Path Integral & Quantum Mechanics: Beyond the Speed of Light?

In summary: But the path integrals in non-relativistic QM are more complicated, and the prof just used them to show the consistency of QFT and causality without going into too much detail.
  • #1
touqra
287
0
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.
 
Physics news on Phys.org
  • #2
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
it is said that a particle can take all sorts of paths, each with a certain probability.
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.
 
  • #3
jostpuur said:
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.

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 ?
 
  • #4
touqra said:
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 ?

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.
 

1. What is the path integral approach in quantum mechanics?

The path integral approach in quantum mechanics is a mathematical formulation used to describe the behavior of quantum particles. It involves summing over all possible paths that a particle can take, incorporating both classical and quantum mechanical principles.

2. How is the path integral approach different from traditional quantum mechanics?

The path integral approach differs from traditional quantum mechanics in that it considers the particle's entire trajectory instead of just its initial and final states. It also allows for the inclusion of interactions and the calculation of transition amplitudes between states.

3. Can the path integral approach be used to describe particles traveling faster than the speed of light?

No, the path integral approach, like traditional quantum mechanics, still follows the principles of special relativity and does not allow for particles to travel faster than the speed of light. It is a mathematical tool used to study the behavior of particles within the framework of special relativity.

4. What are the applications of the path integral approach in physics?

The path integral approach has many applications in physics, including in quantum field theory, condensed matter physics, and statistical mechanics. It is also used to study systems with strong interactions, such as in nuclear physics and high energy physics.

5. What are the limitations of the path integral approach in quantum mechanics?

The path integral approach has some limitations, such as the difficulty in dealing with more complex systems with many particles. It also requires some mathematical approximations and may not always provide exact solutions. Additionally, it does not fully explain the underlying mechanisms of quantum phenomena, but rather describes their behavior mathematically.

Similar threads

  • Quantum Physics
Replies
4
Views
312
Replies
6
Views
648
Replies
8
Views
1K
  • Quantum Physics
Replies
13
Views
712
  • Quantum Physics
Replies
13
Views
521
  • Quantum Physics
Replies
3
Views
1K
Replies
12
Views
1K
  • Quantum Physics
Replies
3
Views
1K
Replies
1
Views
898
Back
Top