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.
An energy-time uncertainty essentially means therefore a distance-time uncertainty [although more complicated than simply in the minimum, in order of h-cross].
So the non-commutation of energy-time must also be, I think, translated into distance-time. [– I have shown this in my OPERA draft and other analysis articles]
The fact that such an uncertainty exists between distance and time is exactly worked out by me, but any non-commutation nature between distance and time is not checked. [are they non-commuting? I am saying they were never checked perhaps but they are non-commuting. This follows from their dependence definition from energy. But stand-alone we may be using commutation of distance and time]
So there is a correlation between various variables in a physical system, and commuting variables may actually due to this correlation behave like non-commuting ones, leading to new forms of uncertainty relations, not necessarily and most likely not; in terms of h-cross.
This may actually be a clarion call for the remaining inconsistencies of Quantum-Relativity. [This one eg explains OPERA anomaly, fuzziness of pictures taken by camera etc .. much explained in the last 1 year on my website ]
I have also worked out the uncertainty relations from wave-particle duality [mostly]