4.3 Notes on Other Examples
67
corresponding to phase spaces of particles with nonvanishing masses have the same
topological and symplectic structure as in the case (i). The action Ad
∗
(G) is, however, different from that of the Galilean case; with this are connected also different
interpretations of coordinates determined by the mutually corresponding generators
in cases (i) and (ii). The dimensionality of orbits O ϕ of unitary irreducible representations U(G) corresponding to nonzero masses is the same as in (i). Also here,
we obtain 7- and 9-dimensional contact manifolds the contact two-form
◦ on them
coincides with the standard two-form of classical relativistic mechanics (which, in
the case of dim O ϕ = 7, comes from the restriction of d p μ ∧ dq
μ defined on T
∗
R
4
onto the submanifold p
2
0 −
j p
2
j = (mass)
2
).
4.3.5 Remark. Any symplectic manifold can be trivially extended to a contact
manifold by taking the direct product with R. If M is a symplectic phase space of some
physical system, then the added dimension in R × M can be interpreted as the ‘time
variable’ t. Let be the symplectic form on M, π : R × M → M be the canonical
projection and σ A the Hamiltonian vector field on M with Hamiltonian function f A ,
i.e. i(σ A )) = −d f A . The contact two-forms
◦
:= π
∗
, resp.
A
:=
◦
− d f A ∧
dt on the manifold R × M have characteristic vector fields δ t (defined by dt (δ t ) = 1
and d f (δ t ) = 0 for any function f of the form f := π
∗ f
, where f
∈ C
∞
(M)),
resp. σ
◦
A := π
∗
σ A + δ t (with the identification T (R × M) = T R × T M in the sense
of vector bundle isomorphisms). Clearly π ∗ σ
◦
A = σ A . For a time-independent vector
field σ A this procedure is trivial, if we have no possibility to distinguish various points
of the fibres R = π
−l
(x) (x ∈ M) by some measurements, i.e. if time is homogeneous
with respect to the considered physical system. This is the case of classical mechanics
determined by (M; ) and f A ∈ C
∞
(M).
The situation is different for contact orbits O ϕ ⊂ P(H). Each point of O ϕ corresponds to a quantum mechanically clearly distinguishable physical state: by measuring of also quantities other than expectations of generators of U (G), we can
empirically distinguish various points of the same fibre, on which all the expectations of the generators in U (g) are constant. This fact breaks, in a certain sense, the
homogeneity of time on contact orbits of the representations, which contain also time
evolution of the system as a one parameter subgroup.
4.3.6 Identical particles.
If the physical system consists of N mutually distinguishable, but otherwise equal
subsystems, it is described in QM by the N -fold tensor product Hilbert space H N :=
H ⊗ H ⊗ · · · ⊗ H with the Hilbert space H describing a single subsystem. If the
‘basic observables’ of a single subsystem are determined by a representation U (G)
in H, observables of the whole compound system might be determined by the representation U N of the N -fold direct product group G N := G × G × · · · × G, i.e. for
ϕ := ϕ 1 ⊗ ϕ 2 ⊗ · · · ⊗ ϕ N ∈ H N , ϕ j ∈ H, we set U N (g 1 × g 2 × · · · × g N )ϕ :=
U (g 1 )ϕ 1 ⊗ U (g 2 )ϕ 2 ⊗ . . . U (g N )ϕ N for all g j ∈ G, and extend U N onto H N by
linearity and continuity. This is the case, e.g. of the example in Sect. 4.1. Then we
can construct in the usual way orbits O ϕ := U N (G N )ϕ in P(H N ) and corresponding
symplectic manifolds M ϕ . We shall write also U (G N ) := U N (G N ).
67
corresponding to phase spaces of particles with nonvanishing masses have the same
topological and symplectic structure as in the case (i). The action Ad
∗
(G) is, however, different from that of the Galilean case; with this are connected also different
interpretations of coordinates determined by the mutually corresponding generators
in cases (i) and (ii). The dimensionality of orbits O ϕ of unitary irreducible representations U(G) corresponding to nonzero masses is the same as in (i). Also here,
we obtain 7- and 9-dimensional contact manifolds the contact two-form
◦ on them
coincides with the standard two-form of classical relativistic mechanics (which, in
the case of dim O ϕ = 7, comes from the restriction of d p μ ∧ dq
μ defined on T
∗
R
4
onto the submanifold p
2
0 −
j p
2
j = (mass)
2
).
4.3.5 Remark. Any symplectic manifold can be trivially extended to a contact
manifold by taking the direct product with R. If M is a symplectic phase space of some
physical system, then the added dimension in R × M can be interpreted as the ‘time
variable’ t. Let be the symplectic form on M, π : R × M → M be the canonical
projection and σ A the Hamiltonian vector field on M with Hamiltonian function f A ,
i.e. i(σ A )) = −d f A . The contact two-forms
◦
:= π
∗
, resp.
A
:=
◦
− d f A ∧
dt on the manifold R × M have characteristic vector fields δ t (defined by dt (δ t ) = 1
and d f (δ t ) = 0 for any function f of the form f := π
∗ f
, where f
∈ C
∞
(M)),
resp. σ
◦
A := π
∗
σ A + δ t (with the identification T (R × M) = T R × T M in the sense
of vector bundle isomorphisms). Clearly π ∗ σ
◦
A = σ A . For a time-independent vector
field σ A this procedure is trivial, if we have no possibility to distinguish various points
of the fibres R = π
−l
(x) (x ∈ M) by some measurements, i.e. if time is homogeneous
with respect to the considered physical system. This is the case of classical mechanics
determined by (M; ) and f A ∈ C
∞
(M).
The situation is different for contact orbits O ϕ ⊂ P(H). Each point of O ϕ corresponds to a quantum mechanically clearly distinguishable physical state: by measuring of also quantities other than expectations of generators of U (G), we can
empirically distinguish various points of the same fibre, on which all the expectations of the generators in U (g) are constant. This fact breaks, in a certain sense, the
homogeneity of time on contact orbits of the representations, which contain also time
evolution of the system as a one parameter subgroup.
4.3.6 Identical particles.
If the physical system consists of N mutually distinguishable, but otherwise equal
subsystems, it is described in QM by the N -fold tensor product Hilbert space H N :=
H ⊗ H ⊗ · · · ⊗ H with the Hilbert space H describing a single subsystem. If the
‘basic observables’ of a single subsystem are determined by a representation U (G)
in H, observables of the whole compound system might be determined by the representation U N of the N -fold direct product group G N := G × G × · · · × G, i.e. for
ϕ := ϕ 1 ⊗ ϕ 2 ⊗ · · · ⊗ ϕ N ∈ H N , ϕ j ∈ H, we set U N (g 1 × g 2 × · · · × g N )ϕ :=
U (g 1 )ϕ 1 ⊗ U (g 2 )ϕ 2 ⊗ . . . U (g N )ϕ N for all g j ∈ G, and extend U N onto H N by
linearity and continuity. This is the case, e.g. of the example in Sect. 4.1. Then we
can construct in the usual way orbits O ϕ := U N (G N )ϕ in P(H N ) and corresponding
symplectic manifolds M ϕ . We shall write also U (G N ) := U N (G N ).