Speculation of Time and Space as Properties of a Particle and Quantum Mechanics

In special relativity, energy and momentum are part of a 4-vector, but are also properties of a particle. X and t also form a 4-vector, but are not properties of a particle, but rather are measured relative to an external ruler or clock. Shifting the origin of the ruler/clock changes their value, but does not change the value of velocity and hence momentum.In (1), we called x and t running variables. 

    Special relativity deals with observations based on measurers in different frames moving at constant velocities. The ruler or clocks used by these people, however, may be shifted so there is no absolute property of time or x in this case either. Shifting a ruler or clock does not change the value of velocity, it is a property of the particle (assuming there are equally space gradations and clock intervals). In order to make x and t properties of a particle, we argue (as in (1)) that one may use A= -Et+px, a Lorentz invariant. For -EE+pp, (c=1), another Lorentz invariant, there is a link between all frames with rest mass squared (c=1). Within a specific frame (E,p exact numbers), -Et+px creates a link between special degenerate (in terms of A) x and t points, i.e.  x+ n constant/p and t + n constant/E. These keep A constant due to the minus sign. Thus E and p create an internal clock and ruler for a particle as we have argued in (1). This means that n constant/p are special “x” points (and the same with n*constant /E and time) and suggests a probability distribution in x with a period of constant/p. Thus E and p interacting with x,t allow the latter to be properties of a particle, but at the cost of introducing probability. 

      Given the notion of probability, this suggests that entropy should come into play. One may ask: Is entropy maximized subject to a constraint? If so, what constraint should one use? We argue for a constraint of ipx associated with the period constant /p, the “i” being associated with periodicity. In such a case, the Lorentz invariant -Et+px leads to exp(ipx) as a dynamic probability without the use of an eigenvalue equation.