2.3 Quantum Mechanics as a Classical Hamiltonian Field Theory
33
evolution equation corresponding to the one-parameter flow F
A
t on P(H) generated
by the Hamiltonian A, i.e. the Schrödinger equation
i
d
dt
x(t) = Ax(t), x(0) := x ∈ H,
(2.3.22)
projected onto P(H), with the help of the chart (N y ; y ; [y]
⊥
), x ∈ N y . Let us
denote by c : t → c(t) a differentiable curve in P(H), and by ˙
c(t) its tangent
vector: ˙
c(t) ∈ T c(t) P(H). The curve c will be a solution of our problem, if for some
x ∈ P(H) : c(t) = F
A
t x for all t ∈ R. For x ∈ P D(A), we then obtain by differentiation
˙
c(t) = σ A (c(t)),
(2.3.23)
which is an abstract form of Hamilton equations on P(H) corresponding to the
Hamiltonian function f A , cf. (2.3.13). The correspondence with (2.3.22) consists in
that, that c(t) = x(t) if c(0) = x, where x(t) (∈ x(t)) is the solution of (2.3.22)
with the initial value x ∈ x. Let us fix y ∈ H, y = 1, and choose the chart
(N y ; y ; [y]
⊥
) defined in (2.1.8). Denote
(t) := y (c(t)) for a curve c in N y .
(2.3.24)
The curve in [y]
⊥ will correspond to a solution c of (2.3.23) iff it satisfies the
equation
i
d
dt
(t) = [A − (y, A(y + (t)))](y + (t)), ,(0) ∈ [y]
⊥
.
(2.3.25)
The equation (2.3.25) describes the wanted projection of (2.3.22) onto P(H) in
the chart y . It is a nonlinear (field-) equation in the Hilbert space [y]
⊥ , in which
different vectors correspond to different physical states.
If we denote by v y the representative of a vector v ∈ T x P(H) for x ∈ N y in the
chart y (y = 1), then the symplectic form in this chart has the expression:
x (v, w) = −2 T r(P x P y ) Im(v y , (I − P x )w y ).
(2.3.26)
Remember, that v y , w y ∈ [y]
⊥
:= (I − P y )H.
Let us write f
t
:= f ◦ F
A
t for any differentiable function f on P(H). Then, for
x ∈ P D(A), we obtain the wanted form of the Schrödinger equation:
d
dt
f
t
(x) = { f A , f
t
}(x) := x (σ A , σ f ),
(2.3.27)
33
evolution equation corresponding to the one-parameter flow F
A
t on P(H) generated
by the Hamiltonian A, i.e. the Schrödinger equation
i
d
dt
x(t) = Ax(t), x(0) := x ∈ H,
(2.3.22)
projected onto P(H), with the help of the chart (N y ; y ; [y]
⊥
), x ∈ N y . Let us
denote by c : t → c(t) a differentiable curve in P(H), and by ˙
c(t) its tangent
vector: ˙
c(t) ∈ T c(t) P(H). The curve c will be a solution of our problem, if for some
x ∈ P(H) : c(t) = F
A
t x for all t ∈ R. For x ∈ P D(A), we then obtain by differentiation
˙
c(t) = σ A (c(t)),
(2.3.23)
which is an abstract form of Hamilton equations on P(H) corresponding to the
Hamiltonian function f A , cf. (2.3.13). The correspondence with (2.3.22) consists in
that, that c(t) = x(t) if c(0) = x, where x(t) (∈ x(t)) is the solution of (2.3.22)
with the initial value x ∈ x. Let us fix y ∈ H, y = 1, and choose the chart
(N y ; y ; [y]
⊥
) defined in (2.1.8). Denote
(t) := y (c(t)) for a curve c in N y .
(2.3.24)
The curve in [y]
⊥ will correspond to a solution c of (2.3.23) iff it satisfies the
equation
i
d
dt
(t) = [A − (y, A(y + (t)))](y + (t)), ,(0) ∈ [y]
⊥
.
(2.3.25)
The equation (2.3.25) describes the wanted projection of (2.3.22) onto P(H) in
the chart y . It is a nonlinear (field-) equation in the Hilbert space [y]
⊥ , in which
different vectors correspond to different physical states.
If we denote by v y the representative of a vector v ∈ T x P(H) for x ∈ N y in the
chart y (y = 1), then the symplectic form in this chart has the expression:
x (v, w) = −2 T r(P x P y ) Im(v y , (I − P x )w y ).
(2.3.26)
Remember, that v y , w y ∈ [y]
⊥
:= (I − P y )H.
Let us write f
t
:= f ◦ F
A
t for any differentiable function f on P(H). Then, for
x ∈ P D(A), we obtain the wanted form of the Schrödinger equation:
d
dt
f
t
(x) = { f A , f
t
}(x) := x (σ A , σ f ),
(2.3.27)