Wicks theorem

Last updated Aug 5, 2023

Wick’s theorem relates time ordered products of free fields to all possible contractions.

Definition: Contractions

A contraction pairs two free fields in a product of of free fields, e.g.: 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 $$[\phi_-(x_1),\phi_+(x_2)]\Theta(t_1 - t_2) + [\phi_-(x_2),\phi_+(x_1)]\Theta(t_2 -t_1)$$

The idea is to commute operators until they are normal ordered:

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 and annihilation operators acting on the vacuum states vanish: $$\braket{0|:\phi_0(x_1)\dotsc \phi_0(x_n):|0}$$

Wick’s theorem

The time ordering product of free fields is the normal ordering of the fields plus all possible contractions: $$T{\phi_0(x_1)\dotsc \phi_0(x_n)} = ; :\phi_0(x_1) \dotsc \phi_0(x_n) + \begin{matrix} \text{all possible} \\ \text{contractions} \end{matrix}:$$

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). We can show with the expression for the propagator from above that the time ordered product takes the form of Wicks theorem: $$T{\phi_0(x_1)\phi_0(x_2)} = ; :\phi_0(x_1)\phi_0(x_2): + D_F(x_1,x_2)$$ This can be generalized to time ordered products of arbitrary many fields by an iduction type proof:

1. We proved Wicks theorem for n-1=2 fields.
2. Assume Wicks theorem holds for n-1 fields
3. 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.

Since normal ordered products vanish in matrix elements