B Multiple Histories theory by Richard Feynman

[Rosegoldtaken]

TL;DR Summary

There is some thing i did not get about Multiple History theory

Summary: There is some thing i did not get about Multiple History theory

Summary: There is some thing i did not get about Multiple History theory

I am reading big answers to the big questions from Stephan Hawking. He has mentioned very little about Multiple History theory. I could not understand it clearly. I searched about it but it seems really complicated and advance. Can anyone explain it to me in very clear way, please?

 

[jedishrfu]

You can read about it here on PF:

https://www.physicsforums.com/threads/multiple-histories.240/
or some more on wikipedia:

https://en.wikipedia.org/wiki/Multiple_histories
Its part of the path integral approach Feynman used where he sums over the possible tracks of a particle to determine the most likely path.

https://en.wikipedia.org/wiki/Path_integral_formulation
It follows the notion of Least Action principle often used in Classical Mechanics where you find the path of least resistance ie where the difference between a particle's KE and PE are zero which will be the path the particle takes.

The brachistochrone problem is a great example of this showing two balls one rolling down a slide and another down a curved slide of a specific shape. The curved slide ball reaches the bottom faster.

 

[Rosegoldtaken]

You can read about it here on PF:

https://www.physicsforums.com/threads/multiple-histories.240/
or some more on wikipedia:

https://en.wikipedia.org/wiki/Multiple_histories
Its part of the path integral approach Feynman used where he sums over the possible tracks of a particle to determine the most likely path.

https://en.wikipedia.org/wiki/Path_integral_formulation
It follows the notion of Least Action principle often used in Classical Mechanics where you find the path of least resistance ie where the difference between a particle's KE and PE are zero which will be the path the particle takes.

The brachistochrone problem is a great example of this showing two balls one rolling down a slide and another down a curved slide of a specific shape (a catenary curve). The curved slide ball reaches the bottom faster.

Thanks a lot! I am checking out all these!

 

[Ranku]

It follows the notion of Least Action principle often used in Classical Mechanics where you find the path of least resistance ie where the difference between a particle's KE and PE are zero which will be the path the particle takes.

Least action principle implies that there is one ideal path. Sum over histories considers all possible paths which integrates to the actual path. Would it be correct to say: while all the possible paths do not themselves adhere to the least action principle, but the actual sum over history path does?

 

[jedishrfu]

Yes, i think that's fair to say. However, as I’m a robot and not a physicist, it would be better to have @ZapperZ or @PeterDonis say something here.

 

[Ranku]

However, as I’m a robot

But I thought you're the Jedi!

 

[jedishrfu]

I'm a robot Jedi, C3PO and R2D2 were my teachers.

 

[Michael Price]

Least action principle implies that there is one ideal path. Sum over histories considers all possible paths which integrates to the actual path. Would it be correct to say: while all the possible paths do not themselves adhere to the least action principle, but the actual sum over history path does?

No.
The principle of least action says classical mechanics follows the path of least action.
In quantum mechanics/QFT the path of least action is irrelevant. You add up the phases (~action) of all the possible paths (histories) to determine the likelihood that anyone outcome will occur.

 

[sophiecentaur]

The rolling cylinders get there at the same time down a Cycloid and so do masses on springs 'cos that's Simple Harmonic Motion. But simple pendulums don't work exactly because their curves are circles (wrong curve). The motion of a mass on a spring is described by a Cosine function of time and I was wondering how that relates to the Principle of least action (if it does).
Calling all nerds. ?

 

[Michael Price]

The brachistochrone illustrates the principle of least time - so it is vaguely analogous to the principle least action.
Action = KE - PE summed over time.

 

[PeterDonis]

while all the possible paths do not themselves adhere to the least action principle, but the actual sum over history path does?

There is no such thing as "the actual sum over history path". The sum over histories sums over all paths. It doesn't pick one out and say that one is the "actual" one.

The connection between the sum over histories and the least action path is that if the actions along the paths are much, much larger than $\hbar$, then paths which are not very, very close to the least action path will cancel out in the sum, because the action will change very rapidly from one path to the next and so the sum will be over complex exponentials with rapidly varying phase, which cancel out by destructive interference. Only near the least action path (more precisely, near a path where the action is stationary) will the action be changing slowly enough that nearby paths will constructively interfere and give a significant nonzero sum. But this argument still does not pick out the least action path as the only actual one; it just explains why, in the classical approximation, which involves actions which are many, many orders of magnitude larger than $\hbar$, the least action path can be treated to a good approximation as if it were the only actual path.