Feynman Paths

The Feynman path integral (PI) and Schrödinger's equation (SE) are completely equivalent formulations of QM in the sense that they give the same time evolution of an initial state. They have exactly the same information content. It's true that you can derive SE from the PI, while the reverse derivation isn't very natural. On the other hand, the PI is mathematically completely non-rigorous (roughly, the space of paths is too large) while SE evolution can be made precise.

Practically, the PI cannot be used to solve almost anything except the harmonic oscillator. This is a serious handicap in QM, since SE can be used to solve many problems exactly. But in quantum field theory, all the calculations are perturbations around harmonic oscillators, so the PI can be very useful.

Many physicists would agree that the PI is more "fundamental" because it's gives insight into QFT and theoretical physics. But the distinction is largely a matter of taste.

The way Feynman expresses the flow of amplitude to a certain point given a prior configuration is as a weighted sum over space of sums over path weights. The sum over space is simply weighted by the amplitude distribution of the given configuration and the weight of each path is but itself a sum over time of a quantity called Lagrangian (more precisely the complex exponential of this quantity but whatever) along said path.

Since this quantity is the difference between kinetic and potential energy, it normally should only depends on the position and time derivatives along the path. In that sense the path integral formalism for a finite number of particles is independent of the derivative of the amplitude distribution itself and thus of Schrödinger equation.

If one now goes to a situation with an infinite number of degrees of freedom, that is a field, and tries to implement there also a path integral formalism, then the equation changes slightly. Amplitude doesn't flow from one point to the other but rather between field configurations. In that case the second sum is not over all possible paths in between two points but over all possible field configurations in between two field configurations. Doing so, the quantity used to weight configurations now depends on the amplitude and space derivatives of the field everywhere.

And if one fancies a Schrödinger equation for a quantum field, then in the interacting and non-relativistic case this equation turns out to be nonlocal and nonlinear.

Feynman paths would basically correspond to points in spacetime configuration space, ie, histories, rather than points on plain ole position configuration space, wouldn't it?

(Actually, summing over histories is basically one way of explaining WHY in GR things follow geodesics. Think of metric as being analogous to refractive index, affecting the "optical" path length, so you end up getting the same idea as principle of least action.)

Ever since I read QED a few years ago, I've wanted to write a Quantum Ray-Tracing package that would use a discrete version of this summation over arrows to render scenes composed of a 3D grid of particles. It would have the advantage that certain classical ray-tracing problems having to do with questions of what, exactly, is a surface and its normal would go away. It would also correctly render diffraction gratings, butterfly wings and oil slicks, just given their physical arrangements.

On the negative side, it would require some serious R&D into rendering algorithms to get the computation times down to acceptable levels. Alas, I've never had the leisure to spend that kind of time on the problem.

And when you add up all the ways the photon can go from S to P, you find that most of the amplitude comes from the middle part of the mirror - the contributions from other parts of the mirror tend to mostly cancel each other out, as shown at the bottom of Feynman's figure.

Eliezer, one thing that is confusing me is that you are trying to show that the billiard ball and the "particles have identities" analogy is wrong. At the same time you keep speaking from "the photon". In the quotation the impression I get is that "the photon" splits up into the different paths it travels. Why does it split up in the first place? Again the "splitting up" assumes that there is a particle(alias small billiard ball) but your writing seems to imply this.

Btw, I have posted a question to your last entry "The quantum arena" which unfortunately wasn't answered and has to do with this confusion.

Thanks, Roland

PS: I'm no physicist and from reading the other comments I have the impression that most who are following this are physicists or at least have quite an advanced knowledge of QM. Please don't subestimate the inferential distance for those of us who don't have all that knowledge.

As Jess says, Schrödinger and Feynman are formally equivalent: either can be derived from the other. So if the question of which is more "fundamental" can be answered at all, it will have to be from other considerations. My own favorite way to think about the difference between the two pictures is in terms of computational complexity. The Schrödinger equation can be seen as telling us that quantum computers can be simulated by classical computers in exponential time: just write out the whole amplitude vector to reasonable precision, which takes exponentially many floating-point numbers, then update it step by step. The Feynman path integral can be seen as telling us that quantum computers can be simulated by classical computers in polynomial space: just add up the amplitudes of all paths leading to the quantum computer accepting, reusing the same memory from one path to another. Since polynomial space is contained in exponential time, the Feynman picture yields the better simulation -- and on that basis, one could argue that it's the more "fundamental" of the two representations.

Since Scott Aaronson has chimed in, it is worth pointing to this discussion on his blog in which Greg Kuperberg explains the Hilbert space issues from the previous thread.

What Roland's PS said :)

On the subject of 3D rendering:

Treating light as a classical wave can also produce pretty good experimental results on the scale of everyday life. Ray tracing algorithms ignore the properties light shares with classical waves, such as diffraction. I suspect that you don't need "quantum amplitude tracing" algorithm for more accurate 3D rendering, just a "classical wave tracing" algorithm. (Ordinary ray tracing is already rather computationally expensive anyway...)

You know, I enjoyed this post when I first read it, but now upon further thought it doesn't make any sense at all.

We're talking about the fundamental nature of reality, right? Photons are a fundamental thing? Taking all paths from S to P is fundamental? Little rotating arrows corresponding to wavelength is fundamental? OK, fine.

But what the heck is this "mirror" thing you then introduced? I'm supposed to assume that a mirror is a fundamental component of reality too?

No, obviously a mirror is just made up of atoms, which is just a pile of subatomic particles, also. You don't explain how a photon interacts with even a single other particle, but we're supposed to know how it interacts with a mirror? Especially a "flat" mirror, when we know that real physical mirrors must be very bumpy at a subatomic level. And a lot of them are silver; what's so special about silver atoms? Why is a mirror different from a (not very) flat rock?

See, here's the problem: you spend all this time telling us how our macroscopic intuitions are wrong, and we can't trust them, and that QM is the reality of how the universe works. And then, in the explanation of QM, you slip in a "mirror", and rely on our naive pre-QM common-sense understanding of mirrors to complete the example. But you've just told us that those intuitions are false!

I think you need to at least give some QM explanation of what a mirror "is", before using it in this example.

Awh, what are you trying to do here :) Giving me a headache?

If Feynman paths is correct then it invalidates most of what we experience at a every day level. For example there are no photons traveling from the sun to us, and every photon takes every possible path which in fact means that one photon fills up our universe. And how do we get away from that? By saying that they are 'probabilities' :) huh.

Which to my eyes comes very near 'parallel universes' as well as 'many paths universes'. It all seems reasonable but the implications gives me a headache. And if to that add the idea of time as being just 'spacetime intervals' that is not a flow but distinct static 'frames'?

Nah.. However much I like Feynman I wonder what kind of universe his ideas leaves us with.

I'm not a physicist so my question may be really old hat, but whatever.

I can think of two situations in which one ends up with a diffusion equation but in which the underlying physics is quite different.

First, the flow of heat in a solid. Here there is a continuous 'heat flows down a temperature gradient' picture that is mathematically equivalent to a picture in which individual particles follow Brownian motions. Physically, the former is just a sort of averaged version of the latter - some accounting short cuts - while the latter is some way closer to reality; the particles are really diffusing.

Second, the flow of water in an aquifer. Here the Darcian flow is proportional to the pressure gradient. For the sake of argument, imagine a medium that is a perfectly regular and homogenous 3D network of tiny tubes or something. In this case, there is no 'diffusion' of the fluid particles; they flow in a completely deterministic (indeed reversible) way through the network. But of course, one could presumably 'solve' the aquifer equation with a Monte Carlo similation of a diffusion process, if one really wanted to or if that was handy.

So to my question: Does the Feynman path integral purport to represent what's actually going on in any sense? Or is it more in the nature of a device for solving the problem? Or is this one of those things that is not answerable?

Sorry I asked that wrong. I don't mean heat flow in the first case, there are no diffusing particles there. Say concentration of tracer in fluid suspension or something.

"How fast does the little arrow rotate? As fast as the photon's wavelength - that's what a photon's wavelength is. The wavelength of yellow light is ~570 nanometers: If yellow light travels an extra 570 nanometers, its little arrow will turn all the way around and end up back where it started."

Which would seem to make it a ruler as well as a clock. But then, since general relativity made time an axis like space, I have sometimes wondered why we don't measure time in meters or distance in seconds.

I know this is an old post, but I'm hoping someone will see this. I read this a long time ago and have been thinking about QM questions (non-professionally) for a while. Recently, I started to wonder about a specific question regarding this post. Specifically, I'm thinking about the idea that we are summing paths leading to "identical configurations". While the various paths the photon takes in this problem do appear to lead to the same configuration, it seems to me that this is only true if you are just looking at the configuration of the photon and the mirror. The path A takes much more time to be completed than the path G, and it seems to me that during that time, the configuration of the rest of the universe would change as well, so the two configurations aren't the same.

I think this understanding is probably wrong, but I have about twenty guesses as to mistakes I could be making, and no clue which ones are genuine. Can anyone who has studied QM more help me out?

Okay, so where did those arrows come from? I see how the graph second from the top corresponds to the amount of time a particle, were particles to exist, would take if it bounced, if it could bounce, because it's not actually a particle, off of a specific point on the mirror. But how does one pull the arrows out of that graph?

When we sum over all paths some paths are longer than others. The argument says that the phase arrow will move further round because the time is longer. If the time is longer the the path won't end at the destination at the right time to coincide with the other paths. So how can this work?

(PS: I'm given to understand that the Feynman path integral may be more fundamental than the Schrödinger equation: that is, you can derive Schrödinger from Feynman. But as far as I can tell from examining the equations, Feynman is still differentiating the amplitude distribution, and so reality doesn't yet break down into point amplitude flows between point configurations. Some physicist please correct me if I'm wrong about this, because it is a matter on which I am quite curious.)

Feynman really does give you the amplitude for going from one point distribution to another point distribution. The formula for the path integral doesn't involve any derivatives of the amplitude distribution. But your fundamental point is still correct. Nature can't be viewed as classical just by thinking only in terms of point distributions. This is because the point distribution evolves into a non-point distribution. So even if you start out thinking in terms of point distributions you are immediately forced to consider other distributions.

(You might be worried that the point distribution has infinite second derivative, and so can't be evolved using the Schrodinger equation. But if you turn down your rigour dial you can find the solution:

phi = exp[i x^2 / (4t) ]/sqrt[4 pi i t]

(This is the solution for a free particle in one dimension where I've picked the mass hbar/2 for convenience.) One can sort of see how this becomes a point distribution as t tends to zero. The amplitude becomes very oscillatory everywhere except zero, and at zero all those oscillations cancel out. Meanwhile the magnitude increases like 1/sqrt(t) as t tends to zero, so at zero it has the correct value of sqrt(infintiy).)

what does the path integral actually look like?