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This is part of the lecture note of QFT by Tong:

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I have a question on the last part of (6.27). Why does it hold? Actually this question is not about QFT and is about just integral. The typical way to show the integral is zero is to use odd/evenness of the integrand, but $e^{i\mathbf p \cdot (\mathbf x-\mathbf y)}$ spoils everything.

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The integral is zero because the integrand is zero:

$$\left(\delta_{ij}-\frac{p_ip_j}{|\vec p|^2}\right)p_i=p_j-p_j=0.$$

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