  We first present a summary of the quantization of the electromagnetic field
in position space representation, using two main approaches: the Landau-Peierls
approach in the Coulomb gauge and the Bialynicki-Birula approach, based on the
Riemann-Silberstein vector. We describe both in a framework that starts with a
classical Hamiltonian structure and builds the quantum model in a bosonic Fock
space by a precisely defined principle of correspondence. We show that the two
approches are completly equivalent. This is formulated by showing that there is
a unitary map between the Fock spaces that makes them isomorphic. Since all the
physically measurable quantities can be expressed in terms of scalar products,
this implies that the two quantizations lead to exactly the same physical
properties. We show furthemore that the isomorphism is preserved in the time
evolutions. To show the equivalence, we use the concepts of helicity and
frequency operators. The combination of these two operators provides a
formulation that allows one to make the link between these two methods of
quantization in a precise way. We also show that the construction in the
Bialynicki-Birula quantization that avoids the presence of negative eigenvalues
in the Hamiltonian, in analogy with the one for the Dirac equation for
electrons and positrons, can be performed through an alternative choice of the
canonical variables for Maxwell's equations.
