Another Type of an Entangled Solution

Φ[χ] ＝ Σ_p,q Π_j ∫dx_j ∫dt_j G(χ(p_j), p_j; x_j, t_j)F(x_j, t_j)G(χ(q_j), q_j; x_j, t_j)
where I divided the set {1/α, 2/α, 3/α, ・・・} into pairs {(p₁, q₁), (p₂, q₂), (p₃, q₃), ・・・}.
Σ_p,q means the sum over all patterns of division.

I expect that the path integral:
[Π_{j≠j0 and j≠j1} ∫dχ(j/α)]Φ[χ]^*Φ[χ]
yields some function resembling G(χ(j₀/α), j₀/α; χ(j₁/α), j₁/α) when F(x, t) = 1.

∫dx G(x₁, t₁; x, 0)G(x₂, t₂; x, 0) resembles G(x₁, t₁; x₂, t₂) because G(x₂, t₂; x, 0) represents the wavefunction Ψ(x, 0) such that Ψ(x, t₂) = δ(x - x₂).

G(x₂, t₂; x, 0) = ＜x₂|U(t₂, 0)|x＞
= ＜x|U(t₂, 0)†|x₂＞^*
= ＜x|U(t₂, 0)^-1|x₂＞^*
= ＜x|U(0, t₂)|x₂＞^*
= G(x, 0; x₂, t₂)^*
This is the Green's function of Ψ^*.

Ψ^* obeys the time reversed Schrodinger equation, but it is essentialy the same as Ψ when Ψ(x, t₂) = δ(x - x₂).
So,
∫dx G(x₁, t₁; x, 0)G(x₂, t₂; x, 0)
= ∫dx'∫dx G(x₁, t₁; x, 0)G(x', t₂; x, 0)Ψ_*(x', t₂)
is essentially the same as
∫dx'∫dx G(x₁, t₁; x, 0)G(x, 0; x', t₂)Ψ(x', t₂)
= ∫dx' G(x₁, t₁; x', t₂)Ψ(x', t₂)
= G(x₁, t₁; x₂, t₂)
where Ψ(x', t₂) = δ(x' - x₂).

---
Last edited at 2013/05/01/15:29JST