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A functional Φ representing an entangled quantum history obeys the new grammar version of Schrödinger equation:

where α is a new physical constant with the dimension of (time)-1 introduced by Yuichi Uda and χ(□-ε) is a function defined as
∀t∈R; [χ(□-ε)](t) = χ(t-ε).

This equation describes the physical law for a particle with mass m in one-dimensional space.
V is a function representing a potential energy.

I named this equation Uda equation.

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I found the Uda equation by the following schematic deduction which is an application of
the correspondence principle.

Φ[χ] = Πkφ(χ(kε), kε)　[Mnew(Φ) = Mold(φ)]
ih(∂/∂t)φ(x, t) = H(x, -ih∂/∂x)φ(x, t) ・・・ the schrödinger equation as an old equation

(1/ε){Φ[χ(□-ε)] - Φ[χ]}
= (1/ε){Πkφ(χ(kε-ε), kε) - Πkφ(χ(kε), kε)}
= (1/ε){Πkφ(χ((k-1)ε), kε) - Πkφ(χ(kε), kε)}
= (1/ε){Πsφ(χ(sε), (s + 1)ε) - Πkφ(χ(kε), kε)}　(s=k-1)
= (1/ε){Πsφ(χ(sε), sε+ε) - Πsφ(χ(sε), sε)}
= Σks≠kφ(χ(sε), sε)](∂/∂t)φ(χ(kε), t)|t=kε　(product rule)
= (ih)-1Σks≠kφ(χ(sε), sε)]H(x, -ih∂/∂x)φ(x, t)|x=χ(kε), t=kε
= (ih)-1ΣkH(χ(kε), -ih∂/∂χ(kε))Πsφ(χ(sε), sε)
= (ihε)-1εΣkH(χ(kε), -ih∂/∂χ(kε))Φ[χ]
→ (ihε)-1∫dt H(χ(t), -ihεδ/δχ(t))Φ[χ]　　[εΣk→∫dt, ∂/∂χ(kε)→εδ/δχ(t)]

∴ (ihε)(1/ε){Φ[χ(□-ε)] - Φ[χ]} = ∫dt H(χ(t), -ihεδ/δχ(t))Φ[χ]

By setting ε for 1/α in the above result, we see the Uda equation as a new equation.

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