不変デルタ関数と +iε

; draft

Minkowski 計量は diag⁡(+−−−)\operatorname{diag}(+---) とする.

Klein–Gordon 方程式

(□+m2)ϕ(x)=0(□+m^2) ϕ(x) = 0

の Green 関数を調べよう.

場の積の期待値を

Δ+(x−y)≡⟨0∣ϕ(x)ϕ(y)∣0⟩,Δ−(x−y)≡⟨0∣ϕ(y)ϕ(x)∣0⟩Δ_+(x-y) ≡ ⟨0|ϕ(x)ϕ(y)|0⟩, \quad Δ_-(x-y) ≡ ⟨0|ϕ(y)ϕ(x)|0⟩

で定義する. これらの

(□+m2)Δ+(x)=(□+m2)⟨0∣ϕ(x)ϕ(0)∣0⟩=⟨0∣[(□+m2)ϕ(x)]ϕ(0)∣0⟩=0,(□+m2)Δ−(x)=(□+m2)⟨0∣ϕ(0)ϕ(x)∣0⟩=⟨0∣ϕ(0)[(□+m2)ϕ(x)]∣0⟩=0.\begin{gathered} (□+m^2) Δ_+(x) = (□+m^2) ⟨0|ϕ(x)ϕ(0)|0⟩ = ⟨0| [(□+m^2)ϕ(x)] ϕ(0) |0⟩ = 0, \\ (□+m^2) Δ_-(x) = (□+m^2) ⟨0|ϕ(0)ϕ(x)|0⟩ = ⟨0| ϕ(0) [(□+m^2)ϕ(x)] |0⟩ = 0. \\ \end{gathered}

交換子関数と反交換子関数は

Δ(x−y)≡⟨0∣[ϕ(x),ϕ(y)]∣0⟩=Δ+(x−y)−Δ−(x−y),Δ1(x−y)≡⟨0∣{ϕ(x),ϕ(y)}∣0⟩=Δ+(x−y)+Δ−(x−y).\begin{gathered} Δ(x-y) ≡ ⟨0|[ϕ(x),ϕ(y)]|0⟩ = Δ_+(x-y) - Δ_-(x-y), \\ Δ_1(x-y) ≡ ⟨0|\{ϕ(x),ϕ(y)\}|0⟩ = Δ_+(x-y) + Δ_-(x-y). \\ \end{gathered}

Klein–Gordon 演算子の線形性よりただちに,

(□+m2)Δ(x)=0,(□+m2)Δ1(x)=0(□+m^2) Δ(x) = 0, \quad (□+m^2) Δ_1(x) = 0 ΔF(x−y)≡⟨0∣T[ϕ(x)ϕ(y)]∣0⟩=θ(x0−y0)Δ+(x−y)+θ(y0−x0)Δ−(x−y)Δ_{\rm F}(x-y) ≡ ⟨0|T[ϕ(x)ϕ(y)]|0⟩ = θ(x^0-y^0) Δ_+(x-y) + θ(y^0-x^0) Δ_-(x-y) (□+m2)ΔF(x)=−δ4(x)(□+m^2) Δ_{\rm F}(x) = - δ^4(x) (□+m2)ΔF(x)=(∂02+m2)θ(x0)Δ+(x)+(∂02+m2)θ(−x0)Δ−(x)(□+m^2) Δ_{\rm F}(x) = (∂_0^2+m^2) θ(x^0) Δ_+(x) + (∂_0^2+m^2) θ(-x^0) Δ_-(x)