32
2 Geometry of the State Space of Quantum Mechanics
is densely defined. If, moreover, the operator i[A, B] is selfadjoint and D is its core
5
then, according to (2.3.10), we have
{ f A , f B } = f i[A,B] .
(2.3.18)
Remember that this is a quantummechanical formula corresponding to (1.3.8).
2.3.9 Assume that a weakly continuous unitary representation U of a connected Lie
group G in the Hilbert space H is given:
U (g 1 g 2 ) = U (g 1 )U (g 2 ), g 1 , g 2 ∈ G.
(2.3.19)
Then U is projected onto a weakly continuous realization of G by a group of
symplectic isometries U(g) (g ∈ G) of (P(H), ,). To any element ξ of the Lie
algebra g of G corresponds the selfadjoint generator X ξ of the one-parameter subgroup U (exp(tξ)) :
X ξ x := i
d
dt
t=0
U (exp(tξ))x, x ∈ D(X ξ ),
(2.3.20)
and U (exp(tξ)) = exp(−it X ξ ). By a use of the adjoint representation Ad : G →
L(g),
Ad(g)ξ :=
d
dt
t=0
[g exp(tξ)g
−1
]
we obtain:
[X ξ , X η ] := X ξ X η − X η X ξ = i X [ξ,η] .
(2.3.21)
The mapping ξ → X ξ is linear. It is known (compare [13]), that the Gårding
domain D G , as well as the analytic domain A G of the representation U (G) are
common dense invariant sets of all the generators X ξ (ξ ∈ g) and they are also
common cores of all these selfadjoint operators (cf. also 3.1.1). Let us define the
vector fields σ ξ (ξ ∈ g) on PD G ⊂ P(H) corresponding to the flows U(exp(tξ))
on P(H) according to the definition of σ A in 2.3.5. Let f ξ (x) := T r(P x X ξ ) for
x ∈ D G . Then 2.3.8 is applicable to these quantities. All the formulas of 1.3.7 are
valid on PD G . Difference w.r.t. the classical case is that neither P(H) nor PD G are
homogeneous spaces even for irreducible U(G).
2.3.10 Up to now, we used charts (N x ; x ; [x]
⊥
) for identification of T x P(H) with
[x]
⊥ , and for each point x ∈ P(H) it was used its own chart. Let us rewrite now the
5 A core D ⊂ H of a closable operator C is such a subset D ⊂ D(C) ⊂ H, that the closure of the
restriction C D = C, cf. also [37, C1].
2 Geometry of the State Space of Quantum Mechanics
is densely defined. If, moreover, the operator i[A, B] is selfadjoint and D is its core
5
then, according to (2.3.10), we have
{ f A , f B } = f i[A,B] .
(2.3.18)
Remember that this is a quantummechanical formula corresponding to (1.3.8).
2.3.9 Assume that a weakly continuous unitary representation U of a connected Lie
group G in the Hilbert space H is given:
U (g 1 g 2 ) = U (g 1 )U (g 2 ), g 1 , g 2 ∈ G.
(2.3.19)
Then U is projected onto a weakly continuous realization of G by a group of
symplectic isometries U(g) (g ∈ G) of (P(H), ,). To any element ξ of the Lie
algebra g of G corresponds the selfadjoint generator X ξ of the one-parameter subgroup U (exp(tξ)) :
X ξ x := i
d
dt
t=0
U (exp(tξ))x, x ∈ D(X ξ ),
(2.3.20)
and U (exp(tξ)) = exp(−it X ξ ). By a use of the adjoint representation Ad : G →
L(g),
Ad(g)ξ :=
d
dt
t=0
[g exp(tξ)g
−1
]
we obtain:
[X ξ , X η ] := X ξ X η − X η X ξ = i X [ξ,η] .
(2.3.21)
The mapping ξ → X ξ is linear. It is known (compare [13]), that the Gårding
domain D G , as well as the analytic domain A G of the representation U (G) are
common dense invariant sets of all the generators X ξ (ξ ∈ g) and they are also
common cores of all these selfadjoint operators (cf. also 3.1.1). Let us define the
vector fields σ ξ (ξ ∈ g) on PD G ⊂ P(H) corresponding to the flows U(exp(tξ))
on P(H) according to the definition of σ A in 2.3.5. Let f ξ (x) := T r(P x X ξ ) for
x ∈ D G . Then 2.3.8 is applicable to these quantities. All the formulas of 1.3.7 are
valid on PD G . Difference w.r.t. the classical case is that neither P(H) nor PD G are
homogeneous spaces even for irreducible U(G).
2.3.10 Up to now, we used charts (N x ; x ; [x]
⊥
) for identification of T x P(H) with
[x]
⊥ , and for each point x ∈ P(H) it was used its own chart. Let us rewrite now the
5 A core D ⊂ H of a closable operator C is such a subset D ⊂ D(C) ⊂ H, that the closure of the
restriction C D = C, cf. also [37, C1].