# How is this change of basis performed?

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.

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.