| MadSci Network: Physics |
You're on to something here -- a seeming contradiction between quantum mechanics and relativity -- but you posed the question backwards...
The answer to your question is simply that at relativistic speeds, momentum does not equal mass times velocity. The relativistic equation (for a massive particle) is:
momentum (p) = m gamma v ; gamma = 1/(1 - v^2/c^2)^0.5
So as v approaches c, gamma gets very big and the momentum gets very big as well. Therefore there is no "maximum" momentum; a massive particle travelling at c has an inifinite momentum! So making p get a little bigger through the uncertainty principle is no problem...
But, as I said, you're onto something here -- you just phrased the question backwards. Instead, imagine that momentum is fixed and it is POSITION which is uncertain. In this case there is a chance that the particle will suddenly "teleport" some sizable distance, and the net effect might be to make it appear to travel faster than light!
Feynman not only resolved this "paradox", but he wrote this great argument about how the above example proves that antimatter must exist! (He gave a lecture in 1986 titled "The Reason For Antimatter.")
Quantum mechanics does demand that some particles travel faster than light - - we call these virtual particles. Turns out you can't transmit information with them, so relativity is okay (see the link for details). But in some reference frames the virtual particle seems to be going one way, and in other reference frames it looks like it's going the other way -- it appears to have time-reversed! Because antimatter is equivalent to matter going backwards in time, a virtual particle can be matter or antimatter depending on how it is viewed. Since virtual particles can appear to be antimatter, it is natural to expect real antimatter particles as well.
Therefore, far from causing a paradox, merging quantum mechanics and relativity actually explains the existence of antimatter.
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