Cracow, le 19 Dec 2009

Dear Guys, About the equality

∫_{ℂ} |z⟩ W (|z|^{2}) ⟨z| dz = I  (1)

At this point of our discussion, one has at hand two definitions. We denote H_{G} our “untouchable” ground Hilbert space with basis of states (|n⟩)_{n∈ℕ}.

First definition. — After our dis-

cussion with Kus. In fact the left hand side of (2) is a weak operator integration.

(∀n,m ∈ ℕ)(∫_{ℂ} W (|z|^{2}) ⟨n|z⟩⟨z|m⟩ dz) = ⟨n|m⟩  (2)

“weak operator equality”.

Second definition. — From yester-

day. In fact the left hand side of (3) is a strong operator integration.

(∀ϕ,ψ ∈ H_{G})(∫_{ℂ} W (|z|^{2}) ⟨ϕ|z⟩⟨z|ψ⟩ dz) = ⟨ϕ|ψ⟩  (3)

“strong operator equality”. Let us prove that, in our case the weak definition implies the strong definition and then what was computed yesterday

∫_{ℂ} W (|z|^{2}).Osc(|z|^{2}) ⟨ϕ|z⟩⟨z|ϕ⟩ dz  (4)

will always amount to zero as W (|z|^{2}).Osc(|z|^{2}) = W_{1}(|z|^{2}) − W_{2}(|z|^{2}).