Berkala MIPA, 173, September 2007
36
polarization is referred to as product polarization. The structure of quantizable
algebras of symplectic products of two symplectic manifolds with respect to the
product polarization in the sense of the geometric quantization is studied. The
structure of the so called GQ-consistent kinematical algebra of symplectic product of
symplectic manifolds is investigated. The above mentioned scheme can be applied for
instance to a physical system consisting of a massive particle with spin.
2. GQ-CONSISTENT
KINEMATICAL ALGEBRA
Let M, ω
be a quantizable symplectic manifold of dimension 2n and P
a strong integrable complex polarization so that 0
≤ dimD
≤ n, where D
C
= P ∩
P is
the isotropic distribution associated with Psee Rosyid, 2003. In the -12+i
γ -P-
densities quantization, a function f ∈
C
∞
M,R is said to be quatizable if the Hamiltonian vector field X
f
generated by f preserves the polarization P in the sense
2
of [X
f
,P] ⊂
P. Let
F
P
M,R be the set of all real quantizable functions and
F
M,R;D := {f ∈
F
P
M,R |[X
f
, D] ⊂
D} as the set of all quantizable functions
preserving the distribution D.
Let Z be a vector field on M. The maping
π
D
Z defined by π
D
Z π
D
m = π
D
|
m
Z determines a smooth vector field on MD if and only if Z preserves the
distribution D, i.e. [Z,D] ⊂
D see Rosyid, 2005c. It is clear therefore that a
quantizable function f belongs to
F
M, R; D if and only if there exists a differentiable
vector field X on MD, so that X
f
and X are
2
The expression [X
f
, P] ⊂
P means that [X
f
,Y] is a cross section of P for every cross section Y of P. A
cross section Y of P is a vector field on M so that Ym ∈
Pm for every m ∈
M.
π
D
-related, i.e. π
D
X
f
is a differentiable vector field on MD.
The set
X
M;D of all vector fields on M preserving the distribution D forms a Lie
subalgebra of
X
M, the set of all smooth vector fields on M. Now let
X
F
M;D denote the set of all Hamiltonian vector fields on M generated by
the functions in the set
F
M,R;D and
F D
∗
π
be defined as the restriction of the Lie
homomorphism
∗ D
π
:
X
M;D →
X
MD to
X
F
M;D. The set
X
F
M;D is clearly a Lie subalgebra of
X
M;D and
F D
∗
π
is a Lie homomophism.
Let
X
F
~
M;D denote the quotient algebra of
X
F
M;D relative to the kernel
K
F D
∗
π
of
F D
∗
π
and [.,.]
~
be the quotient bracket in
X
F
~
M;D. If
X
F
~ c
M;D is the subset of
X
F
~
M;D defined by
X
F
~ c
M;D = {[X
f
] ∈
X
F
~
M;D |
∗ D
π
[X
f
] ∈
X
c
MD}, where
X
c
MD is the set of all complete vector fields on MD, then, in general, the identity
∗ D
π
X
F
~
M;D =
X
c
MD, is not respected, where
∗ D
π
[X
f
] is defined as
∗ D
π
X
f’
for arbitrary X
f’
∈ [X
f
]. Furthermore, the maping
c D
∗
π
:
X
F
~ c
M;D →
X
c
MD which is defined by
c D
∗
π
[X
f
] =
∗ D
π
X
f’
for arbitrary X
f’
∈ [X
f
] is an injective partial Lie homomorphism.
Define now
X
as the assignment X
f
∈
X
F
M;D to f in
F
M,R;D and let
K
F D
∗
π
denote the kernel of the Lie homomorphism
F D
∗
π
in the set
X
F
M;D. Further, let
F
K
M,R;D be the inverse image of
K
F D
∗
π
under the maping
X
. If g is contained in
F
K
M,R;D, then
F D
∗
π
X
g
vanishes. Therefore, there exists uniquely a real smooth function
ζ
g
∈ C
∞
MD,R so that g =
ζ
g
? π
D
. However, the converse is in
Rosyid, M.F., On the Structure of Quantizable Algebras
37
general not correct. If C
∞ D
M,R is the set of all real smooth functions on M which are
constant on every leaf of D and P is real so that D = D
-
, then
F
K
M,R;D = C
∞ D
M,R. Since, C
∞ D
M,R is equal to
∗ D
π
C
∞
MD,R := {
ζ ?
π
D
∈ C
∞
M,R| ζ
∈ C
∞
MD,R}, it follows also that
F
K
M,R;D =
∗ D
π
C
∞
MD,R. Now let
Ξ :
F
K
M,R;D →
C
∞
MD,R be the injection defined by
Ξ g =
ζ
g
and define for every equivalent class [X
f
] ∈
X
F
~ c
M;D a linear operator
] [
f
X
L
on the algebra
F
K
M,R;D so that
] [
f
X
L
= X
f’
g,
for every g ∈
F
K
M,R;D and arbitrary X
f’
∈ [X
f
]. Let
S
GQ
M;D be the semidirect sum
F
K
M,R;D ⊕
s
X
F
~ c
M;D with Lie bracket [
⋅ ,
⋅ ]
s
defined by [g
1
,[X
f
], g
2
,[X
f’
]]
s
=
] [
f
X
L
g
2
−
] [
f
X
L
g
1
, [[X
f
],[X
f’
]]
~
, for all [X
f
], [X
f’
] ∈
X
F
~ c
M;D with [[X
f
],[X
f’
]]
~
∈
X
F
~ c
M;D and all g
1
, g
2
∈
F
K
M,R;D. The pair
S
GQ
M;D, [ ⋅
, ⋅
]
s
is called GQ-consistent kinematical algebra in
M, ω
,P. Define a maping, denoted by
Ξ⊕
s c
D ∗
π
, from the algebra
S
GQ
M;D into the kinematical algebra
S
MD = C
∞
MD,R ⊕
s
X
c
MD through Ξ⊕
s c
D ∗
π
g, [X
f
] = Ξ
g,
c D
∗
π
[X
f
], for every g, [X
f
] ∈
S
GQ
M;D. The maping Ξ⊕
s c
D ∗
π
is clearly a partial Lie homomor- phism. The GQ-consistent kinematical algebra
S
GQ
M;D is said to be almost complete whenever the maping
Ξ⊕
s c
D ∗
π
is a partial Lie isomorphism. Furthermore,
S
GQ
M;D is said to be complete if it is almost complete and
X
F
~ c
M;D is equal to
X
F
~
M;D. The ideal
F
K
M,R;D in the algebra
S
GQ
M;D is associated with the localization of the physical system or particle in its
configuration space. The n, the elements of
F
K
M,R;D represent position of the physical system.
Symplectic Product
Let M
1
, ω
1
and M
2
, ω
2
be two symplectic manifolds of dimension 2n
1
and 2n
2
respectively. Furthermore, let π
1
: M
1
× M
2
→ M
1
and π
2
: M
1
× M
2
→ M
2
be the canonical projections.
Definition 1 : The symplectic manifold M,
ω ,
where M = M
1
× M
2
and ω
= π
1 ∗
ω
1
+ π
2 ∗
ω
2
is
called symplectic product of M
1
, ω
1
and M
2
, ω
2
. Let P
1
be a reducible polarization on M
1
, ω
1
and P
2
a reducible polarization on M
2
, ω
2
. Let P
1
× P
2
denote the distribution defined by
2 1
2 1
| |
|
2 1
, 2
1 m
m m
m
P P
P P
⊕ =
×
,
for every m
1
, m
2
∈ M.
Proposition 1 : The distribution P
1
× P
2
is a reducible polarization of M = M
1
× M
2
. The polarization is referred to as
product polarization
. Now we mention an example of
physical systems which admits the above structure. Let G denote the Poincaré group, i.e.
the symmetry group of relativistic elementary particles and
G
the Lie algebra of G. The algebra
G
is spanned by the generators {P
ν
; ν
= 0,1,2,3} of translations and {J
µν
; µ
, ν
= 0,1,2,3}
Berkala MIPA, 173, September 2007
38
of Lorentz transformations. Define now P
2
and w
2
as P
2
= η
µν
P
µ
P
ν
and w
2
= η
µν
w
µ
w
ν
, where w
ρ
= 12 ε
ρ µντ
J
µν
P
τ
and η
µν
is the Lorentz metric. The orbit of G in
G
which plays the role of classical phase space of
relativistic elementary particles is characterized by P
2
and w
2
see
Landsman, 1998; Simms and Woodhouse, 1976
. If P
2
and w
2
does not vanish, i.e. if the particles under consideration are massive, the orbit of
G in
G
is diffeomorphic to R
6
× S
2
. The canonical symplectic structure
ω
G
on each orbit is given by Def.1, i.e.
ω
G
= π
R ∗
ω
R
+ π
S ∗
ω
S
, where ω
R
and ω
S
is the canonical symplectic structure of R
6
and S
2
, respectively.
3. GQ-CONSISTENT KINEMATICAL ALGEBRA IN
