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I know how the Dirac's equation about the relativistic quantum mechanics works. Can anyone tell me how can one combine special relativity and quantum mechanics as a whole - special relativity is valid only under constant speed while the speeds of electrons in quantum mechanics vary with positions.

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You assert that

special relativity is valid only under constant speed while the speeds of electrons in quantum mechanics vary with positions

This is a very misleading statement. What is true is that via Poincare transformations, special relativity describes precisely how the observations of different inertial observers are connected. It is also the case that Poincare transformations do not connect the spacetime observations of accelerated observers to those of inertial observers.

However, this does not mean that special relativity has trouble describing the physics of accelerating particles. Quite to the contrary, there is, for example, a relativistic version of Newton's Third Law for massive particles which says that if you choose some inertial frame from which to perform observations, then in such a frame one has \begin{align} \mathbf F = \frac{d\mathbf p}{dt} \end{align} where $\mathbf p = \gamma m\mathbf v$ is the relativistic momentum of a particle of mass $m$ and $\mathbf{F}$ is the net external force experienced by that particle.

As for relativity and quantum mechanics, I think it would be overly-ambitious to give some sort of a self-contained account of how one combines the two; I'll defer to the wiki on relativistic quantum mechanics?

In my personal opinion, however, if you want to properly describe the interplay of relativity and quantum mechanics, you might as well make the conceptual leap to relativistic quantum field theory (QFT). I'll again defer to wiki and the countless other posts on physics.SE detailing this vast subject.

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    $\begingroup$ Yes, this is one of the most annoying memes out there (for me anyway). Special relativity is perfectly fine with accelerating objects. It is even perfectly fine with accelerating observers. What it is not ok with is curvature. The question you need to ask is: "am I doing experiments over small enough a region for gravitational tidal forces not to matter?" If yes, then you have special relativity, no matter how you are moving. This is just the famous Einstein equivalence principle. $\endgroup$ – Michael Brown Aug 15 '13 at 0:57
  • $\begingroup$ Thank you all for the replies. Does that mean in Minkowski space - no significant gravitation, no matter there is acceleration or not, special relativity works? $\endgroup$ – Lisa Lee Aug 15 '13 at 17:45
  • $\begingroup$ @LisaLee Yes, it sure does. $\endgroup$ – joshphysics Aug 15 '13 at 18:30

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