History and origin of amplitude summation in QFT

[BucketOfFish]

In chapter 2.2 of Feynman's book on QFT, he states that the probability amplitude of a particle going from a to b is the sum of contributions from all paths, and that each path contributes the same amplitude, but with a different phase.

My question is, why does Feynman state that this is the rule? How is it known that the amplitude for each path is identical, or that the phase is determined by the action of the path? I have heard that using the same amplitude for each path leads to infinite summations, which I can see would be the case, and that this problem had to be solved with mathematical trickery. So how did these rules come about? Were they experimentally derived?

 

[Matterwave]

Feynman was referring basically to his own formulation of Quantum Mechanics, now known as the path integral formulation, or the sum over histories formulation. He pioneered this approach, and it was the topic of his phD thesis. It has been proven that this formulation is equivalent to the more conventional formulation of quantum mechanics, the Schroedinger wave-function formulation.

Feynman basically made a postulate that the propagator between two points in space can be expressed as a path integral over all paths:

$$K(x',t';x,t)\equiv \left<x',t'|x,t\right>=\mathcal{N}\int_{x''(t)=x}^{x''(t')=x'} e^{iS[x'']/\hbar}\mathcal{D}x''$$

One can prove that this formulation reproduces the Schroedinger equation. The usual way of actually making the integral calculable involves what is called lattice space regularization, where basically you split the path up into tiny straight line paths (like points on a lattice) and take the limit as the lattice points go to 0 separation.

This subject has a pretty rich history and is quite broad.

Perhaps you can look at the wikipedia article to start: http://en.wikipedia.org/wiki/Path_integral_formulation

 

[BucketOfFish]

Thanks for the reply, Matterwave. I think I understand the path integral formulation, but I'm still confused as to how Feynman thought up the contribution amplitude for each path.

For example, in problem 2.6 of the same book, he states that relativistic particles have a different path amplitude, something like (ih)^R, where R is the number of turns the particle makes. I don't have the book with me, so that's probably not the actual formula, but either way it's a lot different from the non-relativistic amplitude C*exp(iS/h).

So how did Feynman think up the form of the path amplitude?

 

[Matterwave]

I can't say for sure how Feynman thought up the idea. I don't know what exactly his motivations were. Zee's book QFT in a Nutshell, for example has a (probably apocryphal) story of how Feynman started with the double slit experiment and kept drilling holes in the screen until there was no screen left.

But there are many instances in physics where the action appears in an exponential. For example, in the Eikonal approximation, which is very similar to the WKB approximation in quantum mechanics. By unit analysis alone, for example, since Planck's constant has units of action, we can actually see that all the wave functions which are purely oscillatory of the form Ae^(something/h), the "something" also has units of action.

Sadly, I can't point out exactly which route Feynman took to arrive at his conclusion.

 

[atyy]

Here's Feynman's version of the story. http://www.nobelprize.org/nobel_prizes/physics/laureates/1965/feynman-lecture.html

Professor Jehle showed me this, I read it, he explained it to me, and I said, "what does he mean, they are analogous; what does that mean, analogous? What is the use of that?" He said, "you Americans! You always want to find a use for everything!" I said, that I thought that Dirac must mean that they were equal. "No", he explained, "he doesn't mean they are equal." "Well", I said, "let's see what happens if we make them equal."

...

Professor Jehle's eyes were bugging out - he had taken out a little notebook and was rapidly copying it down from the blackboard, and said, "no, no, this is an important discovery. You Americans are always trying to find out how something can be used. That's a good way to discover things!" So, I thought I was finding out what Dirac meant, but, as a matter of fact, had made the discovery that what Dirac thought was analogous, was, in fact, equal.

 

[bhobba]

My question is, why does Feynman state that this is the rule?

Its pretty easy to derive.

Its the standard calculus thing - by splitting the interval and going to zero you get an integral in the action.

You start out with <x'|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 so rearranging you get
∫...∫c1...cn e^ i∑Si.

In the limit ∫...∫c1...cn goes to ∫D so you get ∫D e^ i∑Si which is an example of what is called a functional integral. Its not only used here, but in stochastic modelling where you have the Wiener integral:
http://www.scholarpedia.org/article/Path_integral

In fact by transforming to imaginary time they are the same. This is a very deep result right at the foundations of QM:
http://arxiv.org/pdf/1204.0653v1.pdf

There are also deep mathematical issues of convergence etc and mathematical tomes, deeply rooted in functional analysis and so called Rigged Hilbert Spaces (also used in Quantum Physics) have been written sorting it out:
http://mathlab.math.scu.edu.tw/mp/pdf/S20N41.pdf

Anyway returning to the path integral from QM, 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 ∫L dt.

Now Si depends on xi and Δxi. But for a path Δxi depends on the velocity vi = Δxi/Δti so its very reasonable to assume when it goes to the continuum L is a function of x and the velocity v.

Its a bit of fun working through the math with Taylor approximations seeing its quite a reasonable process.

In this way you see the origin of the Lagrangian of classical physics. And by considering close paths we see most cancel and you are only left with the paths of stationary action. Also, by considering a field as a limit of a heap of interacting 'somethings' its probably the most common method of developing Quantum Field Theory.

Thanks
Bill