Notion of Potential From Special Relativity?

    In a previous note (1) we argued that a potential V(x), which exists in Newtonian mechanics, may be written as Sum over k  Vk exp(ikx) (i.e. in a Fourier series). In another note (2) we suggested  that the free particle wavefunction form exp(ipx) may be obtained from special relativity i.e. from the Lorentz invariant -Et+px. Thus in (1) we argued that the potential delivers impulse hits in a quantum mechanical scenario.

    In this note we revisit the emergence of exp(ipx) from special relativity, which like quantum mechanics, does not include acceleration i.e. a frame moves at a constant speed. (In Newtonian mechanics a particle moving within a constant dx is also said to have a constant speed, but this speed is differs in the next dx.) Given exp(ipx) in the context of special relativity, we consider the possibility of a system of impulse hits which may change p to p1 etc i.e. a set of exp(ipx)s. This leads to an average <pp> = {Sum over p a(p)exp(ipx) pp } / {Sum over p a(p) exp(ipx)}. 

   A free particle satisfies: EE = pp + momo  (c=1). If one wishes to retain a constant E (energy), one may replace pp with <pp> and imagine a function V(x) such that -(E-V(x)) (E-V(x)) + <pp> - momo. With p→-id/dx one obtains the Klein-Gordon equation. Taking a nonrelativistic limit yields the Schrodinger equation.

 We suggest the notion of potential may appear directly from special relativity together with the notion of a px dependent dynamic probability, namely exp(ipx) which is a quantum mechanical function. We further argue that the Newtonian approach ignores the information contained in -Et+px and uses a smoothing procedure.