The question of a canonical quantization of the electromagnetic field as a system with two primary constraints is analyzed. Attention is focused on the following aspects of the problem: 1) the conditions on physical states; 2) the ``old'' and ``new'' Gupta formalisms; 3) the interpretation of the operators $\hat\alpha_0$, $\hat\alpha_0^+$; 4) the vacuum in quantum electrodynamics. The starting point for the analysis is Dirac's quantization recipe for systems with constraints. It is shown that the Fermi formalism with a condition on the physical states is completely correct.