Path Integral From Heisenberg Uncertainty?

[bolbteppa]

Two questions about the path integral:

a) Intuitively, how does one (as Landau does) start quantum mechanics from Heisenberg's uncertainty principle, which states there is no concept of the path of a particle, derive Schrödinger equation i \hbar \tfrac{\partial \psi}{\partial t} = H \psi the operator solution \psi = Ae^{\tfrac{-i}{\hbar}Ht}\psi_0 and from this basis express Ae^{\tfrac{-i}{\hbar}Ht} as an integral over all possible paths. In other words, starting from the assumption that no path exists we construct the wave function in terms of an integral over all paths (even though no true path exists) and get the right answer, mathematically how does that make sense?

b) Is there a way to derive the path integral from the quasi-classical approximation \psi = e^{\tfrac{i}{\hbar}S}? I mean by using \psi = e^{\tfrac{i}{\hbar}S} \cdot 1 = e^{\tfrac{i}{\hbar}S} \cdot e^{\tfrac{i}{\hbar}S_1} \cdot e^{-\tfrac{i}{\hbar}S_1} where S_1 is the action for a nearby path, then extending over all actions over all paths, or something?

 

[moriheru]

We are talking about probability not about determinisn. The equations only involve probability amplitudes . And then what feynman did was reform the hole equation to a simple exponential.

 

[bolbteppa]

?

 

[moriheru]

I see how my post could confuse, it was badly expressed(sorry). I was simply noting that the equations have the form of <xl U(t) l x. for a particle traveling from x1 to x2 and at t1 to t2. What I am saying is that it is not an integral over all the paths but a integral over the probabilitys the paths.

 

[Jilang]

Have you read Feynman's book Quantum Mechanics and Path Integrals? I would recommend it.

 

[atyy]

In quantum field theory, it can be shown that the path integral is equivalent to a Hamiltonian formulation is the Osterwalder-Schrader conditions are obeyed. Then I think (am not sure) we can get the quantum mechanics case by treating it as 0+1 dimensional field theory.

http://www.einstein-online.info/spotlights/path_integrals

 

[bolbteppa]

Yeah I have looked in Feynman but it does not answer my question, I have read the derivations in Shankar, Feynman, Galitski etc... many times and forget them because I cannot see how they sync with Landau's Heisenberg QM, and also this quasi-classical approximation idea is interesting, so I'm looking for just enough of a reason to get to those derivations from what I've said in a consistent fashion :D

 

[atyy]

There is a brief discussion (not derivation) of the Osterwalder-Schrader conditions in http://www.rivasseau.com/resources/book.pdf (p17-18).

I think there is also a good discussion in https://www.amazon.com/dp/9812835229/?tag=pfamazon01-20.

 

[bolbteppa]

I'm just talking about quantum mechanics, starting from Heisenberg, Schrödinger, Quasi-classical and maybe a propagator, no field theory or axioms. Do you guys have any ideas?

 

[naima]

Paths in path integral are continuous. Take the discrete case:
We have a set of N sites. the source is on one of those sites (equiprobability) at time 0. it propagates to the other ones. At time 1 you have for each couple of sites (i,j) an amplitude $M_{i j}$ At time 2 You have $M_{j k}$ and so on.
IF you want to compute the result 0 -> t you have to multiply the matrices
$\Sigma (i,j) (j,k) (k,l) ...$
This is matrix mechanics. Heisenberg discovered how it works. Paths integrals look like a generalization in the continuous limit.