Free particle on SU(2) group manifold
George Chavchanidze
Department of Theoretical Physics,
A. Razmadze Institute of Mathematics,
1 Aleksidze Street, Tbilisi 0193, Georgia
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
$S^2$ sphere is considered
and Hamiltonian reduction of coset model is carried out on both classical and quantum level.
Dynamics on group manifold; Quantization on group manifold;
70H33; 70H06; 53Z05
Lagrangian description
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
$$
\frac{d}{dt}g^{− 1}ġ = 0
$$
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 transformations
$$
g → h_1g\\
g → gh_2
$$
where $h_1, h_2 ∈ SU(2)$
According to the Noether's theorem these symmetries lead to the matrix valued conserved quantities
$$
C = g^{− 1}ġ \frac{d}{dt}C = 0
$$
and
$$
S = ġg^{− 1} \frac{d}{dt}S = 0
$$
To construct integrals of motion out of $C$ and $S$ let us introduce the basis of
$su(2)$ algebra — three matrices:
$$
T_1 =
\matrix{
\row{i}{0}
\row{0}{− i}
}
T_2 =
\matrix{
\row{0}{− 1}
\row{1}{0}
}
T_3 =
\matrix{
\row{0}{i}
\row{i}{0}
}
$$
The elements of $su(2)$ are traceless anti-hermitian matrices, and any
$A ∈ su(2)$ can be parameterized in the following way
$$
A = A^nT_n n = 1, 2, 3
$$
Scalar product $$
AB = 〈AB〉 = − ½Tr(AB)
$$ ensures that
$$
A^n = 〈AT_n〉 (〈T_nT_m〉 = δ_{nm})
$$
Now we can introduce six functions
$$
C_n = 〈T_nC〉 n = 1, 2, 3 C = C^nT_n\\
S_n = 〈T_nS〉 n = 1, 2, 3 S = S^nT_n
$$
which are integrals of motion.
Conservation of $C$ and $S$ leads to general solution of Euler-Lagrange equations
$$
\frac{d}{dt}g^{− 1}ġ = 0 ⇒ g^{− 1}ġ = const\\
g = e^{Ct}g(0)
$$
These are well known geodesics on Lie group.
Hamiltonian description
Working in a first order Hamiltonian formalism we can construct new Lagrangian
which is equivalent to the initial one
$$
Λ = 〈C(g^{− 1}ġ − v)〉 + ½〈v^2〉
$$
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
$$
Λ = 〈Cg^{− 1}ġ〉 − ½ 〈C^2〉
$$
where function
$$
H = ½〈C^2〉
$$
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
$$
X → i_Xω
$$
For any smooth $SU(2)$ valued smooth function
$f ∈ SU(2)$ one can define Hamiltonian vector field $X_f$ by
$$
i_{X_f}ω = − df
$$
where $i_Xω$ denotes the contraction of $X$ with $ω$.
According to its definition Poisson bracket of two functions is
$$
\{f , g\} = L_{X_f}g = i_{X_f}dg = ω(X_f , X_g)
$$
where $L_{X_f}g$ 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
$$
X_n = X_{C_n} = ([C ,T_n] , gT_n)\\
Y_m = X_{S_m} = ([C , gT_mg^{− 1}] , T_mg )
$$
and give rise to the following commutation relations
$$
\{S_n , S_m\} = − 2ε_{nm}^k S_k\\
\{C_n , C_m\} = 2ε_{nm}^k C_k\\
\{C_n , S_m\} = 0\\
\{C_n , g\} = gT_n\\
\{S_m , g\} = T_mg
$$
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
$$
ġ = \{H , g\} = gR
$$
$$
Ċ = \{H , C\} = 0
$$
Quantization
Let's introduce operators
$$
Ĉ_n = \frac{i}{2}L_{X_n}
$$
$$
Ŝ_m = − \frac{i}{2}L_{Y_m}
$$
They act on the square integrable functions (see Appendix A) on $SU(2)$ and satisfy quantum
commutation 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 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
$Ĥ, Ĉ_a$ and $Ŝ_b$ have the form
$$
Ĥψ_{jsc} = j(j + 1)ψ_{jsc}
$$
where $j$ takes positive integer and half integer values
$$
j = 0, \frac{1}{2}, 1, \frac{3}{2}, 2 ...
$$
$$
Ĉ_aψ_{jsc} = cψ_{jsc}
$$
$$
Ŝ_bψ_{jsc} = sψ_{jsc}
$$
with $c$ and $s$ taking values in the following range
$$
− j, − j + 1, ... , j − 1, j
$$
Further we construct the corresponding eigenfunctions
$ψ_{jsc}$. 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
$$
ψ_{jsc} =
Ŝ_{−}^{j − s}Ĉ_{−}^{j − c}〈Tg〉^{2j}
$$ in the manner described in Appendix B.
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
$$g → h(t)g
$$
Where $h(t) ∈ U(1) ⊂ SU(2)$ is an element of $U(1)$. Without loss of generality we can take
$$
h = e^{β(t)T_3}
$$
Since $T_3$ 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
$$
\frac{d}{dt}g → ∇g = (\frac{d}{dt} + B)g
$$
where $B$ can be represented as follows
$$
B = bT_3 ∈ su(2)
$$
with transformation rule
$$
B → hBh^{− 1} − \frac{dh}{dt}h^{− 1}
$$
or in terms of $b$ variable
$$
b → b − \frac{dβ}{dt}
$$
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 $B$ using Lagrange equations
$$
\frac{∂L_G}{∂B} → b = − 〈ġg^{− 1}T_3〉
$$
put it back in
(45) and rewrite last obtained Lagrangian in terms of gauge invariant variables.
$$
L_G = 〈(g^{− 1}ġ − S_3T_3)^2〉
$$
It's obvious that the following
$$
Z = g^{− 1}T_3g ∈ su(2)
$$
element of $su(2)$ algebra is gauge invariant. Since $Z ∈ su(2)$ it can be parameterized as follows
$$
Z = z^aT_a
$$
where $z^a$ are real functions on $SU(2)$
$$
z_a = 〈ZT_a〉
$$
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
$$
〈Z^2〉 = 〈g^{− 1}T_3gg^{− 1}T_3g〉 = 〈T_3^2〉 = 1
$$
otherwise
$$
〈Z^2〉 = 〈z^aT_az^bT_b〉 = z^az_a
$$
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
$$
L_G = ¼〈Z^{− 1}ŻZ^{− 1}Ż〉
$$
This Lagrangian describes free particle on the sphere. Indeed,
since $Z = z^aT_a$ it's easy to show that
$$
L_G = ¼〈Z^{− 1}ŻZ^{− 1}Ż〉 =
¼〈ZŻZŻ〉 = ½ż^aż_a
$$
So $SU(2)/U(1)$ coset model describes free particle on $S^2$ manifold.
Quantization of the coset model.
Working in a first order Hamiltonian formalism one can introduce equivalent Lagrangian
$$
Λ_G = 〈C(g^{− 1}ġ − u)〉 + ½ 〈(u + g^{− 1}Bg)^2〉
$$
variation of $u$ provides
$$
C = u + g^{− 1}Bg \\
u = C − g^{− 1}Bg
$$
Rewriting $Λ_G$ in terms of $C$ and $g$ leads to
$$
Λ_G = 〈Cg^{− 1}ġ〉 − ½ 〈C^2〉 − 〈BgCg^{− 1}〉 = 〈Cg^{− 1}ġ〉 \\
− ½ 〈C^2〉 − b〈gCg^{− 1}T_3〉 =
〈Cg^{− 1}ġ〉 − ½ 〈C^2〉 − bS_3
$$
Due to the gauge invariance of $Λ_G$ we obtain constrained Hamiltonian system,
where $〈Cg^{− 1}dg〉$ is symplectic potential, function
$$
H =
½〈C^2〉
$$ plays the role of Hamiltonian and
$b$ is a Lagrange multiple leading to the first class constrain
$$
φ = 〈gCg^{− 1}T_3〉 = 〈ST_3〉 = S_3 = 0
$$
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 $SU(2)$,
to the following operator constrain
$$
Ŝ_3|ψ〉 = 0
$$
Hilbert space of the initial system, that is linear span of
$$
ψ_{jcs} j = 0, \frac{1}{2}, 1, \frac{3}{2}, 2, ...
$$
wave functions, reduces to
the linear span of
$$
ψ_{jc0} j = 0, 1, 2, 3, ...
$$
wave functions. Indeed,
$Ŝ_3ψ_{jcs} = 0$ 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
$$
ψ_{jc0} ∼ J_{jc}
$$
One can check the following
$$
ψ_{jc0} ∼ Ŝ_{−}^jĈ_{−}^{j − c} 〈Tg〉^{2j}
∼ Ĉ_{−}^{j − c} 〈T_{+}g^{− 1}T_3g〉^j\\
∼ Ĉ_{−}^{j − c}sin^jθe^{ijθ}
∼ Ĉ_{−}^{j − c}J_{jj} ∼J_{jc}
$$
This is an example of using large initial model in quantization of
coset model.
Appendix A
Scalar product in Hilbert space is defined as follows
$$
〈ψ_1|ψ_2〉 =
\stackrev{∫}{SU(2)}
\stackrev{\stackrel{3}{∏}}{a = 1}
〈g^{− 1}dgT_a〉(ψ_1)^{†}ψ_2
$$
It's easy to prove that under this scalar product operators
$Ĉ_n$ and $Ŝ_m$ are hermitian.
Indeed
$$
〈ψ_1|Ĉ_nψ_2〉 =
\stackrev{∫}{SU(2)}
\stackrev{\stackrel{3}{∏}}{a = 1}
〈g^{− 1}dgT_a〉(ψ_1)^{†}
(\frac{i}{2}L_{X_n}ψ_2) \\
= \stackrev{∫}{SU(2)}
\stackrev{\stackrel{3}{∏}}{a = 1}
〈g^{− 1}dgT_a〉(\frac{i}{2}L_{X_n}ψ_1)^{†}ψ_2
$$
Where integration by part has been used and the additional term coming from measure
$$
\stackrev{\stackrel{3}{∏}}{a = 1} 〈g^{− 1}dgT_a〉
$$
vanished since
$$
L_{X_n}〈g^{− 1}dgT_a〉 = 0
$$
For more transparency one can introduce the following parameterization of
$SU(2)$. For any $g ∈ SU(2)$.
$$
g = e^{q^aT_a}
$$
Then the symplectic potential takes the form
$$
〈Cg^{− 1}dg〉 = C_adq^a
$$
and scalar product becomes
$$
〈ψ_1|ψ_2〉 =
\stackrev{\stackrel{2π}{∫}}{0}
\stackrev{\stackrel{2π}{∫}}{0}
\stackrev{\stackrel{2π}{∫}}{0}
d^3q(ψ_1)^{†}ψ_2
$$
that coincides with
(65) because of
$$
dq_a = 〈g^{− 1}dg T_a〉
$$
Appendix B
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
$$
Ĥψ = Eψ\\
Ĉ_3ψ = cψ\\
Ŝ_3ψ = sψ
$$
it is easy to show that eigenvalues of $Ĥ$ are non-negative $E ≥ 0$
and conditions
$$
E − c^2 ≥ 0\\
E − s^2 ≥ 0
$$
are satisfied. Indeed, operators $Ĉ$ and $Ŝ$ are selfadjoint so
$$
〈ψ|Ĥ|ψ〉 = 〈ψ|Ĉ^2|ψ〉 = 〈ψ|Ĉ_aĈ^a|ψ〉 =
〈ψ|(Ĉ_a)^{†}Ĉ^a|ψ〉 =\\
〈Ĉ_aψ|Ĉ^aψ〉 = ∥Ĉ_aψ∥ ≥ 0
$$
To prove
(74) we shall consider
$Ĉ_1^2 + Ĉ_2^2$ and
$Ŝ_1^2 + Ŝ_2^2$ 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 $E − c^2 ≥ 0$.
Now let's introduce new operators
$$
Ĉ_{+} = iĈ_1 + Ĉ_2 Ĉ_{−} =
iĈ_1 − Ĉ_2
$$
$$
Ŝ_{+} = iŜ_1 + Ŝ_2 Ŝ_{−} =
iŜ_1 − Ŝ_2
$$
These operators are not selfadjoint, but $(Ĉ_{−})^{†} = Ĉ_{+}$ and
$(Ŝ_{−})^{†} = Ŝ_{+}$
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 $•$ 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 :
$$
Ĥψ_{λcs} = λψ_{λcs}\\
Ŝ_3ψ_{λcs} = sψ_{λcs}\\
Ĉ_3ψ_{λcs} = cψ_{λcs}
$$
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
$$
c, c ± 1, c ± 2, c ± 3, ...\\
s, s ± 1, s ± 2, s ± 3, ...
$$
but conditions
(74) give restrictions on a possible range of eigenvalues.
Namely we must have
$$
λ − c^2 ≥ 0\\
λ − s^2 ≥ 0
$$
In other words, in order to interrupt
(84) sequences we must assume
$$
Ŝ_{+} ψ_{λcj} = 0 Ŝ_{−}ψ_{λc, − j} = 0\\
Ĉ_{+}ψ_{λks} = 0 Ĉ_{−}ψ_{λ, − ks} = 0
$$
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 values
should be integer, $j$ and $k$ should take integer or half integer values
$$
j = 0, \frac{1}{2}, 1, \frac{3}{2}, 2, ...\\
k = 0, \frac{1}{2}, 1, \frac{3}{2}, 2, ...
$$
Now using commutation relations we can rewrite $Ĥ$ in terms of
$Ĉ_{±}, Ĉ_3$ operators
$$
Ĥ = Ĉ_{+} Ĉ_{−} + Ĉ_3^2 + Ĉ_3
$$
and it is clear that $j = k$ and $λ = j(j + 1) = k(k + 1)$
References
-
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