Manuscript Em03-G@1 · Em03-G-v1.tex · section · 0bb3a8a3

Finite displacement branches

Ordered noncommuting source dynamics › Finite displacement branches · lines 615–642

Em03-G@1#eq:branches

For signs $z, x \in {+ 1, - 1}$ , put

P_{z}^{Z} = (I + z Z) /2, P_{x}^{X} = (I + x X) /2,

B_{x z} = W (- x a_{2}) W (- z a_{1}) = e^{i x z θ /2} W (- z a_{1} - x a_{2}) .

For $z = + 1$ identify $∣ z ⟩ = ∣ 0 ⟩$ , and for $z = - 1$ identify $∣ z ⟩ = ∣ 1 ⟩$ . Let

ϵ_{z} = (1 - z) /2, m_{x z} = ⟨ x ∣_{X} z ∣ = ⟩ x^{ϵ_{z}} / 2 .

Then

U_{G} = x, z \sum P_{x}^{X} P_{z}^{Z} \otimes B_{x z}, U_{G} (∣ z ⟩ \otimes Ω) = x \sum m_{x z} ∣ x ⟩_{X} \otimes B_{x z} Ω.

These branch phases are part of the dynamics. Dropping them before interference calculations is not valid.

Displayed formulas in this passage

Em03-G@1#eq:branches · line 632
$U_{G} = x, z \sum P_{x}^{X} P_{z}^{Z} \otimes B_{x z}, U_{G} (∣ z ⟩ \otimes Ω) = x \sum m_{x z} ∣ x ⟩_{X} \otimes B_{x z} Ω.$
unlabeled · line 618
$P_{z}^{Z} = (I + z Z) /2, P_{x}^{X} = (I + x X) /2,$
unlabeled · line 621
$B_{x z} = W (- x a_{2}) W (- z a_{1}) = e^{i x z θ /2} W (- z a_{1} - x a_{2}) .$
unlabeled · line 627
$ϵ_{z} = (1 - z) /2, m_{x z} = ⟨ x ∣_{X} z ∣ = ⟩ x^{ϵ_{z}} / 2 .$