Study on Uncertainty Relations as a Consequence of the Lorentz Transformations

Abstract

A macroscopic object equipped with synchronized clocks is examined. General physical relations are directly derived from Lorentz transformations for the case of one-dimensional motion (along the X axis) – the uncertainty relation of the object's x coordinate and the projection of its impulse along the X axis, p_x, and the uncertainty relation of the object's observation time, t, and its energy, E. The relations take the form: \(\Delta\)p_x\(\Delta\)x \(\ge\) H and \(\Delta\)E\(\Delta\)t \(\ge\) H. The H value in the relation has action dimensions and is dependent upon the precision of the rod's clocks and its mass. It is shown that if the macroscopic object in and of itself performs the function of an ideal physical clock, the relations derived in the limiting case then take the form of \(\Delta\)p_x\(\Delta\)x \(\ge\) h and \(\Delta\)E\(\Delta\)t \(\ge\) h, where h is the Planck constant.