Quantum Field Theory

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Non Relativistic QFT.

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