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I am studying the notes of David Tong on Statistical Field Theory (https://www.damtp.cam.ac.uk/user/tong/sft/sft.pdf). I don't understand how to formally get the result after Eq. 2.11, that is the following step: $$ \prod_{\textbf{k}}\Big[\mathcal{N}\int d\phi_{\textbf{k}}d\phi^*_{\textbf{k}}\exp{\Big(-\frac{\beta}{2V}(\gamma k^2+\mu^2)\Big)|\phi_{\textbf{k}}|^2}\Big]=\prod_{\textbf{k}}\mathcal{N}\sqrt{\frac{2\pi TV}{\gamma k^2+\mu^2}}\,, $$ where $\phi_\textbf{k}$ is a variable in a reciprocal lattice, ($\textbf{k}$ is a momentum vector in a discretized space). In particular the author says that since $\phi_{\textbf{k}}$ and $\phi^*_{\textbf{k}}$ are not independent the result is a square root. I don't understand how to compute the integral over two dependent variables.

As I can read here Complex Gaussian integral the result should be without the square root. It seems to me that to have the previous result one should integrate on $d|\phi_\textbf{k}|$ (indeed the presence of $d\phi_\textbf{k}$ is due to a previous step in which the author uses a Dirac delta $\delta^D(k_1+k_2)$ so that $k_2=-k_1$, and rename $k_1$ in $k$).

Ruth Murphy
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You are right when you point out that no square root should appear in $$ I_\mathbf{k} := \int_\mathbb{C} \mathrm{d}\phi_\mathbf{k}^*\mathrm{d}\phi_\mathbf{k}\, e^{\lambda|\phi_\mathbf{k}|^2} = \frac{\pi}{\lambda}, $$ because it is basically a two-dimensional gaussian integral. However, the problem comes from the index $\mathbf{k}$ in the product $\prod_\mathbf{k}$. Indeed, the author follows an implicit convention where no $\mathbf{k}$ appears twice. Yet, $\mathrm{d}\phi_\mathbf{k}^* = \mathrm{d}\phi_{-\mathbf{k}}$ implies that $I_\mathbf{k}I_{-\mathbf{k}} = I_\mathbf{k}^2$ and $$ \prod_{\mathrm{all}\,\mathbf{k}} I_\mathbf{k} = \prod_\mathbf{k} I_\mathbf{k} \cdot \prod_\mathbf{k} I_{-\mathbf{k}} = \prod_\mathbf{k} I_\mathbf{k}^2. $$ In consequence, one has $$ \prod_\mathbf{k} I_\mathbf{k} = \prod_{\mathrm{all}\,\mathbf{k}} \sqrt{I_\mathbf{k}} $$ and this is how the square root reappears.

Abezhiko
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