A Hilbert space for the classical electromagnetic field

The synthetic Maxwell equation, uniting all Maxwell equations within the framework of a Clifford algebra, can be treated as a first-order wave equation. A Hilbert space of its solutions describing classical free electromagnetic fields is introduced. This Hilbert space can be called classical, which means that the Planck constant is absent. The scalar square of an element of this space is the total energy of the field. The time independence of the scalar product is demonstrated. The time and space translation generators are found; they are shown to not coincide with the energy and momentum operators.