68
4 Examples of Classical Mechanical Projections
In physics, however, ‘equal (micro-)subsystems’ are indistinguishable. If the N
subsystems are indistinguishable (identical), then for any permutation π ∈ N (:=
the permutation group of N elements) the product-vectors ϕ := ϕ 1 ⊗ ϕ 2 ⊗ · · · ⊗
ϕ N and π · ϕ := ϕ π(1) ⊗ ϕ π(2) ⊗ · · · ⊗ ϕ π(N ) , as well as their linear combinations
(the permutations π ∈ N act here also as linear operators on H N ) are physically
indistinguishable. There were discovered in the particle and statistical physics two
kinds of particles: Bose particles—bosons (e.g. photons, mesons) specified by
their integer particle spin, and Fermi particles— fermions (e.g. electrons, protons,
neutrinos) having half-integer spins. Collections of N identical particles of each
of these kinds behave according of their own specific ‘statistics’: Bose, resp. Fermi
statistics. The two ‘statistics’ are formalized by two different symmetry properties
of multiparticle wave functions of corresponding collections of particles. In the case
of Bose (resp. Fermi) statistics the only physically realizable states correspond to
totally symmetric (resp. totally antisymmetric) vectors ϕ ∈ H N :
π · ϕ = + (π)ϕ, + (π) := 1, for all π ∈ N ,
(4.3.17a)
in the case of Bose statistics, resp.
π · ϕ = − (π)ϕ, − (π) := ±1 := parity of π ∈ N .
(4.3.17b)
in the case of Fermi statistics.
4
Let P + (resp. P − ) be the orthogonal projector in H N onto the subspace H
+
N
(resp. H
−
N ) of the totally symmetric (4.3.17a) (resp. totally antisymmetric (4.3.17b))
vectors. Now we intend to project the above mentioned orbits O ϕ ⊂ P(H N ) into
P(H
+
N ), resp. into P(H
−
N ). To make the procedure more transparent we shall divide
it to more steps then it is, perhaps, necessary. For a U (G N )-analytic vector ϕ ∈
H N (ϕ = 0) let
O ϕ := U (G N )ϕ, so that O ϕ := P
O ϕ . We shall denote by P : H
→
P(H
), ϕ → P ϕ , the natural projection in all the cases of H
:= H N , H
+
N , H
−
N . Let
O
+
ϕ := P +
O ϕ ,
O
−
ϕ := P −
O ϕ be subsets of H
+
N (resp. H
−
N ).
(4.3.18)
Assume, for definiteness, that P + ϕ = 0, and concentrate ourselves to the Bosonic
case (the formal procedures are similar with the fermions). Let K
ϕ be the stability
group of ϕ with respect to U (G N ). Considerations similar to those of Sect. 3.1 show
that
O ϕ , as an immersed submanifold of H N , is diffeomorphic to G N /K
ϕ
. We shall
consider
O ϕ with the differentiable manifold structure of G N /K
ϕ
. The restricted
mapping of P + :
P
ϕ
+ :
O ϕ → H
+
N , ψ → P + ψ, ψ ∈
O ϕ ,
(4.3.19)
is (infinitely) differentiable. Hence the set
4 This relation between spin and statistics can be obtained as a consequence of mathematical axiomatics of relativistic quantum field theory, cf. e.g. [301].
4 Examples of Classical Mechanical Projections
In physics, however, ‘equal (micro-)subsystems’ are indistinguishable. If the N
subsystems are indistinguishable (identical), then for any permutation π ∈ N (:=
the permutation group of N elements) the product-vectors ϕ := ϕ 1 ⊗ ϕ 2 ⊗ · · · ⊗
ϕ N and π · ϕ := ϕ π(1) ⊗ ϕ π(2) ⊗ · · · ⊗ ϕ π(N ) , as well as their linear combinations
(the permutations π ∈ N act here also as linear operators on H N ) are physically
indistinguishable. There were discovered in the particle and statistical physics two
kinds of particles: Bose particles—bosons (e.g. photons, mesons) specified by
their integer particle spin, and Fermi particles— fermions (e.g. electrons, protons,
neutrinos) having half-integer spins. Collections of N identical particles of each
of these kinds behave according of their own specific ‘statistics’: Bose, resp. Fermi
statistics. The two ‘statistics’ are formalized by two different symmetry properties
of multiparticle wave functions of corresponding collections of particles. In the case
of Bose (resp. Fermi) statistics the only physically realizable states correspond to
totally symmetric (resp. totally antisymmetric) vectors ϕ ∈ H N :
π · ϕ = + (π)ϕ, + (π) := 1, for all π ∈ N ,
(4.3.17a)
in the case of Bose statistics, resp.
π · ϕ = − (π)ϕ, − (π) := ±1 := parity of π ∈ N .
(4.3.17b)
in the case of Fermi statistics.
4
Let P + (resp. P − ) be the orthogonal projector in H N onto the subspace H
+
N
(resp. H
−
N ) of the totally symmetric (4.3.17a) (resp. totally antisymmetric (4.3.17b))
vectors. Now we intend to project the above mentioned orbits O ϕ ⊂ P(H N ) into
P(H
+
N ), resp. into P(H
−
N ). To make the procedure more transparent we shall divide
it to more steps then it is, perhaps, necessary. For a U (G N )-analytic vector ϕ ∈
H N (ϕ = 0) let
O ϕ := U (G N )ϕ, so that O ϕ := P
O ϕ . We shall denote by P : H
→
P(H
), ϕ → P ϕ , the natural projection in all the cases of H
:= H N , H
+
N , H
−
N . Let
O
+
ϕ := P +
O ϕ ,
O
−
ϕ := P −
O ϕ be subsets of H
+
N (resp. H
−
N ).
(4.3.18)
Assume, for definiteness, that P + ϕ = 0, and concentrate ourselves to the Bosonic
case (the formal procedures are similar with the fermions). Let K
ϕ be the stability
group of ϕ with respect to U (G N ). Considerations similar to those of Sect. 3.1 show
that
O ϕ , as an immersed submanifold of H N , is diffeomorphic to G N /K
ϕ
. We shall
consider
O ϕ with the differentiable manifold structure of G N /K
ϕ
. The restricted
mapping of P + :
P
ϕ
+ :
O ϕ → H
+
N , ψ → P + ψ, ψ ∈
O ϕ ,
(4.3.19)
is (infinitely) differentiable. Hence the set
4 This relation between spin and statistics can be obtained as a consequence of mathematical axiomatics of relativistic quantum field theory, cf. e.g. [301].