Classical limit of the path integral

[exmachina]

In feynman's quantum mechanics and path integrals,

he makes the following claim:

"Now if we move the path by a small amount dx, small on the classical scale, the change in S (the action), is likewise small on the classical scale, but not when measured in the tiny unit of reduced Planck's constant h. These small changes in path will, generally, make enormous changes in phase, and our cosine or sine will oscillate exceedingly rapidly between plus and minus values. The total contribution will then add to zero.

I have difficulty mathematically showing this.

The contribution of the action along a path j, $$\phi_j[x(t)]=e^{(i/h)S_i[x(t)]}$$ , k constant and $$S_j[x(t)]$$ is the action along the path j

$$K(b,a) = \sum_{over all paths}\phi[x(t)] = \phi_1[x(t)]+\phi_2[x(t)]+...+\phi_N[x(t)] = e^{(i/h)S_1[x(t)]}+e^{(i/h)S_2[x(t)]}+ ... +e^{(i/h)S_N[x(t)]}$$

Using Euler's identity,

$$K(b,a) = cos((i/h)S_1[x(t)]) + i*sin((i/h)S_1[x(t)]) + cos((i/h)S_2[x(t)]) + i*sin((i/h)S_2[x(t)])+... + cos((i/h)S_N[x(t)]) + i*sin((i/h)S_N[x(t)])$$ In the classical limit where $$S_i >> h$$ :

Let's assume that the action $$S_1$$ has the least action, and all other paths differ from $$S_1$$ by $$ds$$

Then $$K(b,a) = cos((i/h)S_1[x(t)]) + i*sin((i/h)S_1[x(t)])$$

since Feynman claims that:

$$cos((i/h)S_2[x(t)]) + i*sin((i/h)S_2[x(t)]) ... + cos((i/h)S_N[x(t)]) + i*sin((i/h)S_N[x(t)] = 0$$ as they "cancel each other out"

But what I don't understand is that, because no matter how large S_N becomes in the classical limit (ie for objects of very large mass), $$cos(S_N) \leq 1$$, so how exactly do they cancel out?

So all paths that are far away from the path of least action is NOT important so long as the action S >> h? What exactly is >>, ie is 10^2 bigger, 10^3 bigger? How about the action of the rotational and translational movement for single 8 carbon polymer?

Also, at molecular mass and time scale do we start considering it to be "classic"?

 

[azntoon]

The claim that the paths "cancel each other out" is true, but a more detailed explanation must be given. Essentially, what happens is that for large S (on the classical scale) the phase difference between paths is very large, so the contributions of each path oscillate rapidly between positive and negative values. The net result is that these contributions cancel each other out, and only those paths with relatively small phase differences will contribute to the sum. This is why only paths close to the least action path are important in the classical limit.The size of S required for this effect to kick in depends on the particular problem, but generally speaking it should be much larger than h. For example, for a single 8 carbon polymer, the action S will be on the order of 10^-26 joules, which is much larger than h (10^-34 joules). So in this case, we can safely consider the motion to be "classic".

 

[Entropia]

I would first like to clarify that the classical limit of the path integral is a concept in theoretical physics and is not a proven fact. It is a mathematical approach used to reconcile classical mechanics with quantum mechanics, and its validity is still a topic of ongoing research and debate.

That being said, let me try to address your concerns. In the classical limit, the action S_i becomes much larger than the reduced Planck's constant h. This means that the exponential term in the path integral, e^(iS_i/h), becomes very oscillatory and rapidly changing. This is because the phase of the exponential function is directly proportional to the action, and as the action becomes much larger than h, the phase changes very quickly. In other words, the contribution of each path to the overall path integral becomes highly oscillatory and rapidly changing.

Now, when we add up all these oscillatory contributions from different paths, they can either add up or cancel out, depending on their relative phases. In the classical limit, the paths that deviate significantly from the path of least action will have very different phases compared to the path of least action. This means that their contributions will mostly cancel out, leaving behind only the contribution from the path of least action.

To answer your question about the magnitude of the classical limit, it is not a specific number or value. It is a relative comparison between the action and the reduced Planck's constant. So, the classical limit can vary depending on the system and the scale being considered. Generally, it is considered to be applicable for macroscopic systems with large masses and long timescales.

As for the specific example you mentioned, the action for rotational and translational movements of a single 8 carbon polymer would depend on the specific system and its parameters. It is not possible to give a specific value for the classical limit in this case without knowing more details.

In conclusion, the classical limit of the path integral is a complex concept and its application and validity depend on various factors. It is an ongoing area of research and understanding, and further studies and experiments are needed to fully understand its implications.