64
4 Examples of Classical Mechanical Projections
y
k
(exp(−itτ
j Y j )ϕ) = y
k
(ϕ) cos t + k jm τ
j y
m
(ϕ) sin t
+2τ
k
τ · y(ϕ) sin
2 t
2
,
(4.3.6)
what gives an explicit expression for the sphere S
2
⊂ so(3)
∗
. The r ϕ := radius of
the sphere is equal to the length of y(ϕ),
|y(ϕ)|
2
= y(ϕ) · y(ϕ) = r
2
ϕ .
(4.3.7)
In the case (i) the values of (4.3.7) might be only the numbers J
2
, (J − 1)
2
, (J −
2)
2
, . . . , i.e. the orbits O ϕ ⊂ P(H) are mapped by the association ϕ → F ϕ (cf.
(4.3.2)) onto a finite-number of [J + 1] distinct spheres in the three-dimensional
linear space so(3)
∗ (here [k] is the integer part of k ∈ R + ; if J ∈ Z + one of the
spheres degenerates into a point). But P(H) is a connected manifold and the mapping
ϕ → F ϕ is continuous, hence for J ≥ 1 also the cases (ii) occur and the numbers
(4.3.7) acquire values from a whole interval of R + , if ϕ runs over P(H).
Let us write explicitly the symplectic form
M on the phase space M ϕ = S
2
.
In terms of coordinate functions y k from (4.3.5), we obtain in the region where
y 3 (ϕ) = 0 (indices are written down for convenience):
M
= −
1
y 3
dy 1 ∧ dy 2 , y
2
3 := r
2
ϕ − y
2
1 − y
2
2 .
(4.3.8)
The Poisson bracket of these coordinate functions is
{y k , y m } = − km j y j .
(4.3.9)
The sphere S
2 with this symplectic structure is interpreted as the phase space of
an (isolated) classical spin. It is an example of a compact symplectic manifold.
4.3.3 We can construct now certain combinations of the previous example with
those of Sects. 4.1 and 4.2. Let us distinguish generators X j of the representation of
6N + 1—dimensional Heisenberg group corresponding to coordinates of positions
and momenta of N individual particles. Denote them Q
a
j , P
a
j (a = 1, 2, . . . N ; j =
1, 2, 3) with CCR in the form
[Q
a
j , P
b
k ] = i Iδ ab δ jk , [Q
a
j , Q
b
k ] = [P
a
j , P
b
k ] = 0,
(4.3.10)
for all a, b = 1, 2, . . . N ; j, k = 1, 2, 3. Now we define operators of orbital momenta
(no summation over repeated indices a, b):
Y
a
j := jkm Q
a
k P
a
m , Y j := Y
tot
j :=
a
Y
a
j
(4.3.11)
satisfying (4.3.1) (up to domain specifications) for any upper index (a, or tot).
Relations (4.2.2) have now the form:
4 Examples of Classical Mechanical Projections
y
k
(exp(−itτ
j Y j )ϕ) = y
k
(ϕ) cos t + k jm τ
j y
m
(ϕ) sin t
+2τ
k
τ · y(ϕ) sin
2 t
2
,
(4.3.6)
what gives an explicit expression for the sphere S
2
⊂ so(3)
∗
. The r ϕ := radius of
the sphere is equal to the length of y(ϕ),
|y(ϕ)|
2
= y(ϕ) · y(ϕ) = r
2
ϕ .
(4.3.7)
In the case (i) the values of (4.3.7) might be only the numbers J
2
, (J − 1)
2
, (J −
2)
2
, . . . , i.e. the orbits O ϕ ⊂ P(H) are mapped by the association ϕ → F ϕ (cf.
(4.3.2)) onto a finite-number of [J + 1] distinct spheres in the three-dimensional
linear space so(3)
∗ (here [k] is the integer part of k ∈ R + ; if J ∈ Z + one of the
spheres degenerates into a point). But P(H) is a connected manifold and the mapping
ϕ → F ϕ is continuous, hence for J ≥ 1 also the cases (ii) occur and the numbers
(4.3.7) acquire values from a whole interval of R + , if ϕ runs over P(H).
Let us write explicitly the symplectic form
M on the phase space M ϕ = S
2
.
In terms of coordinate functions y k from (4.3.5), we obtain in the region where
y 3 (ϕ) = 0 (indices are written down for convenience):
M
= −
1
y 3
dy 1 ∧ dy 2 , y
2
3 := r
2
ϕ − y
2
1 − y
2
2 .
(4.3.8)
The Poisson bracket of these coordinate functions is
{y k , y m } = − km j y j .
(4.3.9)
The sphere S
2 with this symplectic structure is interpreted as the phase space of
an (isolated) classical spin. It is an example of a compact symplectic manifold.
4.3.3 We can construct now certain combinations of the previous example with
those of Sects. 4.1 and 4.2. Let us distinguish generators X j of the representation of
6N + 1—dimensional Heisenberg group corresponding to coordinates of positions
and momenta of N individual particles. Denote them Q
a
j , P
a
j (a = 1, 2, . . . N ; j =
1, 2, 3) with CCR in the form
[Q
a
j , P
b
k ] = i Iδ ab δ jk , [Q
a
j , Q
b
k ] = [P
a
j , P
b
k ] = 0,
(4.3.10)
for all a, b = 1, 2, . . . N ; j, k = 1, 2, 3. Now we define operators of orbital momenta
(no summation over repeated indices a, b):
Y
a
j := jkm Q
a
k P
a
m , Y j := Y
tot
j :=
a
Y
a
j
(4.3.11)
satisfying (4.3.1) (up to domain specifications) for any upper index (a, or tot).
Relations (4.2.2) have now the form: