# Do uncertainty relations for charge or mass exist?

Is there a uncertainty relation for charge $q$ of the form $\Delta q \Delta? \geq \hbar$ in quantum mechanics? From checking the units ($[q] = A\cdot s$) I guess that $?$ would have to be the magnetic flux $[\Phi] = V \cdot s$, so we would have $$\Delta q \Delta \Phi \geq \hbar$$ If I play the same game with mass $m$ the units would suggest a uncertainty relation $$\Delta m \Delta \Phi_g \geq \hbar$$ where $[\Phi_g] = \frac{m^2}{s}$ is the gravitomagnetic flux (which happens to have the same units as the kinematic viscosity, the specific angular momentum, the mass diffusivity and the thermal diffusivity).

Do these two uncertainty relations exist in quantum mechanics?

Regarding electrical charge the answer is definitely negative: In Quantum Mechanics there exists a so-called superselection rule of the charge which requires that the charge is always definite in every quantum state of any quantum physical system carrying electrical charge. So $\Delta q_\psi =0$ in every state $\psi$ and no Heisenberg relations are possible for whatever choice of a conjugated variable of $q$.
• No, exactly the contrary: as $\Delta q=0$, for the conjugate variable we would have $\Delta X = +\infty$ and thus no state at definite $X$ may exist in contradiction with a superselection rule for $X$. – Valter Moretti Aug 20 '16 at 17:26