Conjugate Variables, Noether's Theorem and QM What is the underlying reason that the same pairs of conjugate variables (e.g. energy & time, momentum & position) are related in Noether's theorem (e.g. time symmetry implies energy conservation) and likewise in QM (e.g. $\Delta E \Delta t \ge \hbar$)?
 A: Good observation.
The pairs of variables in Noether's theorem are conjugate momenta. Basically they are generalized version of mechanical momentum. Interesting note is that even without relativity, energy is recognized as the momentum of time. Generalized momenta are extensively used in the Hamiltonian formulation of classical physics. In quantization, basically the Hamiltonian and most related structures are unchanged, so the conjugate pairs remain the same.
A: Both commutation and conservation work in the context of geometry, specifically the generation of transformations. It is important to understand that conjugate pairs like position and momentum relate as generators of translations. If these translations leave the system unchanged, i.e. you have a symmetry generated by the conjugate momentum, then the conjugate momentum must be conserved.
In quantum theory the conjugate pairs are not independent. This is because on a hilbert space the generators of translations and the coordinate they act on relate like the derivative and the coordinate, which don't commute.
So it really comes down to the geometry of the phase space of a system. In classical physics the phase space is just a normal manifold with the usual geometric structure. But in quantum theory the construction using translations on the hilbert space result in a non-commutative geometry. There is no way to avoid this.
