Easy Physics: Wick's theorem

Wick's theorem is a theorem which links the time-ordering of fields to the normal-ordering of those.

$$\begin{align} T(\phi_{z_{1} } \phi_{z_{2} } \cdots \phi_{z_{n-1} } \phi_{z_{n } } ) & = \; \; : \phi_{z_{1} } \phi_{z_{2} } \cdots \phi_{z_{n-1} } \phi_{z_{n } } : \; + \; : \phi_{z_{3} } \cdots \phi_{z_{n-1} } \phi_{z_{n } } : \langle 0 | T( \phi_{z_{1} } \phi_{z_{2} }) | 0 \rangle \; + \; : \phi_{z_{2} } \phi_{z_{4} } \cdots \phi_{z_{n-1} } \phi_{z_{n } } : \langle 0 | T( \phi_{z_{1} } \phi_{z_{3} }) | 0 \rangle + \cdots \nonumber \\ & \quad \; + \; : \phi_{z_{1} } \phi_{z_{2} } \cdots \phi_{z_{n-3} } \phi_{z_{n-2 } } : \langle 0 | T( \phi_{z_{n-1} } \phi_{z_{n} }) | 0 \rangle \nonumber \\ & \quad \; + \; : \phi_{z_{5} } \cdots \phi_{z_{n-1} } \phi_{z_{n } } : \langle 0 | T( \phi_{z_{1} } \phi_{ z_{2} } ) | 0 \rangle \langle 0 | T( \phi_{z_{3} } \phi_{ z_{4} } ) | 0 \rangle \; + \; : \phi_{z_{5} } \cdots \phi_{z_{n-1} } \phi_{z_{n } } : \langle 0 | T( \phi_{z_{1} } \phi_{ z_{3} } ) | 0 \rangle \langle 0 | T( \phi_{z_{2} } \phi_{ z_{4} } ) | 0 \rangle \; + \cdots \nonumber \\ & \quad \; + \; : \phi_{z_{1} } \phi_{z_{2} } \cdots \phi_{z_{n-5} } \phi_{z_{n -4} } : \langle 0 | T( \phi_{z_{n-3} } \phi_{ z_{n-2} } ) | 0 \rangle \langle 0 | T( \phi_{z_{n-1} } \phi_{ z_{n } } ) | 0 \rangle \nonumber \\ & \quad \; + \; \text{sum over triply contracted terms} \nonumber \\ & \quad \; + \; \text{higher contractions} \; . \end{align}$$ The easiest example is for $n=2$: if $t_{A} > t_{B}$, we have $$\begin{align} {\rm LHS} = \phi_{A+} \phi_{B+} + \phi_{A+} \phi_{B-} + \phi_{A-} \phi_{B+} + \phi_{A-} \phi_{B-} \end{align}$$ and $$\begin{align} {\rm RHS} & = \quad : \phi_{A} \phi_{B} : + \langle 0 | T( \phi_{A } \phi_{ B } ) | 0 \rangle \\ & = \phi_{A+} \phi_{B+} + \phi_{A+} \phi_{B-} + \phi_{B+} \phi_{A-} + \phi_{A-} \phi_{B-} + [\phi_{A-}, \phi_{B+} ] \nonumber \\ & = \phi_{A+} \phi_{B+} + \phi_{A+} \phi_{B-} + \phi_{A-} \phi_{B+} + \phi_{A-} \phi_{B-} \; . \end{align}$$

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We have a lemma:
$$\begin{align} : \phi_{z_{1} } \phi_{z_{2} } \cdots \phi_{z_{n-1} } : \phi_{z_{n} + } & = \quad : \phi_{z_{1} } \phi_{z_{2} } \cdots \phi_{z_{n-1} } \phi_{z_{n} + } : + : \phi_{z_{2} } \cdots \phi_{z_{n-1} } : \langle 0 | T( \phi_{z_{1} } \phi_{z_{n} + } ) | 0 \rangle + : \phi_{z_{1} } \cdots \phi_{z_{n-1} } : \langle 0 | T( \phi_{z_{2} } \phi_{z_{n} + } ) | 0 \rangle + \cdots \nonumber \\ & \quad + : \phi_{z_{1} } \phi_{z_{2} } \cdots \phi_{z_{n-2} } : \langle 0 | T( \phi_{z_{n-1} } \phi_{z_{n} + } ) | 0 \rangle \end{align}$$ The simplest example of this is for $n=2$: $$\begin{align} (LHS) = \quad : \phi_{A} : \phi_{B+} = \phi_{A -} \phi_{B +} + \phi_{A + } \phi_{B + } \end{align}$$ and $$\begin{align} (RHS) = \quad : \phi_{A} \phi_{B+} : + \langle 0 | T ( \phi_{A} \phi_{B +} ) | 0 \rangle = : \phi_{A} \phi_{B+} : + [ \phi_{A- } \phi_{B +} ] = \phi_{A +} \phi_{B+} + \phi_{A - } \phi_{B +} \; . \end{align}$$ So we find that the lemma is true for $n = 2$.
Let us denote $\phi_{A}$ by $A$, and $\phi_{A \pm }$ by $A_{\pm }$. We also check the lemma for $n =3$: $$\begin{align} (LHS) = \quad : AB : Z_{+} = A_{+} B_{+} Z_{+} + A_{-} B_{-} Z_{-} + A_{+} B_{-} Z_{+} + B_{+} A_{- } Z_{+} \; , \end{align}$$ and $$\begin{align} (RHS) & = \quad : AB Z_{+} : + : A : [B_{-} , Z_{+} ] + : B : [A_{-}, Z_{+ } ] \; , \\ & = Z_{+} A_{+} B_{+} + Z_{+} A_{+} B_{-} + Z_{+} B_{+} A_{- } + Z_{+} A_{- } B_{- } \nonumber \\ & \quad + A_{+} B_{-} Z_{+} - A_{+} Z_{+} B_{-} + A_{-} B_{-} Z_{+} - A_{-} Z_{+ } B_{- } + B_{+} A_{-} Z_{+} - B_{+} Z_{+} A_{-} + B_{-} A_{-} Z_{+} - B_{-} Z_{+} A_{-} \nonumber \\ & = A_{+} B_{+} Z_{+} + A_{-} B_{-} Z_{-} + A_{+} B_{-} Z_{+} + B_{+} A_{- } Z_{+} \end{align}$$