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In David Tong's Gauge Theory notes on page 136 eq. (3.29) he performed the following change of basis, during a momentum cut-off regularization scheme.

$$ \sum_{n} \overline{\phi}_{n}\gamma^5 e^{(\gamma^{\mu}D_{\mu})^2/\Lambda^2}\phi_{n}= \int \frac{d^4k}{(2\pi)^4}\text{Tr}\left(\gamma^5 e^{-ikx}e^{(\gamma^{\mu}D_{\mu})^2/\Lambda^2}e^{ikx}\right) $$

Where $D_{\mu}$ is the covariant derivative, $\phi_{n}$ are eigenspinors of the Dirac operator, and $\Lambda$ is the regulator.

This seems like some kind of Fourier transfor from position to momentum space, but I don't how exactly. How to show this? Or is there a source I could look at where this is computed? Tong elaborates a bit in the next page, but I still don't understand how to get this.

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The sum is over all possible field configurations. These are expanded in plane waves with momentum $k$ so that "sum" becomes an integral over "k". But the field also have spinor indices, so you also need to sum over that, which becomes the trace.

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