} n = 1, 2, 3

Scalar product
AB = 〈AB〉 = − ½Tr(AB)

ensures that
A^{n} = 〈AT_{n}〉 (〈T_{n}T_{m}〉 = δ_{nm})

Now we can introduce six functions
C_{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.S

Conservation of *C* and *S* leads to general solution of Euler-Lagrange equations

g^{− 1}ġ = 0 ⇒ g^{− 1}ġ = const

g = e^{Ct}g(0)

These are well known geodesics on Lie group.
g = e

Working in a first order Hamiltonian formalism we can construct new Lagrangian
which is equivalent to the initial one
*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
*〈Cg*^{− 1}dg〉 is a symplectic potential *θ*.
External differential of *θ* is the symplectic form
*SU(2)* valued smooth function
*f ∈ SU(2)* one can define Hamiltonian vector field *X*_{f} by
*i*_{X}ω denotes the contraction of *X* with *ω*.
According to its definition Poisson bracket of two functions is
*L*_{Xf}g denotes Lie derivative of *g* with respect to vector filed *X*_{f}.
The skew symmetry of *ω* provides skew symmetry of Poisson bracket.

Λ = 〈C(g^{− 1}ġ − v)〉 + ½〈v^{2}〉

in sense that variation of C provides
g^{− 1}ġ = v

and
Λ = 〈Cg^{− 1}ġ〉 − ½ 〈C^{2}〉

where function
H = ½〈C^{2}〉

plays the role of Hamiltonian and
one-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 → i_{X}ω

For any smooth
i_{Xf}ω = − df

where
{f , g} = L_{Xf}g = i_{Xf}dg = ω(X_{f} , X_{g})

where Hamiltonian vector fields that correspond to *C*_{n}, S_{m} and *g* functions are
*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

X_{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
Y

{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

{C

{C

{S

ġ = {H , g} = gR

Ċ = {H , C} = 0

Let's introduce operators
*SU(2)* and satisfy quantum
commutation relations
*Ĥ, Ĉ*_{a} and *Ŝ*_{b} have the form
*j* takes positive integer and half integer values
*c* and *s* taking values in the following range
*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

Ĉ_{n} = L_{Xn}

Ŝ_{m} = − L_{Ym}

They act on the square integrable functions (see Appendix A) on
[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 as
Ĥ = Ĉ^{2} = Ŝ^{2}

and the complete set of observables that commute with each other is
Ĥ, 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
Ĥψ_{jsc} = j(j + 1)ψ_{jsc}

where
j = 0, , 1, , 2 ...

Ĉ_{a}ψ_{jsc} = cψ_{jsc}

Ŝ_{b}ψ_{jsc} = sψ_{jsc}

with
− j, − j + 1, ... , j − 1, j

Further we construct the corresponding eigenfunctions
ψ_{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 transformations
*h(t) ∈ U(1) ⊂ SU(2)* is an element of *U(1)*. Without loss of generality we can take
*T*_{3} is antihermitian *h(t) ∈ U(1)* and since *h(t)* depends on *t* Lagrangian

g → h(t)g

Where
h = e^{β(t)T3}

Since
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
*B* can be represented as follows
*b* variable
*B* using Lagrange equations
*su(2)* algebra is gauge invariant. Since *Z ∈ su(2)* it can be parameterized as follows
*z*^{a} are real functions on *SU(2)*

g → ∇g = ( + B)g

where
B = bT_{3} ∈ su(2)

with transformation rule
B → hBh^{− 1} − h^{− 1}

or in terms of
b → b −

The new Lagrangian
L_{G} = 〈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
∂L_{G}/∂B = 0 → 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 following
Z = g^{− 1}T_{3}g ∈ su(2)

element of
Z = z^{a}T_{a}

where
z_{a} = 〈ZT_{a}〉

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

〈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 form
*Z = z*^{a}T_{a} it's easy to show that
*SU(2)/U(1)* coset model describes free particle on *S*^{2} manifold.

L_{G} = ¼〈Z^{− 1}ŻZ^{− 1}Ż〉

This Lagrangian describes free particle on the sphere. Indeed,
since
L_{G} = ¼〈Z^{− 1}ŻZ^{− 1}Ż〉 =
¼〈ZŻZŻ〉 = ½ż^{a}ż_{a}

So Working in a first order Hamiltonian formalism one can introduce equivalent Lagrangian
*u* provides
*Λ*_{G} in terms of *C* and *g* leads to
*Λ*_{G} we obtain constrained Hamiltonian system,
where *〈Cg*^{− 1}dg〉 is symplectic potential, function
*b* is a Lagrange multiple leading to the first class constrain
*SU(2)*,
to the following operator constrain
*Ŝ*_{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 *ψ*_{jcs} rewriten in terms of gauge invariant
variables up to a constant multiple should coincide with well known
spherical harmonics

Λ_{G} = 〈C(g^{− 1}ġ − u)〉 + ½ 〈(u + g^{− 1}Bg)^{2}〉

variation of
C = u + g^{− 1}Bg

u = C − g^{− 1}Bg

Rewriting u = C − g

Λ_{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 − ½ 〈C

H =
½〈C^{2}〉

plays the role of Hamiltonian and
φ = 〈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 of
quantizing coset model we can subject quantum model that corresponds to the free particle on
Ŝ_{3}|ψ〉 = 0

Hilbert space of the initial system, that is linear span of
ψ_{jcs} j = 0, , 1, , 2, ...

wave functions, reduces to
the linear span of
ψ_{jc0} j = 0, 1, 2, 3, ...

wave functions. Indeed,
ψ_{jc0} ∼ J_{jc}

One can check the following
ψ_{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 of
coset model.∼ Ĉ

Scalar product in Hilbert space is defined as follows
*Ĉ*_{n} and *Ŝ*_{m} are hermitian.
Indeed
*SU(2)*. For any *g ∈ SU(2)*.

〈ψ_{1}|ψ_{2}〉 =
〈g^{− 1}dgT_{a}〉(ψ_{1})^{†}ψ_{2}

It's easy to prove that under this scalar product operators
〈ψ_{1}|Ĉ_{n}ψ_{2}〉 =
〈g^{− 1}dgT_{a}〉(ψ_{1})^{†}
(L_{Xn}ψ_{2})

= 〈g^{− 1}dgT_{a}〉(L_{Xn}ψ_{1})^{†}ψ_{2}

Where integration by part has been used and the additional term coming from measure
= 〈g

〈g^{− 1}dgT_{a}〉

vanished since
L_{Xn}〈g^{− 1}dgT_{a}〉 = 0

For more transparency one can introduce the following parameterization of
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}〉 =
d^{3}q(ψ_{1})^{†}ψ_{2}

that coincides with (65) because of
dq_{a} = 〈g^{− 1}dg T_{a}〉

Without loss of generality we can take
*Ĥ, Ŝ*_{3} and
*Ĉ*_{3} as a complete set of observables.
Assuming that operators *Ĥ, Ŝ*_{3} and *Ĉ*_{3}
have at least one common eigenfunction
*Ĥ* are non-negative *E ≥ 0*
and conditions
*Ĉ* and *Ŝ* are selfadjoint so
*Ĉ*_{1}^{2} + Ĉ_{2}^{2} and
*Ŝ*_{1}^{2} + Ŝ_{2}^{2} operators
*E − c*^{2} ≥ 0.

Ĥψ = Eψ

Ĉ_{3}ψ = cψ

Ŝ_{3}ψ = sψ

it is easy to show that eigenvalues of Ĉ

Ŝ

E − c^{2} ≥ 0

E − s^{2} ≥ 0

are satisfied. Indeed, operators E − s

〈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
〈Ĉ

〈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 Now let's introduce new operators
*(Ĉ*_{−})^{†} = Ĉ_{+} and
*(Ŝ*_{−})^{†} = Ŝ_{+}
and they fulfill the following commutation relations
*•* takes values *+, −, 3* using these commutation relations it is easy to show
that if *ψ*_{λcs} is eigenfunction of
*Ĥ, Ŝ*_{3} and
*Ĉ*_{3} with corresponding eigenvalues :
*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
*j* and *k*, therefore *s* and *c* could take only the following values
*2j + 1* and *2k + 1* respectively. Since number of values
should be integer, *j* and *k* should take integer or half integer values
*Ĥ* in terms of
*Ĉ*_{±}, Ĉ_{3} operators
*j = k* and *λ = j(j + 1) = k(k + 1)*

Ĉ_{+} = 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
[Ĉ_{±} , Ĉ_{3}] = ± Ĉ_{±} [Ŝ_{±} , Ŝ_{3}] = ± Ŝ_{±}

[Ĉ_{+} , Ĉ_{−}] = 2Ĉ_{3} [Ŝ_{+} , Ŝ_{−}] = 2Ŝ_{3}

[Ĉ_{•} , Ŝ_{•}] = 0

where
Ĥψ_{λcs} = λψ_{λcs}

Ŝ_{3}ψ_{λcs} = sψ_{λcs}

Ĉ_{3}ψ_{λcs} = cψ_{λcs}

then Ŝ

Ĉ

c, 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 have
s, s ± 1, s ± 2, s ± 3, ...

λ − c^{2} ≥ 0

λ − s^{2} ≥ 0

In other words, in order to interrupt (84) sequences we must assume
λ − s

Ŝ_{+} ψ_{λcj} = 0 Ŝ_{−}ψ_{λc, − j} = 0

Ĉ_{+}ψ_{λks} = 0 Ĉ_{−}ψ_{λ, − ks} = 0

for some Ĉ

− j, − j + 1, ... , j − 1, j

− k, − k + 1, ... , k − 1, k

The number of values is − k, − k + 1, ... , k − 1, k

j = 0, , 1, , 2, ...

k = 0, , 1, , 2, ...

Now using commutation relations we can rewrite k = 0, , 1, , 2, ...

Ĥ = Ĉ_{+} Ĉ_{−} + Ĉ_{3}^{2} + Ĉ_{3}

and it is clear that - 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 (1994) 388-394 - N. M. J. Woodhouse, Geometric Quantization, Claredon, Oxford, 1992