Nonlocality Imposed by Special Relativity on Quantum Free Particles?

In a previous note (1) we considered a rest mass mo at x=0 and t=to and argued this transforms to E’, p’ with x’/t’=v. Thus a single system may be Lorentz transformed to create particles with mo and all velocities. We suggested that all of these particles are on the same footing in a sense. In this note we try to investigate what this “footing” is. In particular, we suggest nonlocality. A particle with rest mass mo, likely occupies some region of space, but is treated as a point (i.e. x=0) which may be the center of mass.The idea of a particle considered as a point is carried over into special relativity i.e motion. In Newtonian mechanics the particle is also considered to be a point. We argue that even though a particle “system” with mo at x=0 and t=to is a point and that after moving there may still be an x’/t’ = v pertaining to one point representing the system, the localization of mo at x=0 may be transformed into a localization in every constant moving frame. This localization seems to be linked to the idea of x and t being independent in -Et+px just as a particle at x=0 and t=to at rest treats x and t as independent. Thus we argue that the localization is the “same footing” of particles with different v and hence momenta mov. The dot product -Et+px = A is -moto for the rest frame which tends to 0 for to→0 . For any frame, t=b/E and x=b/p yields the same A=0. Furthermore x=b/p and t=b/E is a recipe which holds in all frames and describes nonlocality or a “trapped” length associated with the moving particle. This trapped length is not associated with time because x and t are independent. In other words the system moves with a speed of v=x’/t’, but the particle is “trapped” within a wavelength of the order 1/p in the sense that there is a cycling probability as the particle moves through space ignoring time. One does not simply have a point at rest transformed into a point in motion describing all aspects of the moving particle. If one considers x=0 and x=1/p, particles with different p values may follow the same probability distribution associated with energy flow in the region of one wavelength. If one ignores the motion v=x’/t’, then since time is ignored it is as if one has a particle with different probabilities to be at different x points. In a bound quantum state, time is physically dropped and these probabilities are linked to actual densities. For a moving particle (in one direction) one must consider the point motion v=x’/t’, but also kind of periodic trapping of the particle in space (not linked to time). In order to quantify this for a quantum free particle, one may consider the idea of a particle system with a length proportional to 1/p which is consistent with P(x)=constant. In particular, one may take the square root of 1 so to speak to obtain exp(ipx) and exp(-ipx). This also follows from eigenvalue equations of A= -Et+px i.e. -id/dx exp(ipx) = p exp(ipx). It should be noted (as stated above) that exp(ipx) does not involve time. Thus the trapping is in space and may occur over a long period of time if the velocity is very slow i.e. 1/p is very large. In a sense the particle is trapped in a kind of periodic motion in space. There is locality in the sense of position x’/t’=v, but not in the sense of the trapped periodic motion. The nonlocality seems to exist one opposite scales than the usual point velocity. For example, if a particle has a very small velocity then x’/t’=v→0 i.e. one thinks in terms of a very small x’, but 1/p is very large