3.3 Classical Mechanical Projections of Quantal Dynamics
45
and we shall set
f B ([x]) := f B (x) :=
τ
λ τ f A
τ (x).
(3.3.16)
The adjective ‘generalized’ will be sometimes omitted.
3.3.7 Examples.
(i) All the generators X ξ (ξ ∈ g) are z-classical for all z ∈ PA G .
(ii) If, for some z ∈ P(H) : K z = K
◦
z , (cf. 3.1.1) and f A ∈ C
∞
(O z ), then A is
z-classical.
(iii) If A is z-classical and X ξ , . . . X χ ∈ U (g), then all the symbols [X ξ , [X η , . . .
[X χ , A] . . . ]] represent generalized z-classical operators. We can see this from
3.3.4 and (3.2.11):
f A (g · z) = f A ([g · z]) = f A ([gh · z])
and differentiations and induction give the result.
(iv) Let f A ∈ C
∞
(O z ) and all the K x be symmetry groups of the observable
A : U (h
−1
)AU (h) = A for all h ∈ K x and all x ∈ O z (e.g. if K z is a normal
subgroup of G and K z is a symmetry group of A). Then A is z-classical.
3.3.8 If A is z-classical, then the function f A can be considered as a function on M z
according to (3.3.13) and then f A ∈ C
∞
(M). Denote by σ
M
A the Hamiltonian vector
field on M corresponding to the Hamiltonian function f A : m → f A (m), m ∈ M.
Choose a system σ j ( j = 1, . . . dim M) of vector fields on M forming a basis of
T m M for all m in a neighbourhood of m 0 ∈ M. Since the symplectic form
M
is nondegenerate, the inverse matrix to
M
m (σ j , σ k ) with elements
jk
M (m) ( j, k =
1, 2, . . . dim M) exists:
i
ji
M (m))
M
m (σ i , σ k ) =
i
M
m (σ k , σ i ))
i j
M (m) = δ jk .
(3.3.17)
From the connection between Hamiltonian vector fields and corresponding Hamiltonian functions, we obtain:
σ
M
A (m) =
j,k
jk
M (m)d m f A (σ k )σ j (m).
(3.3.18)
For Poisson brackets of functions f A and f B on M corresponding to z-classical
operators A and B, we obtain with a help of (3.3.17):
{ f A , f B }(m) :=
M
m (σ
M
A , σ
M
B ) = −
j,k
d m f A (σ j ))
jk
M (m)d m f B (σ k ). (3.3.19)
45
and we shall set
f B ([x]) := f B (x) :=
τ
λ τ f A
τ (x).
(3.3.16)
The adjective ‘generalized’ will be sometimes omitted.
3.3.7 Examples.
(i) All the generators X ξ (ξ ∈ g) are z-classical for all z ∈ PA G .
(ii) If, for some z ∈ P(H) : K z = K
◦
z , (cf. 3.1.1) and f A ∈ C
∞
(O z ), then A is
z-classical.
(iii) If A is z-classical and X ξ , . . . X χ ∈ U (g), then all the symbols [X ξ , [X η , . . .
[X χ , A] . . . ]] represent generalized z-classical operators. We can see this from
3.3.4 and (3.2.11):
f A (g · z) = f A ([g · z]) = f A ([gh · z])
and differentiations and induction give the result.
(iv) Let f A ∈ C
∞
(O z ) and all the K x be symmetry groups of the observable
A : U (h
−1
)AU (h) = A for all h ∈ K x and all x ∈ O z (e.g. if K z is a normal
subgroup of G and K z is a symmetry group of A). Then A is z-classical.
3.3.8 If A is z-classical, then the function f A can be considered as a function on M z
according to (3.3.13) and then f A ∈ C
∞
(M). Denote by σ
M
A the Hamiltonian vector
field on M corresponding to the Hamiltonian function f A : m → f A (m), m ∈ M.
Choose a system σ j ( j = 1, . . . dim M) of vector fields on M forming a basis of
T m M for all m in a neighbourhood of m 0 ∈ M. Since the symplectic form
M
is nondegenerate, the inverse matrix to
M
m (σ j , σ k ) with elements
jk
M (m) ( j, k =
1, 2, . . . dim M) exists:
i
ji
M (m))
M
m (σ i , σ k ) =
i
M
m (σ k , σ i ))
i j
M (m) = δ jk .
(3.3.17)
From the connection between Hamiltonian vector fields and corresponding Hamiltonian functions, we obtain:
σ
M
A (m) =
j,k
jk
M (m)d m f A (σ k )σ j (m).
(3.3.18)
For Poisson brackets of functions f A and f B on M corresponding to z-classical
operators A and B, we obtain with a help of (3.3.17):
{ f A , f B }(m) :=
M
m (σ
M
A , σ
M
B ) = −
j,k
d m f A (σ j ))
jk
M (m)d m f B (σ k ). (3.3.19)