I'm am right now reading Chapter 4.3 (Wick's Theorem) in Peskin & Schroeder. It is said that

In the vacuum expectation value, any term in which there remain uncontracted operators gives zero (since $\langle 0 | N (\text{any operator}) | 0 \rangle = 0$).

My question is, why must the vev (vacuum expectation value) of an odd number of operators in Wick's Theorem be 0?


Wick's theorem tells us that $$ \mathcal{T}(\phi_1\dots\phi_N) =\ :\phi_1\dots\phi_N: + :\text{pairwise contractions}:$$ where $:\ :$ is normal ordering. Immediately from the definition of normal ordering (all annihilators to the right, all creators to the left), the expectation value of anything that is normal-ordered and not a constant vanishes because the annihilators on the right and the creators on the left just give zero on the vacuum if there is even a single annihilator or creator in the expression to be normal-ordered.

If $N$ is odd, you'll always have an operator left in the normal-ordered expressions on the right of Wick's theorem, whose expectation value is thus zero.


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