Smeared Field Description of Free Electromagnetic Field

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.