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friend

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For example, the action integral is invariant with respect to time, and this leads to conserved quantity of energy. This means that the energy will remain the same for any time translation. In other words, if we know the energy exactly, then the time variable could be anything and we don't know the time variable with any precision. This sounds like the uncertainty principle between time and energy. Or again, if the action integral is invariant wrt space translations, then the momentum is conserved. And when momentum is conserved, then we know what the momentum is exactly, but the space varaible could be anything. And this sounds like the uncertainty principle between position and momentum.

It seems the uncertainty principle can be derived from commutation relations, as shown here. For the commutator not being zero means we cannot measure both observables at the same tiem, which mean if we know one precisely, then we can't know the other precisely. And as I recall, commutation relations can be derived from symmetries, as shown here. So can commutators and uncertainty principles be developed from any continuous symmetry of the action integral?

For example, could a commutator and an uncertainty principle be derive between electric charge and phase angle? For electric charge is the conserved quantity of phase invariance of the action integral.