Application of the Stacked Daruma Game Formula @ Quantum History Theory @ Problems @ Forum @ www.GrammaticalPhysics.ac

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In this page, I try applying the stacked Daruma game formula to the solution derived in Discretized Quantum Mechanical Model.

Φ[x] ≒ [Π_s=-∞^∞Σ_i(s)=1²∫_-∞^∞de_s] {A₀[e,i] + βA₁[e,i]}Π_k=-∞^∞u_e(k)i(k)(x_k),
A₀[e,i] = Π_k=-∞^∞δ(e_k),
A₁[e,i] = 2mαΣ_k=-∞^∞(e_k + e_k+1)^-1[・・・δ(e_k-1)η_3-i(k)(0,e_k)ξ_3-i(k+1)(0,e_k+1)δ(e_k+2)・・・].

This solution is of the following form.
Φ[x] = Σ_k=-∞^∞ ・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・
That is to say, this is a sum of functions, in each of which only adjoining two degrees of freedom are entangled.

Φ[・・・, x_p-1, a, x_p, ・・・]
= Σ_k≦p-3 ・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・φ(x_p-1)φ(a)φ(x_p)・・・
　+ ・・・φ(x_p-3)ψ(x_p-2, x_p-1)φ(a)φ(x_p)・・・
　+ ・・・φ(x_p-2)ψ(x_p-1, a)φ(x_p)・・・
　+ ・・・φ(x_p-1)ψ(a, x_p)φ(x_p+1)・・・
　+ ・・・φ(x_p-1)φ(a)ψ(x_p, x_p+1)φ(x_p+2)・・・
　+ Σ_k≧p+1 ・・・φ(x_p-1)φ(a)φ(x_p)・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・.

Φ[・・・, x_q-1, b, x_q, ・・・]
= Σ_k≦q-3 ・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・φ(x_q-1)φ(b)φ(x_q)・・・
　+ ・・・φ(x_q-3)ψ(x_q-2, x_q-1)φ(b)φ(x_q)・・・
　+ ・・・φ(x_q-2)ψ(x_q-1, b)φ(x_q)・・・
　+ ・・・φ(x_q-1)ψ(b, x_q)φ(x_q+1)・・・
　+ ・・・φ(x_q-1)φ(b)ψ(x_q, x_q+1)φ(x_q+2)・・・
　+ Σ_k≧q+1 ・・・φ(x_q-1)φ(b)φ(x_q)・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・.

Φ[・・・, x_q-1, b, x_q, ・・・]^* Φ[・・・, x_p-1, a, x_p, ・・・]

= [Σ_k≦q-3 ・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・φ(x_q-1)φ(b)φ(x_q)・・・]^* [Σ_i≦p-3 ・・・φ(x_i-1)ψ(x_i, x_i+1)φ(x_i+2)・・・φ(x_p-1)φ(a)φ(x_p)・・・]
　+ [Σ_k≦q-3 ・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・φ(x_q-1)φ(b)φ(x_q)・・・]^* [・・・φ(x_p-3)ψ(x_p-2, x_p-1)φ(a)φ(x_p)・・・]
　+ [Σ_k≦q-3 ・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・φ(x_q-1)φ(b)φ(x_q)・・・]^* [・・・φ(x_p-2)ψ(x_p-1, a)φ(x_p)・・・]
　+ [Σ_k≦q-3 ・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・φ(x_q-1)φ(b)φ(x_q)・・・]^* [・・・φ(x_p-1)ψ(a, x_p)φ(x_p+1)・・・]
　+ [Σ_k≦q-3 ・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・φ(x_q-1)φ(b)φ(x_q)・・・]^* [・・・φ(x_p-1)φ(a)ψ(x_p, x_p+1)φ(x_p+2)・・・]
　+ [Σ_k≦q-3 ・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・φ(x_q-1)φ(b)φ(x_q)・・・]^* [Σ_i≧p+1 ・・・φ(x_p-1)φ(a)φ(x_p)・・・φ(x_i-1)ψ(x_i, x_i+1)φ(x_i+2)・・・]

　+ [・・・φ(x_q-3)ψ(x_q-2, x_q-1)φ(b)φ(x_q)・・・]^* [Σ_i≦p-3 ・・・φ(x_i-1)ψ(x_i, x_i+1)φ(x_i+2)・・・φ(x_p-1)φ(a)φ(x_p)・・・]
　+ [・・・φ(x_q-3)ψ(x_q-2, x_q-1)φ(b)φ(x_q)・・・]^* [・・・φ(x_p-3)ψ(x_p-2, x_p-1)φ(a)φ(x_p)・・・]
　+ [・・・φ(x_q-3)ψ(x_q-2, x_q-1)φ(b)φ(x_q)・・・]^* [・・・φ(x_p-2)ψ(x_p-1, a)φ(x_p)・・・]
　+ [・・・φ(x_q-3)ψ(x_q-2, x_q-1)φ(b)φ(x_q)・・・]^* [・・・φ(x_p-1)ψ(a, x_p)φ(x_p+1)・・・]
　+ [・・・φ(x_q-3)ψ(x_q-2, x_q-1)φ(b)φ(x_q)・・・]^* [・・・φ(x_p-1)φ(a)ψ(x_p, x_p+1)φ(x_p+2)・・・]
　+ [・・・φ(x_q-3)ψ(x_q-2, x_q-1)φ(b)φ(x_q)・・・]^* [Σ_i≧p+1 ・・・φ(x_p-1)φ(a)φ(x_p)・・・φ(x_i-1)ψ(x_i, x_i+1)φ(x_i+2)・・・]

　+ [・・・φ(x_q-2)ψ(x_q-1, b)φ(x_q)・・・]^* [Σ_i≦p-3 ・・・φ(x_i-1)ψ(x_i, x_i+1)φ(x_i+2)・・・φ(x_p-1)φ(a)φ(x_p)・・・]
　+ [・・・φ(x_q-2)ψ(x_q-1, b)φ(x_q)・・・]^* [・・・φ(x_p-3)ψ(x_p-2, x_p-1)φ(a)φ(x_p)・・・]
　+ [・・・φ(x_q-2)ψ(x_q-1, b)φ(x_q)・・・]^* [・・・φ(x_p-2)ψ(x_p-1, a)φ(x_p)・・・]
　+ [・・・φ(x_q-2)ψ(x_q-1, b)φ(x_q)・・・]^* [・・・φ(x_p-1)ψ(a, x_p)φ(x_p+1)・・・]
　+ [・・・φ(x_q-2)ψ(x_q-1, b)φ(x_q)・・・]^* [・・・φ(x_p-1)φ(a)ψ(x_p, x_p+1)φ(x_p+2)・・・]
　+ [・・・φ(x_q-2)ψ(x_q-1, b)φ(x_q)・・・]^* [Σ_i≧p+1 ・・・φ(x_p-1)φ(a)φ(x_p)・・・φ(x_i-1)ψ(x_i, x_i+1)φ(x_i+2)・・・]

　+ [・・・φ(x_q-1)ψ(b, x_q)φ(x_q+1)・・・]^* [Σ_i≦p-3 ・・・φ(x_i-1)ψ(x_i, x_i+1)φ(x_i+2)・・・φ(x_p-1)φ(a)φ(x_p)・・・]
　+ [・・・φ(x_q-1)ψ(b, x_q)φ(x_q+1)・・・]^* [・・・φ(x_p-3)ψ(x_p-2, x_p-1)φ(a)φ(x_p)・・・]
　+ [・・・φ(x_q-1)ψ(b, x_q)φ(x_q+1)・・・]^* [・・・φ(x_p-2)ψ(x_p-1, a)φ(x_p)・・・]
　+ [・・・φ(x_q-1)ψ(b, x_q)φ(x_q+1)・・・]^* [・・・φ(x_p-1)ψ(a, x_p)φ(x_p+1)・・・]
　+ [・・・φ(x_q-1)ψ(b, x_q)φ(x_q+1)・・・]^* [・・・φ(x_p-1)φ(a)ψ(x_p, x_p+1)φ(x_p+2)・・・]
　+ [・・・φ(x_q-1)ψ(b, x_q)φ(x_q+1)・・・]^* [Σ_i≧p+1 ・・・φ(x_p-1)φ(a)φ(x_p)・・・φ(x_i-1)ψ(x_i, x_i+1)φ(x_i+2)・・・]

　+ [・・・φ(x_q-1)φ(b)ψ(x_q, x_q+1)φ(x_q+2)・・・]^* [Σ_i≦p-3 ・・・φ(x_i-1)ψ(x_i, x_i+1)φ(x_i+2)・・・φ(x_p-1)φ(a)φ(x_p)・・・]
　+ [・・・φ(x_q-1)φ(b)ψ(x_q, x_q+1)φ(x_q+2)・・・]^* [・・・φ(x_p-3)ψ(x_p-2, x_p-1)φ(a)φ(x_p)・・・]
　+ [・・・φ(x_q-1)φ(b)ψ(x_q, x_q+1)φ(x_q+2)・・・]^* [・・・φ(x_p-2)ψ(x_p-1, a)φ(x_p)・・・]
　+ [・・・φ(x_q-1)φ(b)ψ(x_q, x_q+1)φ(x_q+2)・・・]^* [・・・φ(x_p-1)ψ(a, x_p)φ(x_p+1)・・・]
　+ [・・・φ(x_q-1)φ(b)ψ(x_q, x_q+1)φ(x_q+2)・・・]^* [・・・φ(x_p-1)φ(a)ψ(x_p, x_p+1)φ(x_p+2)・・・]
　+ [・・・φ(x_q-1)φ(b)ψ(x_q, x_q+1)φ(x_q+2)・・・]^* [Σ_i≧p+1 ・・・φ(x_p-1)φ(a)φ(x_p)・・・φ(x_i-1)ψ(x_i, x_i+1)φ(x_i+2)・・・]

　+ [Σ_k≧q+1 ・・・φ(x_q-1)φ(b)φ(x_q)・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・]^* [Σ_i≦p-3 ・・・φ(x_i-1)ψ(x_i, x_i+1)φ(x_i+2)・・・φ(x_p-1)φ(a)φ(x_p)・・・]
　+ [Σ_k≧q+1 ・・・φ(x_q-1)φ(b)φ(x_q)・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・]^* [・・・φ(x_p-3)ψ(x_p-2, x_p-1)φ(a)φ(x_p)・・・]
　+ [Σ_k≧q+1 ・・・φ(x_q-1)φ(b)φ(x_q)・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・]^* [・・・φ(x_p-2)ψ(x_p-1, a)φ(x_p)・・・]
　+ [Σ_k≧q+1 ・・・φ(x_q-1)φ(b)φ(x_q)・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・]^* [・・・φ(x_p-1)ψ(a, x_p)φ(x_p+1)・・・]
　+ [Σ_k≧q+1 ・・・φ(x_q-1)φ(b)φ(x_q)・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・]^* [・・・φ(x_p-1)φ(a)ψ(x_p, x_p+1)φ(x_p+2)・・・]
　+ [Σ_k≧q+1 ・・・φ(x_q-1)φ(b)φ(x_q)・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・]^* [Σ_i≧p+1 ・・・φ(x_p-1)φ(a)φ(x_p)・・・φ(x_i-1)ψ(x_i, x_i+1)φ(x_i+2)・・・].

Below, I investigate the case that q is much larger than p.
I assume that
∫dx φ(x)^*φ(x) = 1.

∫Dx Φ[・・・, x_q-1, b, x_q, ・・・]^* Φ[・・・, x_p-1, a, x_p, ・・・]

= Aφ(b)^*φ(a)
　+ φ(b)^*∫dx_p-1 f₁(x_p-1)ψ(x_p-1, a)
　+ φ(b)^*∫dx_p f₂(x_p)ψ(a, x_p)
　+ ∫dx_q-1 f₃(x_q-1)ψ(x_q-1, b)^*φ(a)
　+ ∫dx_q f₄(x_q)ψ(b, x_q)^*φ(a)
　+ B₁∫dx_q-1dx_p-1φ(x_q-1)ψ(x_q-1, b)^*ψ(x_p-1, a)φ(x_p-1)^*
　+ B₂∫dx_q-1dx_pφ(x_q-1)ψ(x_q-1, b)^*ψ(a, x_p)φ(x_p)^*
　+ B₃∫dx_qdx_p-1φ(x_q)ψ(b, x_q)^*ψ(x_p-1, a)φ(x_p-1)^*
　+ B₄∫dx_qdx_pφ(x_q)ψ(b, x_q)^*ψ(a, x_p)φ(x_p)^*

= Aφ(b)^*φ(a)
　+ φ(b)^*∫dx [f₁(x)ψ(x, a) + f₂(x)ψ(a, x)]
　+ ∫dx [f₃(x)ψ(x, b)^*+ f₄(x)ψ(b, x)^*]φ(a)
　+ ∫dx dyφ(x)ψ(x, b)^*[B₁ψ(y, a) + B₂ψ(a, y)]φ(y)^*
　+ ∫dx dyφ(x)ψ(b, x)^*[B₃ψ(y, a) + B₄ψ(a, y)]φ(y)^*.

A = Σ_k≠q-1Σ_i≠p-1∫Dx [・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・]^* [・・・φ(x_i-1)ψ(x_i, x_i+1)φ(x_i+2)・・・].

B₁ = B₂ = B₃ = B₄ = 1.

f₁(x_p-1) = Σ_k≠q-1(・・・∫dx_p-2∫dx_p・・・)[・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・]^* [・・・φ(x_p-2)φ(x_p)・・・],
f₂(x_p) = Σ_k≠q-1(・・・∫dx_p-1∫dx_p+1・・・)[・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・]^* [・・・φ(x_p-1)φ(x_p+1)・・・],
f₃(x_q-1) = Σ_k≠p-1(・・・∫dx_q-2∫dx_q・・・)[・・・φ(x_q-2)φ(x_q)・・・]^* [・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・],
f₄(x_q) = Σ_k≠p-1(・・・∫dx_q-1∫dx_q+1・・・)[・・・φ(x_q-1)φ(x_q+1)・・・]^* [・・・φ(x_k-1)ψ(x_k, x_k+1)φ(x_k+2)・・・],

∴ f₁(x) = f₂(x) = f₃(x)^* = f₄(x)^*.

∫Dx Φ[・・・, x_q-1, b, x_q, ・・・]^* Φ[・・・, x_p-1, a, x_p, ・・・]

= Aφ(b)^*φ(a)
　+ φ(b)^*∫dx [f₁(x)ψ(x, a) + f₁(x)ψ(a, x)]
　+ ∫dx [f₁(x)^*ψ(x, b)^*+ f₁(x)^*ψ(b, x)^*]φ(a)
　+ ∫dx dyφ(x)ψ(x, b)^*[ψ(y, a) + ψ(a, y)]φ(y)^*
　+ ∫dx dyφ(x)ψ(b, x)^*[ψ(y, a) + ψ(a, y)]φ(y)^*

= Aφ(b)^*φ(a)
　+ φ(b)^*∫dx f₁(x)[ψ(x, a) + ψ(a, x)]
　+ ∫dx f₁(x)^*[ψ(x, b)^* + ψ(b, x)^*]φ(a)
　+ ∫dx dyφ(x)[ψ(x, b)^* + ψ(b, x)^*] [ψ(y, a) + ψ(a, y)]φ(y)^*.

If A^-1/2f₁(x) = φ(x)^*, this expression reduces to the following one.

{A^1/2φ(b)^* + A^-1/2∫dx f₁(y)^*[ψ(y, b)^* + ψ(b, y)^*] } {A^1/2φ(a) + A^-1/2∫dx f₁(x)[ψ(x, a) + ψ(a, x)] }.

However, it is not so.

Therefore, the p-th degree of freedom and the q-th degree of freedom are entangled with each other.
Though entanglement exists, ∫Dx Φ[・・・, x_q-1, b, x_q, ・・・]^* Φ[・・・, x_p-1, a, x_p, ・・・] has no dependence on q - p at all, and so this result can not reproduce the result of the old quantum mechanics.

Author Yuichi Uda, Write start at 2015/05/04/22:22JST, Last edit at 2015/06/25/21:02JST