Undergrad Path Integral in QM: Resolving Confusion on Causality

[Silviu]

Hello! I am reading a derivation of the path formulation of QM and I am a bit confused. They first find a formula for the propagation between 2 points for an infinitesimal time $\epsilon$. Then, they take a time interval T (not infinitesimal) and define $\epsilon=\frac{T}{n}$. Then they sum up the propagations for each of these $\epsilon$'s, take the limit $n \to \infty$ in order to find a formula for the propagation between any 2 points in a finite time T. Now I am a bit confused. When you split the initial interval (let's say between $x_i$ and $x_f$) those n intermediate steps don't need to be close to each other (at least this is not implied in the derivation, and from what I understand, any path in the universe can be valid in the summation). So if the space can be arbitrary large but the time is infinitesimally small, isn't relativity violated? And even for the case of finite space and time (so in our case from $x_i$ to $x_f$ traveled in time T), I see nothing to force the time interval to be such that $x_f - x_i < cT$. So how is causality preserved in this case? Thank you!

 

[stevendaryl]

Well, the original path integral was for nonrelativistic physics, and it allowed arbitrary paths. The path integral for relativistic physics is harder to construct, but it turns out that it does include contributions for paths corresponding to faster-than-light motion: https://arxiv.org/pdf/gr-qc/9210019.pdf