Quantum Momentum Scaling px and Special Relativity

The classical action (relativistic or nonrelativistic) may be written with v=x/t and x and t varied independently to yield dA/dx partial = p and dA/dt partial = -E. Thus A = -Et+px is a solution. This expression also appears in classical wave forms and in the electromagnetic description of a photon (from Maxwell’s laws). -Et+px is a Lorentz invariant and so is directly linked to special relativity and different frames moving with constant velocity. In (1) we argued that x and t should be treated as independent in A. This suggests that p is a scaling in space and E in time. Thus there should be physical lengths proportional to 1/p and physical time intervals proportional to 1/E.

   Special relativity considers different frames (moving with constant velocity) and 4-vectors so it seems to be quite different from a scenario with t and x separated and scaling of x by p. We argue that although -Et+px is a Lorentz invariant and as such may be used to find E(v) and p(v), quantum mechanics is usually performed in a special frame, namey one in which V(x) or V(x,v) the potential is at rest. (For electromagnetism one treats (A, phi) as a 4-vector, but this is not the norm.) As a result, energy En found in a rest frame of the potential also represents “rest energy” so En is binding energy which is added to rest mass energy. In other words motion in the rest frame of a potential creates rest energy = rest mass.

  We argue that V(x) may be written as a sum over k Vk exp(ikx) and so no time related terms appear, only momentum linked ones. Thus momentum is the key feature for the interaction, even though velocity and time are still in the picture (just sometimes not directly in the interaction scheme). For example, a two slit interference or single slit diffraction does not depend on time or speed although these are certainly in the problem. For a bound state v and t are also in the problem as one calculates average kinetic energy using pp/2m = .5 p v which is rate dependent, but there is no acceleration which is a time-dependent phenomenon. Rather one deals with an ensemble of exp(ipx)’s because these interact with the potential in a wavelike manner to create the effect of acceleration. exp(ipx) a momentum wave linked to the scaling of x by p in px is not only involved in an interaction with V(x), but represents extra information from the classical mechanical picture. exp(ipx) for a known rest mass mo means one knows v=x/t=dx/dt, yet there is an extra piece of information namely the phase px which is involved in even “free” interactions such as two idendical particle with different phases arriving at the same x at the same t or the average kinetic energy in a bound state. Velocity v is converted into a momentum expression through p/mo in the norelativistic case and so exp(ipx) governs the whole scenario. Acceleration then becomes a mathematical abstraction of the motion of vrms(x) for any quantum bound energy level  with the real physics associated with exp(ipx) being ignored.