46
3 Classical Mechanical Projections of QM
If σ j are Hamiltonian vector fields corresponding to generators X j ∈ U (g), then
we obtain according to (3.3.12)
d m f A (σ j ) = f i[X j ,A] (m)
(3.3.20)
and the Poisson bracket (3.3.19) has the form
{ f A , f B }(m) = −
j,k
f i[X j ,A] (m))
jk
M (m) f i[X k ,B] (m).
(3.3.21)
If the operator B is one of the generators of U (G), B := X ∈ U (g), then the
Poisson bracket (3.3.21) has the expression:
{ f A , f X }(m) = i T r(P x [A, X ]) = f i[A,X ] (m),
(3.3.22)
where x ∈ x ∈ [x] := m ∈ M. The results (3.3.21) and (3.3.22) have to be compared
with 3.2.14. If the orbit O z coincides with the manifold M := M z , then the vector
field σ
M
A in (3.3.18) is the skew-orthogonal projection of σ A (from (2.3.8)) onto M,
the skew-orthogonality being defined by the form on P(H), see Sect. 2.2.
3.3.9 The unitary group U A : t → U A (t) := exp(−it A) does not leave the orbit
O z invariant for a general selfadjoint z-classical operator A. Then we would like
to compare the classical Hamiltonian evolution on M z generated by f A (with the
flow F
A
t ) and the quantal evolution on P(H) described by the flow U A (t). From
the point of view of this work, the ‘quantities of interest’ are generators of the
representation U (G). The evolutions of the corresponding functions f X (X = X
∗
∈
U (g)) are described by
d
dt
f
t
X = { f A , f
t
X } = f
t
i[A,X ]
(3.3.23)
in both cases of the classical flow F
A
t as well as of the quantal evolution U A (t),
compare 3.2.12, (2.3.27) and (3.3.22). The difference is between the two cases in the
meaning of f
t :
(i) In the case of the flow F
A
t on M for any f ∈ C
∞
(M), we define
f
t
(m) := f (F
A
t m), m ∈ M,
(3.3.24)
and the flow F
A
t has to be determined from (3.3.23) (∀X
∗
= X ∈ U (g)).
(ii) In the quantal case, we have given the flow U A on P(H) and for functions f on
(the dense U A -invariant subset of) P(H) we set
f
t
(x) := f (U A (t)x).
(3.3.25)
3 Classical Mechanical Projections of QM
If σ j are Hamiltonian vector fields corresponding to generators X j ∈ U (g), then
we obtain according to (3.3.12)
d m f A (σ j ) = f i[X j ,A] (m)
(3.3.20)
and the Poisson bracket (3.3.19) has the form
{ f A , f B }(m) = −
j,k
f i[X j ,A] (m))
jk
M (m) f i[X k ,B] (m).
(3.3.21)
If the operator B is one of the generators of U (G), B := X ∈ U (g), then the
Poisson bracket (3.3.21) has the expression:
{ f A , f X }(m) = i T r(P x [A, X ]) = f i[A,X ] (m),
(3.3.22)
where x ∈ x ∈ [x] := m ∈ M. The results (3.3.21) and (3.3.22) have to be compared
with 3.2.14. If the orbit O z coincides with the manifold M := M z , then the vector
field σ
M
A in (3.3.18) is the skew-orthogonal projection of σ A (from (2.3.8)) onto M,
the skew-orthogonality being defined by the form on P(H), see Sect. 2.2.
3.3.9 The unitary group U A : t → U A (t) := exp(−it A) does not leave the orbit
O z invariant for a general selfadjoint z-classical operator A. Then we would like
to compare the classical Hamiltonian evolution on M z generated by f A (with the
flow F
A
t ) and the quantal evolution on P(H) described by the flow U A (t). From
the point of view of this work, the ‘quantities of interest’ are generators of the
representation U (G). The evolutions of the corresponding functions f X (X = X
∗
∈
U (g)) are described by
d
dt
f
t
X = { f A , f
t
X } = f
t
i[A,X ]
(3.3.23)
in both cases of the classical flow F
A
t as well as of the quantal evolution U A (t),
compare 3.2.12, (2.3.27) and (3.3.22). The difference is between the two cases in the
meaning of f
t :
(i) In the case of the flow F
A
t on M for any f ∈ C
∞
(M), we define
f
t
(m) := f (F
A
t m), m ∈ M,
(3.3.24)
and the flow F
A
t has to be determined from (3.3.23) (∀X
∗
= X ∈ U (g)).
(ii) In the quantal case, we have given the flow U A on P(H) and for functions f on
(the dense U A -invariant subset of) P(H) we set
f
t
(x) := f (U A (t)x).
(3.3.25)