Quantum Free Particle exp(-iEt+ip dot r) and State Probability Part 3

   In Part 2, we suggested the existence of a free particle probability P(x,t,E,p) and argued that it must be Lorentz invariant as equations and formulas should appear the same as written in terms of the variables of various rest frames moving at constant speeds relative to each other. This led to P(-Et+px) and we saw that energy and momentum conservation followed. 

    Here, we consider the following. Imagine that one wishes to introduce a momentum and energy conserving probability and does not wish to make it Lorentz invariant. In previous notes, we argued that one would be led to exp(iC1 E) and exp(iC2 p) where C1 and C2 are constants related to units and p is along an x-axis. We argued that for time reversal invariance exp(iC2p) → exp(-iC2p) and suggested that this is unphysical.

   Here, we show that using exp(iC1 E) and exp(iC2 p) leads to problems if one compares results from mixed frames (v and -v) to a rest frame. Given that probability exp(iC1E) and exp(iC2 p) only contain E and p, one may formally consider such probabilities as being created using different frames. For example, one may take a rest particle and create p using a frame moving with -v (relative to the rest frame) and -p using a frame moving with v. exp(iC2 p) provides no information about a frame, it is only linked to a momentum consideration. This leads, we argue, to issues, and forces one to include x,t as well as E,p, i.e. use Lorentz invariance when dealing with such probabilities. 

  As a result, we argue that there is no way to avoid introducing x,t into such a probability and as a result, quantum free particle features such as wavelength and frequency emerge. In other words, when one thinks only in terms of E and p, as in Newtonian mechanics, one forces a fixed x,t frame. If, however, one wishes to consider p and E as resulting from different frames and use probabilities,  these must combine p,E and x,t to define the frame. Then one may actually mix probabilities linked with different frames.