To focus on the idea, I will be a bit fast and loose with notation. Please refer to the reference for the rigorous proof.
Firstly we note the mode expansion of \(\phi(x) \) (denote \(\phi_x\)) can be split into two terms $$ \phi_x=\phi_x^+ +\phi_x^-, $$ where \(\phi_x^+=\int [dp] a_p e^{-ipx}\) contains an annihilation operator while \(\phi_x^-=\int [dp] a_p^\dagger e^{ipx}\) contains a creation operator.
Now, in the case \(x^0>y^0\), the time-ordered 2-point function \(T(\phi(x)\phi(y))\) is $$ \phi_x\phi_y=\phi_x^+\phi_y^++\phi_y^-\phi_x^++\phi_x^-\phi_y^++\phi_x^-\phi_y^-+[\phi_x^+,\phi_y^-]=N(\phi_x\phi_y)+[\phi_x^+,\phi_y^-].$$ Everything but the commutator is normal-ordered. When \(y^0>x^0\), the commutator becomes \([\phi_y^+,\phi_x^-]\), with the rest of the terms still being normal-ordered. Thus $$ T(\phi_x\phi_y)=T( N(\phi_x\phi_y+[\phi_x^+,\phi_y^-]) ).$$ The contraction is the Feynman propagator $$T( N( [\phi_x^+,\phi_y^-] ) ) = D_F(x-y). $$ Therefore $$ T(\phi_x\phi_y)=N(\phi_x\phi_y) + D_F(x-y).$$
Moving on to the 3-point function in the case \(x_1>x_2>x_3\), (denote \(x_i\) as i) $$ \phi_1(\phi_2\phi_3)=\phi_1(N(\phi_2\phi_3)+D_F(2,3))=\phi_1D_F(2,3)+N(\phi_1^-\phi_2\phi_3)+\phi_1^+N(\phi_2\phi_3). $$ To make the last term normal-ordered, we need to commute \(\phi_1\) past \(\phi_2\phi_3\) giving all contractions involving \(\phi_1\), thus $$\phi_1\phi_2\phi_3=N(\phi_1\phi_2\phi_3)+\phi_1 D_F(2,3)+\phi_2 D_F(1,3)+\phi_3 D_F(1,2)$$
In general, Wick’s theorem breaks down an m-point correlation function into a sum of terms involving only normal-ordered operators and Feynman propagators (expressed as contractions) $$ T(\phi(x_1),\phi(x_2),\dots,\phi(x_m))=N(\phi(x_1)\phi(x_2)\dots\phi(x_m)+[\phi(x_1),\phi(x_2)]\dots\phi(x_m) \\+\phi(x_1)\phi(x_2)\dots[\phi(x_5),\phi(x_7)][\phi(x_6),\phi(x_8)]\dots\phi(x_m)+\text{all other contractions}).$$ This way, when applied to a vacuum state, only fully contracted terms will survive, reducing the growth of the number of terms in the m-point correlation function from \(2^m\) to at most \(\left(\begin{array}{c} m \\ 2 \end{array}\right)\), each being a product of Feynman propagators.
Reference
- Peskin and Schroeder, Chapter 4.3 Wick’s Theorem, An Introduction to Quantum Field Theory, 1995