Using Wick’s theorem, evaluate the following expressions:
- T($\phi_1 N(\phi_2 \phi_3))$
- $T(N(\phi_1 \phi_2)N(\phi_3 \phi_4))$
For the first expression, I would do the following: $N(\phi_1 N(\phi_2 \phi_3)+ [\phi_1(N(\phi_2\phi_3))])$, where [] represent contractions. Then writing out $N(\phi_2 \phi_3)$ as $\phi_2 \phi_3$ we should get contractions between $\phi_1\phi_2$, $\phi_1 \phi_3$ and $\phi_2 \phi_3$.
For the second expression I would do the similar procedure to get:
$N(\phi_1 \phi_2 \phi_3 \phi_4 + [12]+[13]+[14]+[23]+[24]+[34]+[12][34]+[13][24]+[14][23])$.
However, the solutions do not have the contractions terms of the fields that were normally ordered. Why is that the case?
