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.

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.

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.

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.

(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).)