72
4 Examples of Classical Mechanical Projections
plectic manifold. This means, that Z ϕ± for both signs are locally diffeomorphic (and
symplectomorphic) to M ϕ = O ϕ = R
2Nn
⊂ P(H N ) (Sect. 4.1). In a neighbourhood
of ϕ
∈ O ϕ , the functions ϕ
→ T r(P ϕ X
j
k ) ( j = 1, 2, . . . N ; k = 1, 2, . . . 2n) are
symplectic coordinates. Similarly, in a neighbourhood of s
ϕ
± (ϕ) the functions
f
j
k : s
ϕ
± (ϕ
) → T r(P ϕ X
j
k ) = (ϕ
j , X
j
k ϕ
j ), j = 1, 2, . . . N ; k = 1, 2, . . . 2n,
(4.3.34)
are symplectic coordinate functions on Z ϕ± .
Let us assume now, that ϕ j ’s in (4.3.31) have the form
ϕ j := W x ( j) ϕ 0 for some ϕ 0 ∈ L
2
(R
n
), x
( j)
∈ R
n
,
(4.3.35)
assuming ϕ 0 to be smooth with compact support, and x
( j)
= x
(k)
( j = k) such that
ϕ j , ϕ k have mutually disjoint supports, see 4.1.3 for the notation. On the orbit
O ϕ
in H N , there is also the point
(⊗ϕ 0 )
N
:= ϕ 0 ⊗ ϕ 0 ⊗ · · · ⊗ ϕ 0 .
(4.3.36)
Choose now ϕ equal to (4.3.36) and calculate the values of (4.3.28) in the
points ϕ ± ∈ P(H
±
N ). In the antisymmetric case we obtain zero, since P − ϕ =: ϕ − =
0 (hence ϕ ∈
O
◦
ϕ− , (4.3.20), and P P − ϕ is not defined).
In the case of Bose statistics we have:
ϕ (+) (v η , v ξ ) =
i
N
N
j=1
N
k=1
(ϕ 0 , [X
k
η , X
j
ξ ]ϕ 0 ),
(4.3.37)
where X
j
η ( j = 1, 2, . . . N ) should be considered as operators in L
2
(R
n
), ignoring
the definition (4.3.32) : they act on L
2
(R
n
) regardless of its order in the tensor product
forming the whole Hilbert space H N . The rank of the form (4.3.37) equals to 2n and
the point P P
ϕ
+ (ϕ) does not belong to i
ϕ
+ (Z ϕ+ ) for N ≥ 2, i.e. ϕ is not mapped by s
ϕ
+
into the symplectic manifold Z ϕ+ . We see that, although locally symplectomorhic to
R
2Nn , the both classical phase spaces Z ϕ− and Z ϕ+ of identical particles are globally
different from the standard cotangent bundle T
∗
R
Nn
: in classical projections the
Pauli exclusion principle holds for identical particles, regardless to the kind of their
statistics.
4.3.8 With the notation from 4.3.6, let V N (G) be the unitary representation of G
in H N (reducible for N ≥ 2) defined as the diagonal part of U N :
V N (g) := U N (g × g × · · · × g), for all g ∈ G.
(4.3.38)
The Lie algebra V N (g) is generated by the basis of the form (4.3.30) with X
j
ξ =
X
k
ξ (considered as operators in H) for all j, k = 1, 2, . . . N , ξ ∈ g. Such operators
4 Examples of Classical Mechanical Projections
plectic manifold. This means, that Z ϕ± for both signs are locally diffeomorphic (and
symplectomorphic) to M ϕ = O ϕ = R
2Nn
⊂ P(H N ) (Sect. 4.1). In a neighbourhood
of ϕ
∈ O ϕ , the functions ϕ
→ T r(P ϕ X
j
k ) ( j = 1, 2, . . . N ; k = 1, 2, . . . 2n) are
symplectic coordinates. Similarly, in a neighbourhood of s
ϕ
± (ϕ) the functions
f
j
k : s
ϕ
± (ϕ
) → T r(P ϕ X
j
k ) = (ϕ
j , X
j
k ϕ
j ), j = 1, 2, . . . N ; k = 1, 2, . . . 2n,
(4.3.34)
are symplectic coordinate functions on Z ϕ± .
Let us assume now, that ϕ j ’s in (4.3.31) have the form
ϕ j := W x ( j) ϕ 0 for some ϕ 0 ∈ L
2
(R
n
), x
( j)
∈ R
n
,
(4.3.35)
assuming ϕ 0 to be smooth with compact support, and x
( j)
= x
(k)
( j = k) such that
ϕ j , ϕ k have mutually disjoint supports, see 4.1.3 for the notation. On the orbit
O ϕ
in H N , there is also the point
(⊗ϕ 0 )
N
:= ϕ 0 ⊗ ϕ 0 ⊗ · · · ⊗ ϕ 0 .
(4.3.36)
Choose now ϕ equal to (4.3.36) and calculate the values of (4.3.28) in the
points ϕ ± ∈ P(H
±
N ). In the antisymmetric case we obtain zero, since P − ϕ =: ϕ − =
0 (hence ϕ ∈
O
◦
ϕ− , (4.3.20), and P P − ϕ is not defined).
In the case of Bose statistics we have:
ϕ (+) (v η , v ξ ) =
i
N
N
j=1
N
k=1
(ϕ 0 , [X
k
η , X
j
ξ ]ϕ 0 ),
(4.3.37)
where X
j
η ( j = 1, 2, . . . N ) should be considered as operators in L
2
(R
n
), ignoring
the definition (4.3.32) : they act on L
2
(R
n
) regardless of its order in the tensor product
forming the whole Hilbert space H N . The rank of the form (4.3.37) equals to 2n and
the point P P
ϕ
+ (ϕ) does not belong to i
ϕ
+ (Z ϕ+ ) for N ≥ 2, i.e. ϕ is not mapped by s
ϕ
+
into the symplectic manifold Z ϕ+ . We see that, although locally symplectomorhic to
R
2Nn , the both classical phase spaces Z ϕ− and Z ϕ+ of identical particles are globally
different from the standard cotangent bundle T
∗
R
Nn
: in classical projections the
Pauli exclusion principle holds for identical particles, regardless to the kind of their
statistics.
4.3.8 With the notation from 4.3.6, let V N (G) be the unitary representation of G
in H N (reducible for N ≥ 2) defined as the diagonal part of U N :
V N (g) := U N (g × g × · · · × g), for all g ∈ G.
(4.3.38)
The Lie algebra V N (g) is generated by the basis of the form (4.3.30) with X
j
ξ =
X
k
ξ (considered as operators in H) for all j, k = 1, 2, . . . N , ξ ∈ g. Such operators