In presence of a magnetic field, the relation between velocity and momentum
\begin{equation} \dot{q}^i=\frac{\partial H}{\partial p_i}=\frac{1}{m}\left(p_i+e A_i\right) \end{equation}
is gauge-dependent showing that the conjugated momentum $p$ is no longer a physical quantity but only the kinetic momentum $p+eA(t,q)$.
Why isn't $p$ a physical quantity in the presence of a magnetic field when we can measure it in particle detectors?
