Wicks theorem
To calculate interacting correlation functions, we have to calculate matrix elements of time ordered products of free fields like e.g. $$\braket{0|T{\phi_0(x_1)\phi_0(x_2)\phi_0(x)^3\phi_0(y)^3}|0}$$ Wick’s theorem relates time ordered products of free fields to normal ordered products and contractions
Definition: Normal ordering:
Normal ordering $:\dotsc :$ of operators orders all creation operators $a^\dagger_p$ to the left and all annihilation operators $a_p$ to the right, e.g.: $$:(a^\dagger_p + a_p)(a^\dagger_k + a_k): = a^\dagger_ka_p + a^\dagger_p a_k + a_pa_k + a^\dagger_p a^\dagger_k.$$ This means normal ordered vacuum matrix elements vanish, since the creation operators acting on the vacuum states on the right vanish: $$\braket{0|:\phi_0(x_1)\dotsc \phi_0(x_n):|0}=0$$
Definition: Contractions
A contraction pairs two free fields in a product of of free fields, e.g.: wickcontractionexample.png In general a contraction of two free fields will give a factor of a propagator, for the expression above, e.g.: $$D_F(x_1,x_2)D_F(x_3,x_4) + D_F(x_1,x_3)D_F(x_2,x_4)$$ Formally a contraction is defined as the feynman propagator given by $$D_F(x_1,x_2)=\braket{0|T{\phi_0(x_1)\phi_0(x_2)}|0}=[\phi_-(x_1),\phi_+(x_2)]\Theta(t_1 - t_2) + [\phi_-(x_2),\phi_+(x_1)]\Theta(t_2 -t_1),$$ where $\phi_+$ and $\phi_-$ correspond to the creation and annihilation part of the free field respectively. This can be seen using the commutation relation $[a_k,a^\dagger_p]=(2\pi)^3\delta^3(\vec{p}-\vec{k}).$
Wick’s theorem
The time ordering product of free fields is the normal ordering of the fields plus all possible contractions: $$T{\phi_0(x_1)\dotsc \phi_0(x_n)} = ; :\phi_0(x_1) \dotsc \phi_0(x_n) + \begin{matrix} \text{all possible} \\ \text{contractions} \end{matrix}:$$ All possible contractions means a products of Feynman operators for all possible pairings of fields.
Proof of Wick’s theorem
Exemplary proof for $$T{\phi_0(x_1)\phi_0(x_2)}$$ When splitting $\phi_0(x) = \phi_+(x)+\phi_-(x)$ with the $+$ part corresponding to the creation $a^\dagger_p$, we can write $$D_F(x_1,x_2) = \braket{0|T{\phi_0(x_1)\phi_0(x_2)}|0} = [\phi_-(x_1),\phi_+(x_2)]\Theta(t_1 - t_2) + [\phi_-(x_2),\phi_+(x_1)]\Theta(t_2 -t_1),$$ which can be seen when inserting the commutator $[a_k,a^\dagger_p]$ We write out the time ordering for $t_1 > t_2$ and $t_1 < t_2$ explicitly and commute all terms so that they are normal ordered (this will give terms with commutators): For $t_1>t_2$: $$T{\phi_0(x_1)\phi_0(x_2)}=\phi_+(x_1)\phi_+(x_2) + \phi_+(x_1)\phi_-(x_2) + \phi_-(x_1)\phi_+(x_2)+\phi_-(x_1)\phi_-(x_2).$$ In this expression everything is normal ordered except $\phi_0(x_1)\phi_+(x_2)$. This term we can rewrite as $$\phi_0(x_1)\phi_+(x_2)=\phi_0(x_2)\phi_1(x_1)+[\phi_-(x_1),\phi_+(x_2)].$$ Then the above expression is the normal ordering of the free fields plus the commutator term: $$T{\phi_0(x_1)\phi_0(x_2)}=:\phi_0(x_1)\phi_0(x_2): + [\phi_-(x_1),\phi_+(x_2)], \quad t_1 > t_2.$$ We get the same for $t_2 > t_1$ by exchanging $x_1$ and $x_2$. $$T{\phi_0(x_1)\phi_0(x_2)}=:\phi_0(x_2)\phi_0(x_1): + [\phi_-(x_2),\phi_+(x_1)], \quad t_2 > t_1.$$ Since $:\phi_0(x_1)\phi_0(x_2):=:\phi_0(x_2)\phi_0(x_1):$ and the commutator terms with corresponding Theta functions for the time regimes gives one Feynman propagator $$T{\phi_0(x_1)\phi_0(x_2)} = ; :\phi_0(x_1)\phi_0(x_2): + D_F(x_1,x_2)$$ This can be generalized to time ordered products of arbitrary many fields by an iduction type proof:
- We proved Wicks theorem for n-1=2 fields.
- Assume Wicks theorem holds for n-1 fields
- Add another field $$T{\phi_0(x_1)\dotsc\phi_0(x_n)} = [\phi_+(x_1) + \phi_-(x_1)]:\phi_0(x_2)\dotsc\phi_0(x_n) + \text{contractions}:$$ To include the new field in the normal ordered product we need to account for the terms that result from commuting $\phi_-(x_1)$ to the right. This will result in the commutators of $\phi_-(x_1)$ with $\phi_+(x_k)$, which leads to the contractions aka factors of feynman propagotors for commuting with every other field.
Since normal ordered products vanish in matrix elements what remains when calculating matrix elements are only the contractions aka Feynman propagators.