Minimal Electromagnetic Coupling, Principle

External fields are introduced in the relativistic free-particle operator hy the minimal substitutions (17). One should at this point carefully note that the principle of minimal electromagnetic coupling requires the specification of particle charge. This becomes particularly important for the Dirac equation which describes not only the electron, but also its antiparticle, the positron. We are interested in electrons and therefore choose q = — 1 in atomic units which gives the Hamiltonian... 

Consider now the inclusion of external fields. We do so by the recipe provided by the principle of minimal electromagnetic coupling. The quantized version of the substitutions (114) can be written in covariant notation as... 

Electric and magnetic fields are introduced in terms of their potentials through the principle of minimal electromagnetic coupling [11]... 

For the same reason it is not clear, how to modify the equation for the inclusion of external fields. The principle of minimal coupling p —> p — A, E E + V for the (scalar) square-root Klein-Gordon equation was critizised by J. Sucher [4], who states that there are solutions ip x) and electromagnetic potentials, such that the Lorentz transformed solution is not a solution of the equation with the Lorentz-transformed potentials. Moreover, the nonlocal nature of the equation means that the value of the potential at some point influences the wave function at other points and it is not clear at all how one can interpret this. 

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