2.3 Quantum Mechanics as a Classical Hamiltonian Field Theory
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if the derivative on the right hand side exists. Assume, that (2.3.11) is well defined
for a dense set of vectors ˙
c ∈ T x P(H). The function
d x f : ˙
c → d x f ( ˙
c)
(2.3.12)
is linear. If it is bounded, it can be extended by continuity to the whole T x P(H), hence
it defines an element d x f ∈ T
∗
x P(H) which will be called the exterior differential
of f in x.
2.3.7 Proposition. Let A be a selfadjoint operator on H, f A is given by (2.3.2), and
the corresponding vector field σ A is defined in 2.3.5. Then, for any x ∈ P D(A), the
exterior differential d x f A ∈ T
∗
x P(H) exists, and for all v ∈ T x P(H) we have
x (σ A (x), v) = − d x f A (v),
∀x ∈ P D(A).
(2.3.13)
Proof. Let {v x }
⊥ be defined according to (2.1.18) for v ∈ T x P(H). Define the
selfadjoint B(v) ∈ L(H ):
B(v)y := i(v x , y)x − i(x, y)v x , ∀y ∈ H.
(2.3.14)
Assume x = 1. Then, according to (2.1.14) and (2.1.16), the curve
t → c v (t) := exp(itB(v))x
(2.3.15)
corresponds to v = ˙
c v . Let v be such that v x ∈ D(A). Then it is seen that d x f A (v)
defined in (2.3.11) exists and has the form
d x f A (v) = −i T r(P x [B(v), A]) = − x (σ A (x), v),
(2.3.16)
where, in the second equality, we used (2.3.10) and σ B(v) (x) = −v. Because (I −
P x )D(A) ⊂ D(A) is dense in {x}
⊥ , we have proved (2.3.13) for a dense linear subset
D ⊂ T x P(H), v ∈ D. The boundednes is clear either from our construction, or from
the boundednes of the left hand side of (2.3.13) for a well defined σ A (x).
2.3.8 We can see from the Proposition 2.3.7, how to reconstruct the vector field σ A
from f A with the help of the symplectic form . Hence, σ A is globally Hamiltonian
vector field on (the dense subset of) P(H) corresponding to the Hamiltonian function
f A (compare with 1.3.5—up to domain differences).
Let two selfadjoint A, B have a common dense domain D ⊂ D(A) ∩ D(B). Then
the function (the Poisson bracket)
x → { f A , f B }(x) := x (σ A , σ B ), x ∈ P D,
(2.3.17)