$$
[a_i,a_j^\dagger]_{\pm} = \delta_{i,j} \\
[a_i,a_j]_{\pm} = 0 \\
[a_i^\dagger,a_j^\dagger]_{\pm}=0
$$
+ for Fermions, - for Bosons.
Plane Wave:
$$
\phi_\boldsymbol{k} = \frac{1}{\sqrt{V}} e^{i\boldsymbol{kr}}
$$
Field Operator:
$$
\psi(\boldsymbol{r},t) = \frac{1}{\sqrt{V}} \Sigma_\boldsymbol{k}
a_\boldsymbol{k}(t) e^{i\boldsymbol{kr}}
$$