Uncertainty principle and limit of momentum.

[kof9595995]

To solve one of my textbook problems about uncertainty principle in relativistic case, I found that for every individual measured momentum p, I needed to assume $$p < \gamma mc$$ to get the correct answer, where $$\gamma = \frac{1}{{\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} }}$$.
But I keep suspecting whether the assumption is valid,because I have a vague memory that I heard that a particle can exceed the speed limit c when considering uncertainty principle. Then why can't the momentum exceed this limit?
Are these information enough for you guys? Or do I need to stick my textbook problem in this post?

Actually I was originally going to title this post as Uncertainty principle and FTL, but what I am going to ask is not that fancy so I change the title. :)

 

[Meir Achuz]

In SR, $$p^2=E^2-m^2$$ (with c=1), so p must be less than
$$E=m\gamma$$.

 

[kof9595995]

In SR, $$p^2=E^2-m^2$$ (with c=1), so p must be less than
$$E=m\gamma$$.

But if there's always a uncertainty associated, can some measured momentum exceed the classical limit?It makes sense to me <p> should not exceed the limit, but is it true for all sample points of p?

 

[humanino]

It depends whether you are doing quantum field theory or relativistic wave equations. In quantum field theory, you can have virtual particle "traveling faster than c" but all virtual contributions must be added together to produce a physical answer, and eventually they conspire to have real particles inside the light-cone. If you are doing relativistic wace equations, Meir Achuz's answer applies. The full understanding of the approximations behind would have to wait a little bit.

 

[kof9595995]

Thanks for all the clarifications. I think I get it now.

 

[Halcyon-on]

In relativity is a classical theory in the sense that all the classical postulate hold once that you assume space-time dimension and Lorentz transforms. In quantum mechanics many of the classical postulate are relaxed. So you find
- that particle can travel faster that c (with "small" contribution to the final probability)
- that the variational principle is not exactly satisfied
- the energy conservation is not exactly satisfied
- that the number of particle in an isolated volume is not constant
- there is not a single path linking two points
- that this message to arrive to you after few seconds can passes from alpha centauri (violating c).
and so on.