42
3 Classical Mechanical Projections of QM
F
ξ
t : [x] = p M x → F
ξ
t ( p M x) := p M (U(exp(tξ))x) .
(3.2.16)
Proof. From the definition of σ
M
ξ in 3.2.8 and 3.2.9 the relation (3.2.16) follows.
Differentiation of f ξ according to (3.1.15) and (3.2.2) gives
d f ξ = −i(σ
M
ξ ))
M
,
(3.2.17)
compare (1.3.4). This proves the first statement.
With the usual definition of Poisson brackets on (M;
M
), we obtain the obvious
(compare also (1.3.11) + (1.3.12))
3.2.13 Lemma. { f ξ , f η } = − f [ξ,η] for all ξ, η ∈ g.
3.2.14 This shows, that the action of Ad
∗
(G) is strictly Hamiltonian. Since for the
generators of U (G) in H we have X [ξ,η] = −i[X ξ , X η ], (2.3.21), the Lemma 3.2.13
establishes the usual correspondence between classical and quantal observables
associated with generators of the group action.
3.3 Classical Mechanical Projections of Quantal Dynamics
3.3.1 Let the time evolution of a given system in QM be described by a one parameter
subgroup of U (G) corresponding to an element χ ∈ g. Then, for a given z ∈ P(H),
the flow U(exp(tχ)) leaves the orbit O z invariant. If z ∈ PA G , then this flow is
projected onto the Hamiltonian flow on M z generated by the Hamiltonian function
f χ with the corresponding Hamiltonian vector field σ
M
χ , as it was described above.
Models one frequently encounters are, however, in which the time evolution is given
by a one parameter group of unitaries U A (R) :
U A : t → U A (t) := exp(−it A), A = A
∗
,
(3.3.1)
where the generator A has not the form X χ for any χ ∈ g. The orbits O z are then
in general not invariant with respect to the action of U A (R). We shall be interested
here in the question whether and how such an action U A (R) can be projected onto a
Hamiltonian flow on M z .
3.3.2 Let A be any selfadjoint operator on H and E A the corresponding projectorvalued measure on R. Assume that z ∈ PA G (defined in 3.1.1) and that O z := U(G)z
is contained in the form domain of A, i.e. the integral in
f A (x) := T r(P x A) :=
R
λ T r(P x E A (dλ))
(3.3.2)
3 Classical Mechanical Projections of QM
F
ξ
t : [x] = p M x → F
ξ
t ( p M x) := p M (U(exp(tξ))x) .
(3.2.16)
Proof. From the definition of σ
M
ξ in 3.2.8 and 3.2.9 the relation (3.2.16) follows.
Differentiation of f ξ according to (3.1.15) and (3.2.2) gives
d f ξ = −i(σ
M
ξ ))
M
,
(3.2.17)
compare (1.3.4). This proves the first statement.
With the usual definition of Poisson brackets on (M;
M
), we obtain the obvious
(compare also (1.3.11) + (1.3.12))
3.2.13 Lemma. { f ξ , f η } = − f [ξ,η] for all ξ, η ∈ g.
3.2.14 This shows, that the action of Ad
∗
(G) is strictly Hamiltonian. Since for the
generators of U (G) in H we have X [ξ,η] = −i[X ξ , X η ], (2.3.21), the Lemma 3.2.13
establishes the usual correspondence between classical and quantal observables
associated with generators of the group action.
3.3 Classical Mechanical Projections of Quantal Dynamics
3.3.1 Let the time evolution of a given system in QM be described by a one parameter
subgroup of U (G) corresponding to an element χ ∈ g. Then, for a given z ∈ P(H),
the flow U(exp(tχ)) leaves the orbit O z invariant. If z ∈ PA G , then this flow is
projected onto the Hamiltonian flow on M z generated by the Hamiltonian function
f χ with the corresponding Hamiltonian vector field σ
M
χ , as it was described above.
Models one frequently encounters are, however, in which the time evolution is given
by a one parameter group of unitaries U A (R) :
U A : t → U A (t) := exp(−it A), A = A
∗
,
(3.3.1)
where the generator A has not the form X χ for any χ ∈ g. The orbits O z are then
in general not invariant with respect to the action of U A (R). We shall be interested
here in the question whether and how such an action U A (R) can be projected onto a
Hamiltonian flow on M z .
3.3.2 Let A be any selfadjoint operator on H and E A the corresponding projectorvalued measure on R. Assume that z ∈ PA G (defined in 3.1.1) and that O z := U(G)z
is contained in the form domain of A, i.e. the integral in
f A (x) := T r(P x A) :=
R
λ T r(P x E A (dλ))
(3.3.2)