Wick's theorem

Remarks

The proof is by induction on $n$. One commutes annihilation parts to the right; each commutator $[\phi^+(x), \phi^-(y)]$ is a number, and with time ordering these numbers combine into $\Delta_F(x - y)$.

Taking the vacuum expectation value kills every normal-ordered product with at least one field. So $\langle 0 \rvert T\{\phi(x_1)\cdots\phi(x_n)\} \lvert 0 \rangle$ is $0$ for odd $n$, and for even $n$ it is the sum over perfect matchings of the indices of the products of propagators. Example: for $n = 4$ there are three matchings, so the four-point function is $\Delta_{12}\Delta_{34} + \Delta_{13}\Delta_{24} + \Delta_{14}\Delta_{23}$, with $\Delta_{ij} = \Delta_F(x_i - x_j)$.

The probabilistic form is Isserlis' theorem. For centred jointly Gaussian random variables $X_1, \dots, X_n$, $\mathbb{E}[X_1 \cdots X_n]$ is the sum over perfect matchings of the products of the covariances $\mathbb{E}[X_i X_j]$.

Going deeper

For a student

Wick's theorem is the engine of perturbation theory. Expanding $e^{-iH_{\text{int}}t}$ in the interaction picture gives vacuum expectations of products of free fields, and Wick's theorem turns each one into a sum of products of propagators. Each term is encoded by a Feynman diagram: the fields are the vertices and the pairs are the edges. For fermion fields the same theorem holds with a sign given by the parity of the permutation that brings each pair together.