  Free electromagnetic fields, satisfying Maxwell's equations with no charges
and electric currents, can be described by complex vector fields. In the
standard formulation with fields sharply dependent on position and time, one
obtains integrals that are mathematically ill-defined. This happens for the
massless Pauli--Jordan function, which is used to describe the time evolution
of fields and appears in the Poisson brackets for classical fields. This
difficulty can be solved by introducing smeared fields as linear functionals
with test functions. In this way, the massless Pauli--Jordan function becomes
a~tempered distribution, allowing a mathematically rigorous analysis.
