# Why can we combine the special relativity and quantum mechanics?

The job of special relativity is to deal with the transformation between different inertial frames of reference, namely

$$(x,y,z,t) \rightarrow (x',y',z',t').$$

And we also know in some heavy elements, such as Bi, the velocity of electron lived in these materials are very large (can compare with the speed of light) so we must take into account the effect of relativistic effect. But the evolution of electron is absolutely governed by the quantum mechanics, so now you have to combine these two theories to describe the motion of the electron.

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?

• en.wikipedia.org/wiki/Relativistic_quantum_mechanics
– user126422
Commented Jan 15, 2017 at 0:47
• Dirac equation does the job. And that's the one we use for electrons in atoms Commented Jan 17, 2017 at 13:06

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.

• You mean the Lorentz transformation is just the transformation between the platform of some events, it has nothing to do with the event and also the laws governing the time evolution of these events?
– Jack
Commented Jan 15, 2017 at 3:57
• The simplest LT involves time and space. It's a transformation between two frames of reference. The physical laws should be the same everywhere in the universe. If I was in the 0.99c spaceship and you were on Earth, because of relativistic effects, my measurement of time and space will be way different than yours. But still $F=ma$ should be the same equation for both of us. I will get different results than you, but the LT allows you to adjust the measurements I send back to fit your Earth measurements, in all physical laws.
– user140606
Commented Jan 15, 2017 at 4:38
• Now say a star explodes, that's an event that has nothing to do with me at high speed and you on Earth, but although we will get different measurements, the LT allows us to match them. The LT for space and time is just a conversion table. When the star explodes, the planets around it explode a day later. I will measure a different time difference than you, again the LT connects the measurements. The time evolution, (causality) is invariant, the star blew up first, the planets a day later. The LT don't have anything to do with that, they are just math equations, how could they?
– user140606
Commented Jan 15, 2017 at 4:39
• @Coutnto10. Welcome back. The same? Commented Jan 15, 2017 at 6:29
• Hey, you still get 8'hours of sleep and 8 hours of work or research, as long as you don't eat, go out, etc. Commented Jan 15, 2017 at 7:59