Comment on ref. 8
“It is also known that ”the set of four first-order equations comprising the Dirac equation is generally equivalent to, and may be reduced to, a single fourth order equation” ([V. G. Bagrov and D. M. Gitman. Exact solutions of relativistic wave equations :. Kluwer, 1990, p.28]), but this author failed to find a direct reference to the relevant derivation so far.”
However, later I approached the authors of the referenced book, and Prof. V.G. Bagrov explained that the above quote from his book only relates to currently known rigorous solutions of the Dirac equation and added the following: “I am not aware of a rigorous and complete mathematical proof that the Dirac equation with arbitrary given electromagnetic field can be written as independent partial differential equations of the fourth order, each of which is a partial differential equation for only one function of four variables.”
Therefore, the following result of my preprint may also be new: the Dirac equation with an arbitrary electromagnetic field is equivalent (at least locally) to one partial differential equation of the fourth order for one of the components of the Dirac spinor function (provided a certain function of electromagnetic field does not vanish). The other components of the Dirac spinor function can be expressed as functions of that one component and its derivatives.
Of course, I cannot be sure this result was not previously published elsewhere.
I am grateful to Prof. V.G. Bagrov for the above clarification.