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The content of this page is toward Uda equation of Free Dirac Field and was developed in Uda Equation of Dirac Field @ Quantum History Theory @ Problems. In this page, I treat a system of two real degrees x and y of freedom. I let the Hamiltonian H of this system be as H = a(x^{2} + y^{2}). At first, I define a complex variable z as z = x + iy. Then H = az^{*}z where z^{*} is the complex conjugate of z. Quantization of this system is done as follows. X is an Hermitian operator whose eigenvalue is x. Y is an Hermitian operator whose eigenvalue is y. Z ≡ X + iY. [Z, Z]_{+} ≡ ZZ + ZZ = 0 ∴ Z^{2} = 0. [Z, Z†]_{+} ≡ ZZ† + Z†Z = 1. H = aZ†Z. Z^{2} = 0 ⇔ X^{2}  Y^{2} + iXY + iYX = 0 ⇔ [X, Y]_{+} = i(X^{2}  Y^{2}). ZZ† + Z†Z = 1 ⇔ 2X^{2} + 2Y^{2} = 1 ⇔ X^{2} + Y^{2} = 1/2. H = aZ†Z = a(X^{2} + Y^{2} + iXY  iYX) = a(i[X, Y]_{} + 1/2). Z†Zn> = nn>, <nn> = 1, n ∈ R. n^{2} = n^{2}(n>, n>) = (nn>, nn>) = (Z†Zn>, Z†Zn>) = <nZ†ZZ†Zn> = <nZ†(1  Z†Z)Zn> = <nZ†Zn> ∵ Z^{2} = 0 ∴ n^{2} = n<nn> = n ∴ n = 0 or 1. (Z0>, Z0>) = <0Z†Z0> = 0<00> = 0 ∵ Z†Z0> = 00>. ∴ Z0> = 0. (Z†1>, Z†1>) = <1ZZ†1> = <1(1  Z†Z)1> = 0 ∵ Z†Z1> = 1>. ∴ Z†1> = 0. Z†ZZ†0> = Z†(1  Z†Z)0> = Z†0> ∵ Z0> = 0. (Z†0>, Z†0>) = <0ZZ†0> = <0(1  Z†Z)0> = <00> = 1 ∵ Z0> = 0. ∴ Z†0> = 1>. Z†ZZ1> = 0 = 0Z1> ∵ Z^{2} = 0. (Z1>, Z1>) = <1Z†Z1> = <11> = 1 ∵ Z†Z1> = 1>. ∴ Z1> = 0>. As a summary, Z0> = 0, Z1> = 0>, Z†0> = 1>, Z†1> = 0. The above explanations are about quantum state. Now I start explaining my theory about quantum history. A function which represents a classical history. ν: R → {0,1}. ν(t) = 0 or 1. A basis of the quantum history vector space: ν>. A quantum history vector: ∫Dν Φ[ν] ν>. An operator which acts on a specific time: Z†(t)Z(t)ν> = ν(t)ν>. Uda Equation: i ∴ i ν'(t) = ν''(t) = ν(t) if t ＜ a or t ＞ a + ε. ν'(t) = 0 and ν''(t) = 1 if a ≦ t ≦ a + ε. Uda equation: i Another formulation can stand as follows. At first, I put a condition [Z(t), Z†(t')]_{+} = δ(t  t'), [Z(t), Z(t')]_{+} = 0. This condition realizes that [Z†(t)Z(t), Z†(t')Z(t')]_{} = 0. Then I replace lim_{ε→+0} (1/ε)∫Dν {Φ[ν(□  ε)]  Φ[ν]} ν> with ∫_{∞}^{∞}dt Z†(t)(d/dt)Z(t)∫Dν Φ[ν] ν>. Uda equation: i I removed factor α heuristically by seeing dimensionality. Below I explain what ∫_{∞}^{∞}dt Z†(t)(d/dt)Z(t) Φ> means. ∫_{∞}^{∞}dt Z†(t)(d/dt)Z(t) Φ> = lim_{ε→0} (1/ε)∫_{∞}^{∞}dt Z†(t)[Z(t + ε)  Z(t)] Φ> = lim_{ε→0} (1/ε)∫_{∞}^{∞}dt [Z†(t  ε)  Z†(t)]Z(t) Φ> Z†(t  ε)Z(t) creates a particle at time t  ε after annihilating a particle at time t only if there is a particle at time t. Z†(t)Z(t) creates a particle at time t after annihilating a particle at time t only if there is a particle at time t. So, lim_{ε→0} (1/ε) [Z†(t  ε)  Z†(t)]Z(t) Φ> is a kind of partial differentiation only if there is a particle at time t. lim_{ε→0} (1/ε) [Z†(t  ε)  Z†(t)]Z(t) Φ> = 0 if there is no particle at time t. As for partial differentiation, the following formula holds. lim_{ε→0} (1/ε) [F(x_{1} +ε, ・・・, x_{n} + ε)  F(x_{1}, ・・・, x_{n})] = Σ_{k=1}^{n} lim_{ε→0} (1/ε) [F(x_{1}, ・・・, x_{k1}, x_{k} + ε, x_{k+1}, ・・・, x_{n})  F(x_{1}, ・・・, x_{n})] On the analogy of this, the variation of Φ> caused by shifting all times when there is a particle by ε is thought to be ε∫_{∞}^{∞}dt Z†(t)(d/dt)Z(t) Φ> as ε → 0. If a quantum history is not entangled in timelike direction at all, to shift all times when there is a particle by ε is to replace quantum state at time t with quantum state at time t + ε for all t. This is the very concept of left hand side of Uda equation. 

Author Yuichi Uda, Write start at 2019/05/17/16:03JST, Last edit at 2019/05/20/16:27JST  






