quantum propagator (changes) in nLab

causal propagatorΔ S =Δ +−Δ −\begin{aligned}\Delta_S & = \Delta_+ - \Delta_- \end{aligned}
A−\phantom{A}\,\,\,-
iℏΔ S(x,y)= ⟨[Φ(x),Φ(y)]⟩\begin{aligned} & i \hbar \, \Delta_S(x,y) = \\ & \left\langle \;\left[\mathbf{\Phi}(x),\mathbf{\Phi}(y)\right]\; \right\rangle \end{aligned} Peierls-Poisson bracket advanced propagatorΔ +\Delta_+ iℏΔ +(x,y)= {⟨[Φ(x),Φ(y)]⟩ | x≥y 0 | y≥x\begin{aligned} & i \hbar \, \Delta_+(x,y) = \\ & \left\{ \array{ \left\langle \; \left[ \mathbf{\Phi}(x),\mathbf{\Phi}(y) \right] \; \right\rangle &\vert& x \geq y \\ 0 &\vert& y \geq x } \right. \end{aligned} future part of
Peierls-Poisson bracket retarded propagatorΔ −\Delta_- iℏΔ −(x,y)= {⟨[Φ(x),Φ(y)]⟩ | y≥x 0 | x≥y\begin{aligned} & i \hbar \, \Delta_-(x,y) = \\ & \left\{ \array{ \left\langle \; \left[\mathbf{\Phi}(x),\mathbf{\Phi}(y) \right] \; \right\rangle &\vert& y \geq x \\ 0 &\vert& x \geq y } \right. \end{aligned}past part of
Peierls-Poisson bracket Wightman propagatorΔ H =i2(Δ +−Δ −)+H =i2Δ S+H =Δ F−iΔ −\begin{aligned} \Delta_H &= \tfrac{i}{2}\left( \Delta_+ - \Delta_-\right) + H\\ & = \tfrac{i}{2}\Delta_S + H \\ & = \Delta_F - i \Delta_- \end{aligned} ℏΔ H(x,y) =⟨Φ(x)Φ(y)⟩ =⟨:Φ(x)Φ(y):⟩⏟=0 =+⟨[Φ (−)(x),Φ (+)(y)]⟩\begin{aligned} & \hbar \, \Delta_H(x,y) \\ & = \left\langle \; \mathbf{\Phi}(x) \mathbf{\Phi}(y) \; \right\rangle \\ & = \underset{ = 0 }{\underbrace{\left\langle \; : \mathbf{\Phi}(x) \mathbf{\Phi}(y) : \; \right\rangle}} \\ & \phantom{=} + \left\langle \; \left[ \mathbf{\Phi}^{(-)}(x), \mathbf{\Phi}^{(+)}(y) \right] \; \right\rangle \end{aligned} positive frequency of
Peierls-Poisson bracket,
Wick algebra-product,
2-point function
=\phantom{=} of vacuum state
=\phantom{=} or generally of
=\phantom{=} Hadamard state Feynman propagatorΔ F =i2(Δ ++Δ −)+H =iΔ D+H =Δ H+iΔ −\begin{aligned}\Delta_F & = \tfrac{i}{2}\left( \Delta_+ + \Delta_- \right) + H \\ & = i \Delta_D + H \\ & = \Delta_H + i \Delta_- \end{aligned} ℏΔ F(x,y) =⟨T(Φ(x)Φ(y))⟩ ={⟨Φ(x)Φ(x)⟩ | x≥y ⟨Φ(y)Φ(x)⟩ | y≥x\begin{aligned} & \hbar \, \Delta_F(x,y) \\ & = \left\langle \; T\left( \; \mathbf{\Phi}(x)\mathbf{\Phi}(y) \;\right) \; \right\rangle \\ & = \left\{ \array{ \left\langle \; \mathbf{\Phi}(x)\mathbf{\Phi}(x) \; \right\rangle &\vert& x \geq y \\ \left\langle \; \mathbf{\Phi}(y) \mathbf{\Phi}(x) \; \right\rangle &\vert& y \geq x } \right.\end{aligned}time-ordered product