Noether's theorem vs. Heisenberg uncertainty principle

In continuation of another question about Noether's theorem I wonder whether there exists some kind of relationship between this theorem and the Heisenberg uncertainty principle.

Because both the principle and the theorem relate energy with time, momentum with space, direction with angular momentum. When this is a general fact then e.g. electrical charge and electrostatic potential(*) should be partners in an uncertainty relationship too. Are they?

I feel that these results look so basic and general that I hope that a pure physical reasoning (without math or only with a minimal amout of math) exists.

Also compare this question where again momentum and space are connected, this time through a Fourier transform.

(*) i.e. electric potential and magnetic vector potential combined.

Answer

Noether theorem is as valid in CM(*) as in QM(**). It deals with conservation laws and symmetries. In CM the variables are certain, in QM they may be uncertain.

HUP belongs to QM and gives a limitation on canonically conjugated variable uncertainties in a given state.

If some variable in QM is uncertain, it does not mean its expectation value is not conserved. A superposition of free motions states $e^{ipr}$ is also a free motion state although the momentum, for example, may be uncertain. The dynamics of the momentum expectation value is determined with an external force, like in CM (see the Ehrenfest's equations). No external force, no variation of the expectation value <p(t)>.

So I do not see any relationship between HUP and Noether.

(*) Classical mechanics (**) Quantum mechanics