Many paths and many worlds

[nomadreid]

I get a bit confused seeing where Feynman's many paths fits into the Many-worlds interpretation. Both start off with the fact that there are many possible paths for a particle to take. Then, crudely paraphrasing Feynman's schema, each path will be given a probability amplitude according to the action associated with the path, the amplitudes essentially add up (interfere) until there is one path left, which is the one the particle takes. Yet with the many-worlds interpretation, the path it takes is the one which our world chooses, and in other worlds it may take another path. The two schemata don't seem equivalent to me, so I would enjoy being enlightened. For instance, if, in another "world", the possible paths are the same, and the actions and interference are the same, then if it takes another path than the one prescribed by the minimum action, this would seem to mean that in that other world, the principle of least action does not apply. But I thought the assumption was that in the other worlds the same physical principles apply. The only way out that I can see is that there are several paths who all share the same minimum of the action, and the worlds "choose" amongst them.

 

[bhobba]

I get a bit confused seeing where Feynman's many paths fits into the Many-worlds interpretation. Both start off with the fact that there are many possible paths for a particle to take. Then, crudely paraphrasing Feynman's schema, each path will be given a probability amplitude according to the action associated with the path, the amplitudes essentially add up (interfere) until there is one path left,

That's not quite what it says.

Here is the detail.

You start out with <x'|x> (the square is the probability of it initially being at x' and later being observed at x) then you insert a ton of ∫|xi><xi|dxi = 1 in the middle to get ∫...∫<x|x1><x1|...|xn><xn|x> dx1...dxn. Now <xi|xi+1> = ci e^iSi (the square gives the probability of it being at xi and a very short time later observed to be at xi+1) so rearranging you get
∫...∫c1...cn e^ i∑Si.

Focus in on ∑Si. Define Li = Si/Δti, Δti is the time between the xi along the jagged path they trace out. ∑ Si = ∑Li Δti. As Δti goes to zero the reasonable physical assumption is made that Li is well behaved and goes over to a continuum so you get S = ∫L dt.

What that weird integral says is in going from point A to B it follows all crazy paths and what you get at B is is the sum of all those paths. Now if the parth is long compared to how fast the exponential 'turns', since the integral in those paths is complex most of the time a very close path will be 180% out of phase so cancels out. The only paths we are left with is those whose close paths are the same and not out of phase so reinforce rather than cancel.

Now it does not say that all paths will cancel and you will only get one. No paths may cancel for example if the path is short compared to the frequency e^iSi 'turns' so no close paths will cancel. That is the case, for example, inside a Hydrogen atom.

MW is something entirely different. In that you have the wave-function of the universe - its not really relevant that it can be written as a path integral. When decoherence occurs each possible outcome of the resultant mixed state is interpreted as a new world. No collapse occurs - everything just keeps evolving.

Thanks
Bill

 

[nomadreid]

Thanks, Bill. Well explained. So, although you want to separate the MWI and the path integral formulation, could one mix them a little in saying that in each new world, there is a set of "crazy paths" which is not necessarily the same set as in the other worlds? (That is, they may be the same paths in space but the corresponding probability amplitudes may not be equal). Er?

 

[Matterwave]

Thanks, Bill. Well explained. So, although you want to separate the MWI and the path integral formulation, could one mix them a little in saying that in each new world, there is a set of "crazy paths" which is not necessarily the same set as in the other worlds? (That is, they may be the same paths in space but the corresponding probability amplitudes may not be equal). Er?

The path integral formulation of quantum mechanics is most often thought of as not an interpretation of quantum mechanics like the many world's interpretation, but more often as a different calculational approach to calculating quantum mechanical amplitudes.

It gives us an alternative approach to calculating the relevant amplitudes and therefore probabilities of interest than the wave mechanics of Schrodinger or the matrix mechanics of Heisenberg.

The three are proven to be equivalent in their predictions.

I don't think one can say Feynman's path integral formulation implies particles take all paths in space any more than one can make the statement that Lagrangian mechanics implies that the particle knows where it will end up before choosing a path that minimizes the action between its start and endpoints.

 

[bhobba]

could one mix them a little in saying that in each new world, there is a set of "crazy paths" which is not necessarily the same set as in the other worlds?

I don't think one can say Feynman's path integral formulation implies particles take all paths in space any more than one can make the statement that Lagrangian mechanics implies that the particle knows where it will end up before choosing a path that minimizes the action between its start and endpoints.

Technically the sum over histories approach is a hidden variable theory (the path is the hidden variable) but of a very non trivial type. Since its hidden you don't really have to look on it as in any sense real - just something that aids in calculations.

Of course one can equally think of it as real but I don't think anyone who has gone down that route has actually been able to gain anything by it.

Thanks
Bill

 

[nomadreid]

Thanks for the responses. I have a related question, although perhaps the answer goes no further than computational convenience. In summing up over all the paths (although the last two posts indicate that "histories" might be a better choice of words), why are only continuous paths selected? Is it to ensure their differentiability? In other words, why are not all subsets of the space selected?

 

[bhobba]

Thanks for the responses. I have a related question, although perhaps the answer goes no further than computational convenience. In summing up over all the paths (although the last two posts indicate that "histories" might be a better choice of words), why are only continuous paths selected? Is it to ensure their differentiability? In other words, why are not all subsets of the space selected?

They simply have to converge in that funny integral - they do not have to be continuous.

There are deep issues of existence and convergence that requires deep very advanced mathematics such as so called Hida distributions:
http://arxiv.org/abs/0805.3253

Good luck to you if you want to delve into that sort of stuff - its what mathematicians call non trivial - its a code word for HARD.

Still if its your won't the following may help:
https://www.math.lsu.edu/gradfiles/ngobi.pdf

Or not - depending on your mathematical sophistication.

I was into this sort of stuff at one time, but have now seen the light and don't get too worried by such things.

Thanks
Bill

 

[nomadreid]

Thanks for the links, bhobba. They look interesting. My mathematical sophistication allows me to do two things:
(a) get the main ideas, and (b) know when to skip the details. Before I go further, then, another question about the action S when it is made into an exponent in exp(iS) to give a phase from a particular path: am I right that this is a relative phase, and not a global phase factor?

 

[bhobba]

Before I go further, then, another question about the action S when it is made into an exponent in exp(iS) to give a phase from a particular path: am I right that this is a relative phase, and not a global phase factor?

Its the relative phase between infinitesimally close paths.

I have given a hand-wavy type argument. If you want something with greater rigour you need to investigate the method of steepest decent:
http://www.phys.vt.edu/~ersharpe/spec-fn/app-d.pdf

Have fun - but as far as rigour goes like I said I am over it, so if you have any further questions you will have to seek help elsewhere :tongue::tongue::tongue::tongue:

Thanks
Bill

 

[nomadreid]

Thanks again, bhobba. The hand-waving is actually very helpful. Also, the new link looks clearly laid out -- ah, that's what I like about texts in pure mathematics, they try to be self-contained. In fact, sometimes I would like physics texts to take the hint and include more hand-waving while making the details more self-contained. :shy: