Generator of spatial evolution of the electromagnetic field

Abstract: Starting with Maxwell's equations and defining normal variables in the Fourier space, we write the equations of temporal evolution of the electromagnetic field with sources in the Hamiltonian and Lagrangian forms, making explicit all intermediate steps often omitted in standard textbooks. Then, we follow the same steps to write the equations of evolution of this field along a spatial dimension in the Hamiltonian and Lagrangian forms. In this way, we arrive at the explicit form of the generator of spatial evolution of the electromagnetic field with sources and show that it has a physical meaning of the modulus of momentum transferred through a given plane orthogonal to the direction of propagation. In a particular case of free field this generator coincides with the projection of the full momentum of the field on the propagation direction, taken with a negative sign. The Hamiltonian and Lagrangian formulations of the spatial evolution are indispensable for a correct quantization of the field when considering its spatial rather than temporal evolution, in particular, for a correct definition of the equal-space commutation relations.