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Uda Equation of Dirac Field @ Quantum History Theory @ Problems.

The Hamiltonian:
H = ∫d3x ψ†(x0(-ihkk + mc2)ψ(x).

The quantization conditions:
α(x), ψ†β(y)]+ = δαβδ3(x - y), [ψα(x), ψβ(y)]+ = 0.

The above expressions are of the existing theory.
Now I start explaining my theory of quantum history.

α(x), ψ†β(y)]+ = δαβδ4(x - y), [ψα(x), ψβ(y)]+ = 0.
I assume so by seeing [Z(t), Z†(t')]+ = δ(t - t') and [Z(t), Z(t')]+ = 0 in
Uda Equation of Fermionic Mechanics.

The left hand side of Uda equation:
ih-∞dt∫d3x Σα=14ψ†α(x, t)(∂/∂t)ψα(x, t) |Φ>
= ih-∞d(ct)∫d3x ψ†(x, t)[∂/∂(ct)]ψ(x, t) |Φ>
= ih∫d4x ψ†(x)(∂/∂x0)ψ(x) |Φ> (∵ x0 = ct)
= ih∫d4x ψ†(x)∂0ψ(x) |Φ>.
I assume so by seeing ih-∞dt Z†(t)(d/dt)Z(t) |Φ> in
Uda Equation of Fermionic Mechanics.
The concept is to shift all spacetime points at which there is a particle by -ε in time-like direction.

Because (γ0γ0)αβ = δαβ, the left hand side of Uda equation is
∫d4x ψ†(x)γ0(ihγ00)ψ(x) |Φ>.

Uda Equation:
∫d4x ψ†(x)γ0(ihγ00)ψ(x) |Φ> = ∫-∞dt∫d3x ψ†(x, t)γ0(-ihkk + mc2)ψ(x, t) |Φ>
∴ ∫d4x ψ†(x)γ0(ihγ00)ψ(x) |Φ> = ∫-∞d(ct)∫d3x ψ†(x, t)γ0(-ihγkk + mc)ψ(x, t) |Φ>
∴ ∫d4x ψ†(x)γ0(ihγ00)ψ(x) |Φ> = ∫d4x ψ†(x)γ0(-ihγkk + mc)ψ(x) |Φ>
∴ ∫d4x ψ†(x)γ0(-ihγμμ + mc)ψ(x) |Φ> = 0.
This expression was first published in 15th @ May 2019 @ News at about 2019/05/15/15:00(Japan Standard Time).

In case, below I note down details.

α(x), ψ†β(y)]+ = ψα(x)[ψβ(y)]† + [ψβ(y)]†ψα(x)
= δαβδ(x1 - y1)δ(x2 - y2)δ(x3 - y3)δ(x0 - y0).

α(x), ψβ(y)]+ = ψα(x)ψβ(y) + ψβ(y)ψα(x) = 0.

-∞dx1-∞dx2-∞dx3-∞dx0 Σα=14Σβ=14α(x)]† [-ihΣμ=030γμ)αβμ + mc(γ0)αβ] ψβ(x) |Φ> = 0.

σ1σ2 = -σ2σ1 = iσ3, σ2σ3 = -σ3σ2 = iσ1, σ3σ1 = -σ1σ3 = iσ2,
σ1σ1 = σ2σ2 = σ3σ3 = σ0.

( [γμ, γν]+ )αβ = (γμγν + γνγμ)αβ = 2gμνδαβ,
g00 = 1, g11 = g22 = g33 = -1,
μ≠ν ⇒ gμν = 0.

Author Yuichi Uda, Write start at 2019/05/18/15:19JST, Last edit at 2019/05/20/16:46JST