My question is: How can we do that? I mean due to the quantum uncertainty (we cannot find the position (x,y,z) we cannot make a Lorentz transformation?
A Lorentz transformation does not depend on knowing the position of the particle. Say we have two observers. One of them is on Earth and the other is moving at high speed relative to the first. Assume both perform the two slit experiment, with absolutely identical equipment. The LT allows us to transform the details of one experiment (the physical dimensions of the apparatus used, the width of the fringes,the spacing between the fringes, the time it took for the pattern to develop on the screen ) and match them with a similar set of measurements taken on Earth.
There is invariance of physical laws, so we can use the LT to check and match up the measurements.
But the probabilistic nature of QM means that, even if we did both of the two slit experiments on Earth, they would still differ in the screen pattern because the electrons would located at different points, although the overall pattern looks much the same, no two slit experiment is exactly the same in its screen pattern.
So we don't need to know the positions of the particles to blend QM and SR, but we need SR to ensure the uncertainty principle has been obeyed exactly the same way in both case, and we do this by using SR to compare the different physical quanties involved in checking the uncertainty principle.
We would be very surprised if F did not equal ma in both experiments, after taking the LT into account, because there is no probability involved on a classical scale in that equation, but probabilities are not treated the same way. For example, the second law of thermodynamics is not really a law, it is a statement of strong probability.
