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For the field operators (fermions)

$$\hat{\Psi}^\dagger_\sigma(x) = \dfrac{1}{\sqrt{V}}\sum_k e^{-ikx}~\hat{a}^\dagger_{k,\sigma}$$

$$\hat{\Psi}_\sigma(x) = \dfrac{1}{\sqrt{V}}\sum_k e^{ikx}~\hat{a}_{k,\sigma}$$

I want to prove the following anti-commutator relationship: $$\left\{\hat{\Psi}_{\sigma}(x), \hat{\Psi}^\dagger_{\sigma^\prime}(x^\prime)\right\} = \delta(x-x^\prime)\delta_{\sigma,\sigma^\prime}$$

I have $$ \begin{align}\left\{\hat{\Psi}_{\sigma}(x), \hat{\Psi}^\dagger_{\sigma^\prime}(x^\prime)\right\} &= \dfrac{1}{V}\sum_{k,k^\prime} e^{-ikx}e^{-ik^\prime x^\prime} \{\hat{a}_{k,\sigma}, \hat{a}^\dagger_{k,\sigma}\}\\ &=\dfrac{1}{V}\sum_{k,k^\prime} e^{-ikx}e^{-ik^\prime x^\prime} \delta(k-k^\prime)\delta_{\sigma,\sigma^\prime} \\ &= \dfrac{1}{V}\sum_{k} e^{-ik(x-x^\prime)}\delta_{\sigma,\sigma^\prime}\end{align} $$

But I don't know how to show $$\dfrac{1}{V}\sum_{k} e^{-ik(x-x^\prime)}=\delta(x-x^\prime)\,.$$ I would be thankful for your help!

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Part of the problem, may lie in the fact that this is expressed in term of summations instead of integrals over $k$. If you had integrals, you'd end up with $$ \int \exp[-ik(x-x')] dk = 2\pi\delta(x-x') , $$ which we know from the Fourier properties of a Dirac delta function.

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