Correspondence between Solution and Action

A solution of the new grammar version of the Shrödinger equation:
Φ[χ]＝Π_kG(χ(k/α),k/α;x₀,t₀)
can be cartoonized as a picture of a starfish.

The integrand of the Feynman path integral:
exp{(i/h)S[χ]}＝Π_kG(χ(k/α + 1/α), k/α + 1/α; χ(k/α), k/α)
can be cartoonized as a picture of a centipede.

Using these two pictures as a hint, I made up the following functional as a solution nearly corresponding to the action.
Φ[χ]＝[Π_k∫dx_k∫dt_kG(χ(k/α),k/α;x_k,t_k)]Π_jG(x_j+1, t_j+1; x_j, t_j)

This solution can be called a starfish having a centipede in place of a central disc.

[Π_k∫dξ(0,k/α) G(ξ(ε,k/α),ε;ξ(0,k/α),0)]Φ[ξ(0,□)]
= [Π_k∫dξ(0,k/α) G(ξ(ε,k/α),ε;ξ(0,k/α),0)][Π_s∫dx_s∫dt_sG(ξ(0,s/α),s/α;x_s,t_s)]Π_jG(x_j+1, t_j+1; x_j, t_j)
= [Π_k∫dx_k∫dt_k∫dξ(0,k/α) G(ξ(ε,k/α),ε;ξ(0,k/α),0)G(ξ(0,k/α),k/α;x_k,t_k)]Π_jG(x_j+1, t_j+1; x_j, t_j)
= [Π_k∫dx_k∫dt_k G(ξ(ε,k/α),k/α+ε;x_k,t_k)]Π_jG(x_j+1, t_j+1; x_j, t_j)
= [Π_k'∫dx'_k'∫dt'_k' G(ξ(ε,k'/α-ε),k'/α;x'_k',t'_k')]Π_jG(x'_j+1, t'_j+1; x'_j, t'_j)　where k'=k+αε, x'_k'=x_k, t'_k'=t_k
= [Π_k∫dx'_k∫dt'_k G(ξ(ε,k/α-ε),k/α;x'_k,t'_k)]Π_jG(x'_j+1, t'_j+1; x'_j, t'_j)
= [Π_k∫dx_k∫dt_k G(ξ(ε,k/α-ε),k/α;x_k,t_k)]Π_jG(x_j+1, t_j+1; x_j, t_j)
= Φ[ξ(ε,□-ε)]
∵ Π_j G(x'_j+1, t'_j+1; x'_j, t'_j)
　= Π_j' G(x'_j'+1, t'_j'+1; x'_j', t'_j')
　= Π_j' G(x_j+1, t_j+1; x_j, t_j)
　= Π_j G(x_j+1, t_j+1; x_j, t_j)
　where j'=j+αε

Therefore,
Φ[χ]＝[Π_k∫dx_k∫dt_kG(χ(k/α),k/α;x_k,t_k)]Π_jG(x_j+1, t_j+1; x_j, t_j)
is an approximate solution.

I hope that a path integral of this solution reproduce the same result as a path integral of exp{(i/h)S[χ]}.

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Last edited at 2013/04/21/14:09