Free particle on SU(2) group manifold
George Chavchanidze
Department of Theoretical Physics,
A. Razmadze Institute of Mathematics,
1 Aleksidze Street, Tbilisi 0193, Georgia
Abstract. In the present paper classical and quantum dynamics
of a free particle on SU(2) group manifold is considered.
Poisson structure of the dynamical system and commutation relations for generalized momenta are
derived. Quantization is carried out and the eigenfunctions of the Hamiltonian
are constructed in terms of coordinate free objects.
SU(2)/U(1) coset model yielding after Hamiltonian reduction free particle on
S2 sphere is considered
and Hamiltonian reduction of coset model is carried out on both classical and quantum level.
Keywords: Dynamics on group manifold; Quantization on group manifold;
MSC 2000: 70H33; 70H06; 53Z05
1. Lagrangian description
The dynamics of a free particle on SU(2) group manifold is described by the Lagrangian
L = 〈g− 1ġg− 1ġ〉
(1)
where g ∈ SU(2) and 〈 〉 denotes the normalized trace
〈 · 〉 = − ½Tr( · )
(2)
which defines a scalar product in su(2) algebra. This Lagrangian gives rise to equations of motion
ddtg− 1ġ = 0
(3)
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
(4)
where h1, h2 ∈ SU(2)
According to the Noether's theorem these symmetries lead to the matrix valued conserved quantities
C = g− 1ġ ddtC = 0
(5)
and
S = ġg− 1 ddtS = 0
(6)
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
(7)
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
(8)
Scalar product
AB = 〈AB〉 = − ½Tr(AB)
(9) ensures that
An = 〈ATn〉 (〈TnTm〉 = δnm)
(10)
Now we can introduce six functions
Cn = 〈TnC〉 n = 1, 2, 3 C = CnTn
Sn = 〈TnS〉 n = 1, 2, 3 S = SnTn
(11)
which are integrals of motion.
Conservation of C and S leads to general solution of Euler-Lagrange equations
ddtg− 1ġ = 0 ⇒ g− 1ġ = const
g = eCtg(0)
(12)
These are well known geodesics on Lie group.
2. Hamiltonian description
Working in a first order Hamiltonian formalism we can construct new Lagrangian
which is equivalent to the initial one
Λ = 〈C(g− 1ġ − v)〉 + ½〈v2〉
(13)
in sense that variation of C provides
g− 1ġ = v
(14)
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− 1ġ〉 − ½ 〈C2〉
(15)
where function
H = ½〈C2〉
(16)
plays the role of Hamiltonian and
one-form 〈Cg− 1dg〉 is a symplectic potential θ.
External differential of θ is the symplectic form
ω = dθ = − 〈g− 1dg ∧ dC〉 − 〈Cg− 1 dg ∧ g− 1dg〉
(17)
that determines Poisson brackets, the form of Hamilton's equation
and provides isomorphism between vector fields and one-forms
X → iXω
(18)
For any smooth SU(2) valued smooth function
f ∈ SU(2) one can define Hamiltonian vector field Xf by
iXfω = − df
(19)
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)
(20)
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 )
(21)
and give rise to the following commutation relations
{Sn , Sm} = − 2εnmk Sk
{Cn , Cm} = 2εnmk Ck
{Cn , Sm} = 0
{Cn , g} = gTn
{Sm , g} = Tmg
(22)
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
(23)
Ċ = {H , C} = 0
(24)
3. Quantization
Let's introduce operators
Ĉn = i2LXn
(25)
Ŝm = − i2LYm
(26)
They act on the square integrable functions (see Appendix A) on SU(2) and satisfy quantum
commutation relations
[Ŝn , Ŝm] = iεnmk Ŝk
(27)
[Ĉn , Ĉm] = iεnmk Ĉk
(28)
[Ĉn , Ŝm] = 0
(29)
The Hamiltonian is defined as
Ĥ = Ĉ2 = Ŝ2
(30)
and the complete set of observables that commute with each other is
Ĥ, Ĉa, Ŝb
(31)
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
(32)
where j takes positive integer and half integer values
j = 0, 12, 1, 32, 2 ...
(33)
Ĉaψjsc = cψjsc
(34)
Ŝbψjsc = sψjsc
(35)
with c and s taking values in the following range
− j, − j + 1, ... , j − 1, j
(36)
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 − sĈ−j − c〈Tg〉2j
(37) in the manner described in Appendix B.
4. Free particle on S² as a SU(2)/U(1) coset model
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
(38)
Where h(t) ∈ U(1) ⊂ SU(2) is an element of U(1). Without loss of generality we can take
h = eβ(t)T3
(39)
Since T3 is antihermitian h(t) ∈ U(1) and since h(t) depends on t Lagrangian
L = 〈g− 1ġg− 1ġ〉
(40)
is not invariant under (38) local gauge transformations.
To make (40) gauge invariant we should replace time derivative
with covariant derivative
ddtg → ∇g = (ddt + B)g
(41)
where B can be represented as follows
B = bT3 ∈ su(2)
(42)
with transformation rule
B → hBh− 1 − dhdth− 1
(43)
or in terms of b variable
b → b − dβdt
(44)
The new Lagrangian
LG = 〈g− 1∇gg− 1∇g〉
(45)
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− 1T3〉
(46)
put it back in (45) and rewrite last obtained Lagrangian in terms of gauge invariant variables.
LG = 〈(g− 1ġ − S3T3)2〉
(47)
It's obvious that the following
Z = g− 1T3g ∈ su(2)
(48)
element of su(2) algebra is gauge invariant. Since Z ∈ su(2) it can be parameterized as follows
Z = zaTa
(49)
where za are real functions on SU(2)
za = 〈ZTa〉
(50)
So we have three gauge invariant variables za (a = 1, 2, 3) but it's easy to
check that only two of them are independent. Indeed
〈Z2〉 = 〈g− 1T3gg− 1T3g〉 = 〈T32〉 = 1
(51)
otherwise
〈Z2〉 = 〈zaTazbTb〉 = zaza
(52)
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− 1ŻZ− 1Ż〉
(53)
This Lagrangian describes free particle on the sphere. Indeed,
since Z = zaTa it's easy to show that
LG = ¼〈Z− 1ŻZ− 1Ż〉 =
¼〈ZŻZŻ〉 = ½żaża
(54)
So SU(2)/U(1) coset model describes free particle on S2 manifold.
5. Quantization of the coset model.
Working in a first order Hamiltonian formalism one can introduce equivalent Lagrangian
ΛG = 〈C(g− 1ġ − u)〉 + ½ 〈(u + g− 1Bg)2〉
(55)
variation of u provides
C = u + g− 1Bg
u = C − g− 1Bg
(56)
Rewriting ΛG in terms of C and g leads to
ΛG = 〈Cg− 1ġ〉 − ½ 〈C2〉 − 〈BgCg− 1〉 = 〈Cg− 1ġ〉
− ½ 〈C2〉 − b〈gCg− 1T3〉 =
〈Cg− 1ġ〉 − ½ 〈C2〉 − bS3
(57)
Due to the gauge invariance of ΛG we obtain constrained Hamiltonian system,
where 〈Cg− 1dg〉 is symplectic potential, function
H =
½〈C2〉
(58) plays the role of Hamiltonian and
b is a Lagrange multiple leading to the first class constrain
φ = 〈gCg− 1T3〉 = 〈ST3〉 = S3 = 0
(59)
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
(60)
Hilbert space of the initial system, that is linear span of
ψjcs j = 0, 12, 1, 32, 2, ...
(61)
wave functions, reduces to
the linear span of
ψjc0 j = 0, 1, 2, 3, ...
(62)
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
(63)
One can check the following
ψjc0 ∼ Ŝ−jĈ−j − c 〈Tg〉2j
∼ Ĉ−j − c 〈T+g− 1T3g〉j
∼ Ĉ−j − csinjθeijθ
∼ Ĉ−j − cJjj ∼Jjc
(64)
This is an example of using large initial model in quantization of
coset model.
6. Appendix A
Scalar product in Hilbert space is defined as follows
〈ψ1|ψ2〉 =
∫SU(2)
3∏a = 1
〈g− 1dgTa〉(ψ1)†ψ2
(65)
It's easy to prove that under this scalar product operators
Ĉn and Ŝm are hermitian.
Indeed
〈ψ1|Ĉnψ2〉 =
∫SU(2)
3∏a = 1
〈g− 1dgTa〉(ψ1)†
(i2LXnψ2)
= ∫SU(2)
3∏a = 1
〈g− 1dgTa〉(i2LXnψ1)†ψ2
(66)
Where integration by part has been used and the additional term coming from measure
3∏a = 1 〈g− 1dgTa〉
(67)
vanished since
LXn〈g− 1dgTa〉 = 0
(68)
For more transparency one can introduce the following parameterization of
SU(2). For any g ∈ SU(2).
g = eqaTa
(69)
Then the symplectic potential takes the form
〈Cg− 1dg〉 = Cadqa
(70)
and scalar product becomes
〈ψ1|ψ2〉 =
2π∫0
2π∫0
2π∫0
d3q(ψ1)†ψ2
(71)
that coincides with (65) because of
dqa = 〈g− 1dg Ta〉
(72)
7. Appendix B
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
Ĥψ = Eψ
Ĉ3ψ = cψ
Ŝ3ψ = sψ
(73)
it is easy to show that eigenvalues of Ĥ are non-negative E ≥ 0
and conditions
E − c2 ≥ 0
E − s2 ≥ 0
(74)
are satisfied. Indeed, operators Ĉ and Ŝ are selfadjoint so
〈ψ|Ĥ|ψ〉 = 〈ψ|Ĉ2|ψ〉 = 〈ψ|ĈaĈa|ψ〉 =
〈ψ|(Ĉa)†Ĉa|ψ〉 =
〈Ĉaψ|Ĉaψ〉 = ∥Ĉaψ∥ ≥ 0
(75)
To prove (74) we shall consider
Ĉ12 + Ĉ22 and
Ŝ12 + Ŝ22 operators
〈ψ|Ĉ12 + Ĉ22|ψ〉 =
∥Ĉ1 ψ∥ + ∥Ĉ2 ψ∥ ≥ 0
(76)
and
〈ψ|Ĉ12 + Ĉ22|ψ〉 =
〈ψ|Ĥ − Ĉ32|ψ〉 = (E − c2)〈ψ|ψ〉
(77)
thus E − c2 ≥ 0.
Now let's introduce new operators
Ĉ+ = iĈ1 + Ĉ2 Ĉ− =
iĈ1 − Ĉ2
(78)
Ŝ+ = iŜ1 + Ŝ2 Ŝ− =
iŜ1 − Ŝ2
(79)
These operators are not selfadjoint, but (Ĉ−)† = Ĉ+ and
(Ŝ−)† = Ŝ+
and they fulfill the following commutation relations
[Ĉ± , Ĉ3] = ± Ĉ± [Ŝ± , Ŝ3] = ± Ŝ±
(80)
[Ĉ+ , Ĉ−] = 2Ĉ3 [Ŝ+ , Ŝ−] = 2Ŝ3
(81)
[Ĉ• , Ŝ•] = 0
(82)
where • takes values +, −, 3 using these commutation relations it is easy to show
that if ψλcs is eigenfunction of
Ĥ, Ŝ3 and
Ĉ3 with corresponding eigenvalues :
Ĥψλcs = λψλcs
Ŝ3ψλcs = sψλcs
Ĉ3ψλcs = cψλcs
(83)
then Ĉ±ψλcs and
Ŝ±ψλcs
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
c, c ± 1, c ± 2, c ± 3, ...
s, s ± 1, s ± 2, s ± 3, ...
(84)
but conditions (74) give restrictions on a possible range of eigenvalues.
Namely we must have
λ − c2 ≥ 0
λ − s2 ≥ 0
(85)
In other words, in order to interrupt (84) sequences we must assume
Ŝ+ ψλcj = 0 Ŝ−ψλc, − j = 0
Ĉ+ψλks = 0 Ĉ−ψλ, − ks = 0
(86)
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
(87)
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
j = 0, 12, 1, 32, 2, ...
k = 0, 12, 1, 32, 2, ...
(88)
Now using commutation relations we can rewrite Ĥ in terms of
Ĉ±, Ĉ3 operators
Ĥ = Ĉ+ Ĉ− + Ĉ32 + Ĉ3
(89)
and it is clear that j = k and λ = j(j + 1) = k(k + 1)
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