If I have a Schrodinger Equation, in 4d space, of the form


and set $w=ict$ and $A=-mc^2/2$

then I get


which is equivalent to the Klein Gordon Equation. This means that the Klein Gordon Equation is mathematically the same as a variant of The time-independent Schrodinger Equation in 4 dimensional spatial position space with one of the coordinates being imaginary.

In the time-independent Schrodinger Equation there is a potential operator, which is a function of position space, not a wavefunction, or derivative of a wavefunction, and which describes the interaction between particles. I was wondering if there is something analogous to the potential operator in relativistic quantum mechanics, in the sense of being a function of spacetime, and being neither a wavefunction nor a derivative of a wavefunction. For instance in relativistic quantum mechanics is there an equation of the form

$$B\Psi=\frac{\hbar^2}{2m}\left(\frac{\partial^2\Psi}{\partial{x^2}}+\frac{\partial^2\Psi}{\partial{y^2}}+\frac{\partial^2\Psi}{\partial{z^2}}+\frac{\partial^2\Psi}{(ic)^2\partial{t^2}}\right)+L\Psi$$ with $L$ being a function of spacetime and describing an interaction?


1 Answer 1


A potential term corresponding to a given coordinates function is highly problematic in relativistic dynamics, even at the classical level. The reason is that it would imply an instantaneous action at a distance. The problem could be solved with retarded potentials. However, such a solution is not practical for systems with more than one particle. That's one of the main reasons for reformulating relativistic quantum mechanics in the form of a quantum field theory (QFT). The potential interaction between point-like particles at different space points is substituted by a local interaction between fields at the same point.


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