“For particles to be described by relativistic quantum fields, however, there were no corresponding classical fields. We know of only two classical fields in nature, the electromagnetic and gravitational fields. Where and how do we find the classical fields whose quanta correspond to particles satisfying the KleinโGordon or the Dirac equations? And it is here that we find one of the fundamental conceptual shifts needed to proceed to the next level. Relativistic quantum-mechanical wave equations such as the KleinโGordon and Dirac equations are to be reinterpreted as classical field equations at the same level as Maxwellโs equation for the classical electromagnetic field! This is definitely a leap of faith.
Overnight the wave amplitudes for particles (that is, the particleโwave dualities) were turned into corresponding classical fields and the wave equations of relativistic quantum mechanics were turned into corresponding ‘classical’ equations for the classical fields. No equations were modified and all notations remained intact. The wave amplitude ฯ(x) became the classical field ฯ(x) and relativistic quantum-mechanical equations became wave equations for classical fields. This turned out to be one of the most subtle conceptual switches in the history of physics. This was the first instance โ it would not be the last โ in which matter emulated radiation. This is precisely how we arrived, in the early days of 1930s and 1940s, at the very beginning of quantum field theory of matter โ equations for matter simply emulating those for radiation. So, at this point, every wave equation for matter as well as radiation is a classical wave equation for classical fields, some real (Maxwellโs equations) and others ‘imitations’ (relativistic quantum mechanical wave equations). We have the truly classical field of the electromagnetic field satisfying
โฮผโฮผAฮฝ = 0,
and the ‘imitation’ classical fields, which are the redressed relativistic quantum-mechanical wave equations, satisfying equations such as those of KleinโGordon and Dirac, but now viewed from this point forward as classical field equations:
(โฮผโฮผ + m2)ฯ(x) = 0
and
(i ฮณฮผโฮผ โ m)ฯฮฑ(x) = 0.
Strictly speaking, the classical KleinโGordon or Dirac field does not exist in the macroscopic scale. No signals are transmitted by these ‘fields’ from one point to another in space in the same way radio signals are carried by the classical radiation field. Reinterpretation of these relativistic quantum-mechanical wave equations as classical field equations is the first preliminary step toward establishing the quantum field theory of matter particles. Once these ‘imitation’ classical fields are quantized in exactly the same manner as the electromagnetic field, the resulting theory of matter particles interacting with photons โ quantum electrodynamics โ turned out to be the most successful theory for elementary particles to date. In this sense, the redressing of relativistic quantum-mechanical wave equations into ‘imitation’ classical field equations is one of many examples of ‘the end justifying the means.’ That is its rationalization.”
—The Story of Light by M.Y. Yan