special relativity - How is quantum mechanics compatible with the speed of light limit?

Consider a free electron in space. Let us suppose we measure its position to be at point A with a high degree of accuracy at time 0. If I recall my QM correctly, as time passes the wave function spreads out, and there is a small but finite chance of finding it pretty much anywhere in the universe. Suppose it's measured one second later by a different observer more than one light second away and, although extremely unlikely, this observer discovers that electron. I.e. the electron appears to have traversed the intervening distance faster than light speed. What's going on here?

I can think of several, not necessarily contradictory, possibilities:

1. I'm misremembering how wave functions work, and in particular the wave function has zero (not just very small) amplitude beyond the light speed cone.


2. Since we can't control this travel, no information is transmitted and therefore special relativity is preserved (similar to how non-local correlations from EPR type experiments don't transmit information)


3. Although the difference between positions is greater than could have been traversed by the electron traveling at c, had we measured the momentum instead, we would have always found it to be less than $m_e c$ and it's really the instantaneous momentum that special relativity restricts; not the distance divided by time.


4. My question is ill-posed, and somehow meaningless.

Would anyone care to explain how this issue is resolved?

Answer

Excellent question. You are correct about wavepacket spreading, and in fact you do get superluminal propagation in non-relativistic QM - which is rubbish. You need a relativistic theory.

You should read the first part of Sidney Coleman's lecture notes on quantum field theory where he discusses this exact problem: http://arxiv.org/abs/1110.5013

The short answer is that you need antiparticles. There is no way to tell difference between an electron propagating from A to B, with A to B spacelike separated, and a positron propagating from B to A. When you add in the amplitude for the latter process the effects of superluminal transmission cancel out.

The way to gaurantee that it all works properly is to go to a relativistic quantum field theory. These theories are explicitly constructed so that all observables at spacelike separation commute with each other, so no measurement at A could affect things at B if A and B are spacelike. This causality condition severely constrains the type of objects that can appear in the theory. It is the reason why every particle needs an antiparticle with the same mass, spin and opposite charge, and is partially responsible for the spin-statistics theorem (integer spin particles are bosons and half-integer spin particles are fermions) and the CPT theorem (the combined operation of charge reversal, mirror reflection and time reversal is an exact symmetry of nature).