Wick contraction for same indices (Zee, "QFT in a Nutshell")

[rob_dd]

Hey,

I thought I understood Wick contractions but a formula in Zee's "Quantum Field Theory in a Nutshell" disproved me:

In the section on Feynman Diagrams it is tried to evaluate the "four-point Green's function" in (I.7.10) by the integral $$
\int_{-\infty}^\infty \left ( \prod_m \mathrm{d} q_m \right ) e^{-\frac{1}{2} q \cdot A \cdot q} q_i q_j q_k q_l \left [ 1 - \frac{\lambda}{4!} \sum_n q_n^4 + O(\lambda^2) \right ] / Z(0,0)
$$
which is said to be
$$ (A^{-1})_{ij} (A^{-1})_{kl} + (A^{-1})_{ik} (A^{-1})_{jl} + (A^{-1})_{il} (A^{-1})_{jk} - \lambda \sum_n (A^{-1})_{in} (A^{-1})_{jn} (A^{-1})_{kn} (A^{-1})_{ln} + O(\lambda^2)$$

My question is: In the term $\sim \lambda$ why can't I make contractions like ij kl nn nn ? It seems to be impossible that i make connections nn or at least this term should be zero (?). But in the previous text in this book such connections were calculated and also in the derivation this should be possible I think. Another explanation would be that the diagonal elements of A vanish but I could not explain why.

Thank you for any help,
Robin

 

[eljose79]

Dear Robin,

Thank you for your question. Wick contractions can be a tricky concept, and it's not uncommon for people to have difficulties understanding them at first. Let me try to clarify things for you.

In the expression given in Zee's book, the four-point Green's function is written as a sum of terms, each containing four momentum variables (q_i, q_j, q_k, q_l). Each of these four variables can be contracted with each other, as long as they are in the same term. For example, in the first term of the expression, we have q_i and q_j, which can be contracted with each other. Similarly, in the second term, we have q_i and q_k, which can also be contracted with each other. However, in the third term, we have q_i and q_l, which cannot be contracted with each other because they are not in the same term.

Now, coming to your question about the term ∼λ, let's look at the expression (A^-1)_{in}(A^-1)_{jn}(A^-1)_{kn}(A^-1)_{ln}. Here, we have four different variables, i, j, k, l, and n. We can make contractions between any two of these variables, as long as they are in the same term. For example, we can contract i with n, j with n, k with n, or l with n. However, we cannot contract i with j, i with k, i with l, j with k, j with l, or k with l because they are not in the same term. Therefore, the only possible contractions are in the form of nn, which gives us (A^-1)_{nn}(A^-1)_{nn}(A^-1)_{nn}(A^-1)_{nn}. This is why we have only one term in the ∼λ part of the expression.

I hope this helps clarify things for you. If you have any further questions, please don't hesitate to ask.