PeterDonis said:
The usual position-velocity form of the uncertainty principle assumes non-relativistic QM, in which the rule about the laws of physics looking the same in different inertial frames doesn't really apply. If you really want to combine quantum mechanics with the principle of relativity, you need to look at quantum field theory, in which the formulation of the uncertainty principle requires more care.
Non-relativistic quantum theory is, of course, by construction Galilei invariant, i.e., the observable algebra is constructable by assuming that this algebra allows for unitary ray representations of the Galilei group (semidirect product of temporal and spatial translations, rotations, and boosts).
As turns out from this analysis, the Galilei group can be extended to what I'd call "quantum Galilei group", which is a central extension of the covering group of the classical Galilei group with mass as a non-trivial central charge.
The position-momentum (not velocity!) uncertainty relation follows directly from the definition of momentum as generators of translations, i.e., the (Heisenberg) subalgebra of the corresponding Lie algebra,
$$[\hat{x}_j,\hat{p}_k]=\mathrm{i} \hbar \delta_{jk}.$$
Of course, the position observable has to be derived from the generators of the boosts, but this is pretty straight-forward.
In the relativistic case, it's more subtle. There the construction of position observables is not so straight-forward. It turns out that there exists proper position operators for massive particles (note that in contradistinction from the Galilei group in the case of the Poincare group mass (squared) is a Casimir operator rather than a central charge) and massless particles with spin ##\leq 1/2##. For these cases, of course, the Heisenberg uncertainty relation (for position and momentum, not velocity!) holds. A famous analysis of this uncertainty relation by Peierls and Landau [1] taking into account the finiteness of the speed of light shows that a wave-mechanical interpretation as in the non-relativistic case is inconsistent, and one has to take into account the possibility of creation and annihilation processes in scattering processes at relativistic energies. Massless particles with spin ##\geq 1## do not allow for the definition of a position operator.
[1] L. Landau, R. Peierls, Z. Phys.
69, 56 (1931)
) would be that the HUP also applies to charge-phase in an electronic circuit (in the simplest case a regular LC circuit), i.e. the HUP applies even though there is nothing moving and there are no spatial coordinated involved.
