I don't get this Path Integral stuff

[nhmllr]

I always have a feeling of apprehension posting on the Quantum Physics subforum, because I haven't done any of the math for it

However, a friend recently told men (I think he read it in the Elegant Universe) that if you shine a flashlight on a wall or something, photon takes every possible path from it's origin to it's destination. I looked it up and read a little about it.

Here's the thing I don't understand- the speed of light is constant. And because the photon theoretically is traveling to the next galaxy and back, there's no way it could travel that distance in such a short time. There's something I'm not getting here.

 

[Demystifier]

The point is that the photon does not REALLY takes all possible paths. This is just a mathematical trick in which something more real is calculated in terms of auxiliary mathematical paths that do not really exist.

Roughly, this is like writing
1 apple = 2 apples + (-1 apple)
even though -1 apple does not really exist.

 

[nhmllr]

The point is that the photon does not REALLY takes all possible paths. This is just a mathematical trick in which something more real is calculated in terms of auxiliary mathematical paths that do not really exist.

Roughly, this is like writing
1 apple = 2 apples + (-1 apple)
even though -1 apple does not really exist.

So it's saying that the photon is taking all of these paths except for these certain ones?

 

[Demystifier]

So it's saying that the photon is taking all of these paths except for these certain ones?

No, it's saying that the photon is not taking any paths at all (except in the Bohmian interpretation, but that's another story ...).

 

[nhmllr]

No, it's saying that the photon is not taking any paths at all (except in the Bohmian interpretation, but that's another story ...).

Wait- so it's saying that there's no way to know which path it took, so it _could_ have taken any path, but really it took no path to end up at the destination?

That still doesn't really answer the question of how could a photon "move" faster than the speed of light?

 

[Demystifier]

If photon takes no path, then why do you think it can "move" faster than the speed of light?

 

[unusualname]

The point DM is saying is that the path integral is just a mathematical formalism to correctly calculate observed results from experiments, not a step by step description of a physical process.

You can't talk about "which path" the photon takes, as we have no way of describing such a thing, but we can very accurately describe the probability of its arrival somewhere.

If you want to be a bit more adventurous then you might like to consider a deterministic Bohmian path exists, or alternatively a real ontological path, though probabilistically defined, exists via the Consistent Histories interpretation (Modern Copenhagen view)

 

[muppet]

I think it's fair to say Feynman believed his formulation was as good a description of the physical picture of QM as the standard "no paths" interpretation.

The physical picture you usually associate with the canonical formalism of QM is something like this: you can't generally say that an object *has* a definite position to start off with, only that you have a range of probability (density) amplitudes associated with finding it within various regions in space. If you want to talk about moving from x to y in time t, you need to take position measurements at x and y separated by time t (or prepare the system so that there is a particle at x and then measure its position t seconds later, etc.). If you assume the particle is localised at x, then you can compute the probability that t seconds later you'll find it at y; but as the particle isn't said to have a definite position between the two measurements, then it certainly makes no sense to say that "first it was at x, then it moved here, then here, then... then eventually ended up at y".

The way this "sum over paths" works is like this.

1. You know that your particle is (roughly) at x to start off with.
2. You consider an infinitesmally small time t later, and you can work out a probability that the particle will be somewhere, call it x+a, really really close to x. Draw a (very very short) straight line between x and x plus a; and say that the probability of finding the particle at x+a is the same as the probability that it has traveled there along the straight line you just drew. (I don't know if you know any calculus, so it's worth saying that the reason you can do this is that if you think about a small enough stretch of any curve it will look like a straight line, in the same way as the Earth looks flat to us even though we know it's round).
3. Now pretend that the particle is at x+a, and you can repeat the trick to work out the probability of going from x+a to x+a+b. You multiply this probability amplitude by the the one you got for moving from x to x+a.
4. If you repeat this trick going through x+a+b+c, ... until you end up at y, then you get a probability of the particle "moving" from x to y having been through each of the points x+a, x+a+b, x+a+b+c,... etc. In other words, you get a probability associated with the path through all these points.
5. Now by summing over all possible values you could use for a, b, c... at each step in this calculation, you can sum over all possible paths. The end result agrees with the answer you get by never thinking about paths at all.

In this procedure you pick up paths that loop around the moon and back, and do similarly strange things, but the thing you're really interested in is whether this whole business gives you sensible answers for getting from x to y, and sure enough these absurd paths make such tiny contributions to the overall sum that the chance of net movement at superluminal velocities is really completely negligible.

It turns out, however, that steps 4 and 5 can be performed without reference to standard methods at all; you could start from your answer to step 4 and reconstruct the whole conventional apparatus of QM. So I'm not sure it's fair to say that these "paths" are completely fictitious. What I hope is clear is that it's an "all or nothing" question- you either have to take no paths into account, or you have to take all of them, and you can't single out any of them as being "the" path taken by the particle. The idea that it simply doesn't have a position or momentum is perhaps easier to get your head around than trying to picture a single particle moving along lots of paths simultaneously, but both pictures are useful heuristic tools, both mathematical formalisms make the same (experiment-passing) predictions, and neither probably stands up to much philosophical scrutiny.

It's maybe also worthwhile pointing out that classically, photons don't exist. When you construct the quantum theory of the EM field by these methods, you don't do it by summing over photon trajectories; rather, you sum over classical field configurations that some initial field might move through to end up in some final state.

 

[maverick_starstrider]

The point is that the photon does not REALLY takes all possible paths. This is just a mathematical trick in which something more real is calculated in terms of auxiliary mathematical paths that do not really exist.

Roughly, this is like writing
1 apple = 2 apples + (-1 apple)
even though -1 apple does not really exist.

The Path Integral formulation and a Schrodinger's Equaton formulation are mathematically identical and can be derived one from the other. Thus, there is no logical reason to say one is a description of reality and the other is a mathematical trick. Is a wave REALLY a single function or is it REALLY an infinite sum of cos and sin functions (a la Fourier) with varying coefficients? It's a moot point, there is no way one can motivate an answer. Feynman's path integrals are an equally valid and invalid description of quantum reality as any other given that there could never be an experiment that could distinguish between the two.

Though I for one, as a matter of taste, do find it a formalism with a greater... I dunno... economy of concepts to swallow, than a wave equation conjured and motivated from the proverbial ether. But that's just taste.

 

[Demystifier]

The Path Integral formulation and a Schrodinger's Equaton formulation are mathematically identical and can be derived one from the other. Thus, there is no logical reason to say one is a description of reality and the other is a mathematical trick. Is a wave REALLY a single function or is it REALLY an infinite sum of cos and sin functions (a la Fourier) with varying coefficients? It's a moot point, there is no way one can motivate an answer.

So maybe an apple is REALLY 2 apples and -1 apple?

 

[maverick_starstrider]

So maybe an apple is REALLY 2 apples and -1 apple?

Could there every be a situation where you could prove otherwise? Going back to a Fourier decomposition, if one sees a random wave "out in the wild" of the ocean they may say that its form is the fundamental thing. However, one could take a whole bunch of specialized wave emitters, each which produces a wave of only one frequency (and with a knob to adjust amplitude) and using this array of emitters one could produce the exact same wave form and claim that the individual component waves are the fundamental thing. Similarly think of JPEG image compression; is the image map the thing or its wavelet components? Are BMPs more or less of a representation of an image than a JPEG (ignoring information loss for compression purposes)?