To Physicists: do you think “it would be a new invention that uncertainty relations do not (necessarily) follow from non-commutation, the non-commutation leading to uncertainty relations was just one possible way of proving that simple relations between such non-commuting variables [e.g. momentum and position] exist.
One of the reasons for this could be; such uncertainty relations might generally also exist between commuting variables [such as distance and energy ], maybe nobody has checked if distance and energy also do not commute ? in consistency with the non-commutation of energy-time ?
eg an energy variable has in it already all of the following: distance, time, their ratio = velocity and momentum [and mass]. Which means automatically, there is a contribution of each of those variables into energy, hence an uncertainty of energy automatically extends to same in distance and momentum and so on.