Path integral

touqra

In the path integral interpretation of quantum mechanics, it is said that a particle can take all sorts of paths, each with a certain probability. So, does this mean that there is also a very tiny probability, the particle can take paths which requires it to speed up more than the speed of light? Because I see nowhere in the formulation of path integral, that speed of light, is a limiting speed in its derivation.

jostpuur

In non-relativistic quantum mechanics a particle has non-zero amplitudes to travel along paths that exceed the speed of light. In fact you could also form a wave packet that travels faster than light, but that would of course be using a non-relativistic theory in a way in which it does not give correct results anymore. If the wave packets represent slow particles, then in the path integrals the contribution to the propagator from those high speed paths becomes very small, and things make sense to some extent.

Relativistic quantum mechanics should be a different matter, but it is more confusing.

Few words of critisism on this
The path integral approach still uses wave functions like the SE approach does too. In path integrals the time evolution of the wave function is defined with a propagator (instead of a PDE), and the propagator is defined with a functional integral that sums the quantity [tex]e^{iS/\hbar}[/tex] over all possible paths. But it is not so clear that the particle actually went through all these paths. It is a some kind of philosophical interpretation of this all, but it is not the most important thing if you want to just calculate the time evolution of the wave function.

touqra

First off, what is PDE?
Secondly, we learn path integral during our quantum field theory class. And the prof just introduce the path integral, and after that, straight on to use it in QFT and show that it is consistent with causality and SR. If path integral is non-relativistic QM, how could the prof just use it like that ?

jostpuur

PDE means partial differential equation, and a Shrodinger's equation is an example of such.

If you were talking about path integrals in QFT, then my answer wasn't a best possible. You can also formulate non-relativistic quantum mechanics with path integrals, and I was talking about them.