Extended Faddeev–Jackiw canonical quantization for the Maxwell electrodynamics

By using the extended Faddeev–Jackiw (FJ) formalism, we develop the canonical quantization for the Maxwell electrodynamics. Besides, we consider the results found by means of the application of the Dirac formalism to this theory. Consequently, we note that these formalisms generate the same constraints and generalized brackets. Thus, we find that the extended FJ and the Dirac formalisms are equivalent, at least for the present theory. Therefore, we believe that this fact allows us to think of a possible unified canonical quantization formalism between these formalisms. In this way, the Dirac formalism would be the Hamiltonian version of this unified formalism and the extended FJ one, its Lagrangian version. On the other hand, because the extended FJ formalism takes into account the classification of constraints as first and second-class, it is possible to propose the Feynman path integral quantization through this formalism, in accordance with the Faddeev or the Faddeev–Senjanovic procedures. In this situation, as with the canonical quantization, we believe that it is possible that there exists a unified Feynman path integral quantization formalism between the Hamiltonian version based on the Dirac formalism and the Lagrangian one based on the extended FJ formalism. For this proposal to make sense, we must show that both versions produce the same diagrammatic structure. In future papers, we will consider this last issue for different gauge field theories.