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In quantum mechanics, the position and momentum of a particle cannot be known simultaneously. In special relativity, the idea of simultaneity loses its meaning. Would it be possible to measure first the position of a particle and then the momentum of a particle, and do a Lorentz transformation into a frame where those two measurements are made simultaneously? If not, why not? If that is possible, does this violate the Heisenberg uncertainty principle?

Furthermore, would it be possible for an observer who makes those two measurements at different points in time to transmit that information into a Lorentz frame in which those two measurements are made simultaneously?

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  • $\begingroup$ Since (I assume from the context) the measurement events are time-like separated (have time-like interval), there is no inertial reference frame (IRF) in which the measurement events are simultaneous. $\endgroup$ – Alfred Centauri Feb 13 '16 at 3:58
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    $\begingroup$ So, this is not something I've thought about much, but would it be necessary for these kind of measurements to have a timelike separation? $\endgroup$ – Dargscisyhp Feb 13 '16 at 4:20
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    $\begingroup$ If the measurement events are space-like separated, inertial observers can disagree on the chronological order of the measurements. $\endgroup$ – Alfred Centauri Feb 13 '16 at 4:26
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Firstly, it is not that the position and the momentum of a system cannot be measured by an apparatus simultaneously in time. It can be, only that their standard deviations will have to satisfy the uncertainity principle. Here the fact that in QM the position and momentum of a particle cannot be known simultaneously means that position and momentum operators acting on a system don't have simultaneous eigenstates, since they don't commute. This has no reference with time-simultaneity in relativity. Please see this answer too for more details Simultaneously measurement in quantum mechanics?,

In practice for any measurement we make, we always make a sort of a "combined" measurement. We never use infinitely large detectors for measuring momentum and similar cases for measuring position. So simultaneous measurements in time can always be made, but simultaneous measurements of eigenkets of non-commuting observables cannot be made.

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The question of the Heisenberg uncertainty principle and special relativity has been nicely discussed in a paper by Rosen and Vallarta in 1932. (PhysRev.40.569).

They essentially concluded that when special relativity is taken into account, the Heisenberg relation has an upper and lower bound. Therefore the Heisenberg uncertainty principle is a possibility but not a necessity when relativistic invariance is taken into account.

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