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The original forms(equation0, equation1) of the Uda equation have the following problem.
It is written by using [δ/δχ(t)]2, so it can not have a solution Φ corresponding to a solution ψ of pre-Uda Schrodinger equation as follows.
Φ[χ] = exp[α∫dt φ(χ(t), t)]
where ψ(x, t) = exp φ(x, t) and ih(∂/∂t)ψ(x, t) = Hψ(x, t) and H is a hamiltonian of pre-Uda quantum mechanics.
[δ/δχ(t)]exp[α∫dτ φ(χ(τ), τ)] = α[∂φ(χ(t), t)/∂χ(t)]exp[α∫dτ φ(χ(τ), τ)]
∴ [δ/δχ(t)]2exp[α∫dτ φ(χ(τ), τ)] = α2[∂φ(χ(t), t)/∂χ(t)]2exp[α∫dτ φ(χ(τ), τ)] + α exp[α∫dτ φ(χ(τ), τ)] [δ/δχ(t)] [∂φ(χ(t), t)/∂χ(t)].
However [δ/δχ(t)] [∂φ(χ(t), t)/∂χ(t)] does not exist.
That is to be said as follows.
[δ/δχ(t)]2 hits twice exactly same one time point.
Because of it, any unentangled quantum history can not be a solution of the Uda equation.

This may be one of the physics which the Uda equation newly brings.
Exact Solution of Uda Equation (3) and in Exact Solution of Uda Equation (4), I researched along this possibility.)
However, it may not be a physics, and it may be a mere harm from rigorism.
[I heard that the existing quantum field theory can not stand rigorously. In fact, eigenvalues of ψ†(x)ψ(x) of the existing quantum field theory are 0 and δ(0). Then if we can use unrigorous notion δ(0), we have only to think that [δ/δχ(t)] [∂φ(χ(t), t)/∂χ(t)] = δ(0)∂2φ(χ(t), t)/∂χ(t)2.]
Then, in this page, I develop a method to avoid the above difficulty.
Even if the above difficulty is not a mere harm from rigorism but physically significant, the method developed below gives us a bypass to another significant physics than the above difficulty without going through unessential complexity forest of rigorism.

Functional representing quantum history in Uda's grammar:
Φ[χ] = Πk φk(χ(k/α)).

Eigenfunction uE of hamiltonian in pre-Uda quantum mechanics:
{ [-h2/(2m)](d/dx)2 + V(x) } ue(x) = e ue(x)
where e is a non-negative real number.

Basis ΦE of functional space in Uda's grammar:
ΦE[χ] = Πk uE(k/α)(χ(k/α))
where E is a function.

General functional Φ is expanded as follows.
Φ = ΣE F[E] ΦE
that is to say Φ[χ] = ΣE F[E] ΦE[χ]
where E is a function and F is a complex valued functional.

ΦE[χ(□ - 1/α)]
= Πk uE(k/α)(χ(k/α - 1/α))
= Πk' uE(k'/α + 1/α)(χ(k'/α))
= Πk uE(k/α + 1/α)(χ(k/α))
= Πk uE'(k/α)(χ(k/α))
= ΦE(□+1/α)[χ]
where E' = E(□ + 1/α).

Φ[χ(□ - 1/α)] - Φ[χ]
= ΣE F[E] ΦE[χ(□ - 1/α)] - ΣE F[E] ΦE[χ]
= ΣE F[E] ΦE(□+1/α)[χ] - ΣE F[E] ΦE[χ]
= ΣE F[E] { ΦE(□+1/α)[χ] - ΦE[χ] }

Σk{ [-h2/(2m)][∂/∂χ(k/α)]2 + V(χ(k/α)) } ΦE[χ]
= Σk{ [-h2/(2m)][∂/∂χ(k/α)]2 + V(χ(k/α)) } Πk' uE(k'/α)(χ(k'/α))
= Σk E(k/α) Πk' uE(k'/α)(χ(k'/α))
= Σk E(k/α) ΦE[χ].

Σk{ [-h2/(2m)][∂/∂χ(k/α)]2 + V(χ(k/α)) } Φ[χ]
= ΣE F[E] Σk E(k/α) ΦE[χ]

Uda equation is rewritten as follows.
ihΣE F[E] [ΦE(□+1/α) - ΦE] = (1/α)ΣE F[E] Σk E(k/α) ΦE.
(ih/α)∫DE F[E] limε→0E(□+ε) - ΦE]/ε = ∫DE F[E] ∫dt E(t) ΦE.

limε→0E(□+ε) - ΦE]/ε = ∫dt [dE(t)/dt] [δ/δE(t)] ΦE.
(ih/α)∫DE F[E] limε→0E(□+ε) - ΦE]/ε
= (ih/α)∫DE F[E] ∫dt [dE(t)/dt] [δ/δE(t)] ΦE.
= -(ih/α)∫DE ∫dt [dE(t)/dt] { δF[E]/δE(t) } ΦE

-(ih/α)∫DE ∫dt [dE(t)/dt] { δF[E]/δE(t) } ΦE = ∫DE F[E] ∫dt E(t) ΦE
∴ -(ih/α)∫-∞dt [dE(t)/dt] [δ/δE(t)] F[E] = F[E] ∫-∞dt E(t) .

This equation says that F[E] must be zero if dE(t)/dt = 0 for all t and E(t) ≠ 0.
We can not accept this requirement because it restricts solutions too much.
Then I need to make the following correction to the above result.

In pre-Uda quantum mechanics, a wave-function is thought of as a representation of an abstract state vector representing a quantum state while it is also thought of as a representation of a wave.
The former way of understanding is what Paul Adrien Maurice Dirac taught to us and is called the abstract notation of quantum mechanics.
The latter is what Erwin Rudolf Josef Alexander Schrodinger thought.
In Dirac type understanding, it is thought that wave-functions different only in phase factor represent a same quantum state.
If it is right, in a stationary state solution of the Schrodinger equation:
ψ(x, t) = exp[-i(e/h)t] ue(x),
phase factor exp[-i(e/h)t] is not only removable but also replaceable with any other phase factor.
This is a direct denial of understanding as a wave-function represents a wave because it rejects metachronic vibration.
Then we must choose one of the two understandings, Dirac's or Schrodinger's.
We can not obey both understandings.
At the first glance, the phase factor problem above seems a fault of Dirac type understanding.
However Dirac type understanding brought us the successful gauge theory.
Moreover the difficulty of Uda equation mentioned above can not be overcome by another understanding than Dirac's.
Then now I choose the Dirac type understanding, and obey it below.
Dirac has worked many times after he died, how great he is!

Twist Remove Normalization:
ih(∂/∂t)ψ(x, t) = Hψ(x, t)
ψ'(x, t) = eiθ(t)ψ(x, t), θ(t) is real.
ih(∂/∂t)ψ'(x, t) = ih(∂/∂t)[eiθ(t)ψ(x, t)] = -h[dθ(t)/dt]eiθ(t)ψ(x, t) + eiθ(t)ih(∂/∂t)ψ(x, t)
 = -h[dθ(t)/dt]ψ'(x, t) + eiθ(t)Hψ(x, t) = -h[dθ(t)/dt]ψ'(x, t) + Hψ'(x, t)
 = [H - h dθ(t)/dt]ψ'(x, t)
ψ' is a solution if such θ exists.

Uda equation is corrected as follows.
∃C∈R; (ih/α) limε→0 {Φ[χ(□ - ε)] - Φ[χ] }/ε = [ C + ∫-∞dt { [1/(2m)][(-ih/α)δ/δχ(t)]2 + V(χ(t)) } ] Φ[χ]
Here C = -hθ(∞) + hθ(-∞).

Energy representation of Uda equation is corrected as follows.
∃C∈R; -(ih/α)∫-∞dt [dE(t)/dt] [δ/δE(t)] F[E] = F[E] [C + ∫-∞dt E(t)].

Unentangled Solution.
F[E] = exp[α∫dt f(E(t), t)], f(e, t) = -(i/h)et + g(e) + h(t)
-ih∫dt [dE(t)/dt][∂f(E(t), t)/∂E(t)]F[E] = F[E][C + ∫dt E(t)].
-ih∫dt [dE(t)/dt]∂f(E(t), t)/∂E(t) = C + ∫dt E(t).
-ih∫dt [dE(t)/dt][-(i/h)t + g'(E(t))] = C + ∫dt E(t).
-ih∫dt (d/dt)g(E(t)) - ∫dt [dE(t)/dt]t = C + ∫dt E(t).
-ih[g(E(∞)) - g(E(-∞))] - ∫dt (d/dt)[tE(t)] + ∫dt E(t) = C + ∫dt E(t).
-ih[g(E(∞)) - g(E(-∞))] - ∫dt (d/dt)[tE(t)] = C.
-ih[g(E(∞)) - g(E(-∞))] - [tE(t)]-∞ = C.
F[E] = δ(C + limt→+∞[ihg(E(t)) + tE(t)] - limt→-∞[ihg(E(t)) + tE(t)])exp{α∫-∞dτ[-(i/h)τE(τ) + g(E(τ)) + h(τ)] }.

Solution with Certain Energy.
∫DW exp{iα∫-∞dt W(t)[E(t) - e]} ~ Πtδ(E(t) - e).
Fe[E] = ∫DW exp{iα∫-∞dt W(t)[E(t) - e]}.
E(t)Fe[E] = eFe[E].
δFe[E]/δE(t) = iα∫DW W(t)exp{iα∫-∞dτ W(τ)[E(τ) - e]}.
iα∫-∞DW(t) W(t)exp{i W(t)[E(t) - e]} ~ δ'(E(t) - e).
E(t')δFe[E]/δE(t) = eδFe[E]/δE(t) if t' ≒ t.
[E(t) - e]δFe[E]/δE(t) = -Fe[E] ∵ xδ'(x) = -δ(x).

∴ E(t)δFe[E]/δE(t) = eδFe[E]/δE(t) - Fe[E].
I suppose that E(t')δFe[E]/δE(t) = eδFe[E]/δE(t) - δ(t' - t)Fe[E].
[dE(t')/dt']δFe[E]/δE(t) = δ'(t' - t)Fe[E].
[dE(t)/dt]δFe[E]/δE(t) = δ'(t - t)Fe[E] = 0.
C = -∫-∞dt e.

Author Yuichi Uda, Write start at 2018/05/11/13:13JST, Last edit at 2018/05/23/20:01JST