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.
 A: 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
A: Expanding on Marek's comment, they are related, but not in a deep way. They are related by the notion from Hamiltonian mechanics that every dynamical variable can be interpreted as an infinitesimal generator of some canonical transformation, or the quantum mechnical notion that every Hermitian operator generates a unitary transformation.
The Heisenberg principle is true of any variable with a continuous spectrum and the infinitesimal generator of translations in that variable, just because these variables always have a nonzero commutator in every possible state. Position and momentum, angle and angular momentum, charge and phase, these are all conjugates in classical mechanics. The charge operator generates infinitesimal rotations in the phase of charged-particle wavefunctions, not changes in potential (you were probably thinking of the effect of a gauge transformation on a potential, but a global gauge transformation, the kind that gives you Noether's theorem for charge, does absolutely nothing to the potential).
The Noether theorem states that when translations of a certain variable are a symmetry, the infinitesimal generator of those translations is conserved. So translations in x, translations in angle, and translations in phase give conservation of momentum, angular momentum, and charge. But these generators obey the HUP with their conjugate variables.
The relationship is that both HUP and Noether talk about canonically conjugate pairs.
A: Oh yes they are related and there is a series of papers on this by Uffink and others.Though they didn't prove it from Noether's theorem rather they managed to do it from symmetries.Here is the link of the paper: https://www.sciencedirect.com/science/article/abs/pii/0920563289904477
