The Role of Time and Lorentz Invariances in the Quantum Wavelength

  In general, x and t are considered free variables taking on all real values (with t>0). A particle’s velocity may be calculated from x(t). For the Lorentz invariant A= -Et+px one may consider E for a free particle to follow from Integral F dx and  p from Integral Fdt. If one wishes to put these on an equal footing, then Fdx requires a dt and Fdt a dx. This means t is replaced with dt and x with dx in -Et+px. What does it mean to have t and x intervals associated with a specific E,p and continuous x(t) motion? There does not seem to be either an x or t interval. There are, however, dx and dt intervals associated with the creation of E and p from p=0.In other words, these dt, dx intervals are physically real properties just like p and E and are manifested experimentally.

   If one imposes time reversal symmetry, not only should these intervals represent a kind of probability, but they should be symmetric. Thus, x=0 and t=0 are starting points such that A=0. A is a Lorentz invariant and so the end point should also have A=0 for time reversal to apply. This time reversal leads to two symmetric distributions, one in time with hbar/E as a period and one in space with hbar/p as a wavelength.This, however, does not show the direction of motion, say in the spatial distribution. Thus the symmetric x-probability distribution should ultimately be linked to both p and -p i.e should have a vector nature, although this has to be formally proved.

    A full x-probability distribution linked with p should show the direction of motion. One may write F(px) for the even symmetric function described above and G(px) and G(-px) for skewed full probability distributions such that G(px) + G(-px) = 2F(px). This implies G(px) = F(xp) + B(px) where B is an odd function. We discussed the idea of G(px) being the sum of an odd and even function in previous notes, but did not try to give a reason why F(px) should be an even function with probability endpoints of 0. In this note, we argue that this is due to time reversal invariance. 

   A free Newtonian particle follows x= vt and this should hold on average for a quantum particle which travels through space measured by an external ruler and clock. Above, however, we discussed probability distributions due to p. In fact their sharpness d/dx is proportional to p. How does one eliminate x dependence? As we argued in a previous note (2), one may take the AND condition  G(px)G(-px) which should remove all motion.This should be a constant.  It is not just motion which is removed, but overall impulse which is the cause of the x-distributions. The square nature of G(px)G(-px) explains why quantum mechanics deals with square root probabilities. These are the probabilities which are linked to impulse, hence to interactions, but the spatial distribution cos(px) from exp(ipx), does not describe the motion x=vt which is also real.The problem, as we show, is that G(px) cannot be real. This forces one to adopt a two dimensional probability and ultimately obtain exp(ipx) i.e. the two dimensional vector one might expect.