On the Role of the Subsidiary Condition in Quantum Electrodynamics

The $S$-matrix in quantum electrodynamics may be calculated alternatively from the Hamiltonian density $-j^{\mu }\mathrm{}\left(x\right)\mathrm{}{A}_{\mu }\mathrm{}\left(x\right)\mathrm{}$ and a Hamiltonian in which the Coulomb interaction of the charges and the interaction of the currents with the transverse field are separated. Both procedures are equivalent. Care must be taken, however, to define correctly the initial states, in particular the vacuum state. The definition of the vacuum state in both representations is discussed and the equivalence of the corresponding $S$-matrices is proven.