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Principle of Least Action

  1. Jan 21, 2007 #1
    I'm not sure whether this question belongs in the quantum section or here, since it kind of involves both areas. I'm not even sure I can ask the question sensibly since I'm not a professional physicist, but I'll try:

    Suppose we look at a particle, say a photon, moving along in space time. According to Feynman's "QED, the Strange Theory of Light and Matter", I can "explain" its motion by adding up the complex numbers from various alternative "trajectories" for the particle. Now some of those alternative trajectories will have wiggled way off the striaight line - maybe they went light years away. According to the principles of relativity, no information can be transmitted faster than light. How then do those way off-path alternative histories all get combined to produce the straight line ?

    To be a bit more specific, if I want to send a beam of light to the moon, how does the light "know" that there isn't a shorter path via Alpha Centauri ? It can't know unless it's been there and had a look and brought the information back with it, and that's impossible.

    I've seen people draw pictures where they wiggle a little bit off a straight line, but that doesn't help me understand the enormously diverted alternative paths.
     
  2. jcsd
  3. Jan 28, 2007 #2
    I think Feynman wants to illustrate something called "Huyghens' Principle".
    Another way to illustrate this, is a hologram. A hologram can be cut in half, and each part still carries the whole picture.
    That's because a hologram is the record of a light wave, and the information in a wave is spread over all of the space it occupies.

    More practically, when you focus your beam on the moon, there will be diffraction at the lens, and a tiny fraction of the light will in fact go to Alpha Centauri (taking ~4 years to get there...)

    Now Huyghens says that if you want to calculate the amplitude on the moon, you have to take into account the amplitudes at all other points, even Alpha Centauri.

    In other words, if something happens to the light on Alpha Centauri, you SHOULD be able to see the effect on the moon. After 4 more years, of course...
     
  4. Jan 28, 2007 #3
    I think at a philosophical level, most of the confusion in relativity and quantum theory is that we are not ready to accept that everything is predetermined. In this example, the photon does not need to find out where is alpha-centauri, it just follows the mathematical rules laid down, and as if it has already gone to the moon. To me, that somehow unifies quantum theory and relativity.
     
  5. Jan 28, 2007 #4
    Indeed, this is my current understanding.

    Moreover, if the enormously diverted path isn't a turning point for the "action" (that is, if by slightly changing the enormously diverted path, you can find two groups of very similar paths that happen respectively to have slightly less AND slightly more "action", or divertion), then when you add all Feynmann's phase terms for all paths they just tend not to constructively interfere (except where the straight line path is..).
     
  6. Jan 29, 2007 #5

    pervect

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    As I understand it (not all that well), those paths simply don't contribute much to the path integral. The only paths with significant contributions are paths that are close to the path of minimum action. What's somewhat amazing to me is that the path integral converges at all. I'm not quite sure how this convergence is managed in detail, unfortunately, and this is probably what you're really trying to figure out.
     
  7. Jan 29, 2007 #6
    sheaf,

    The partial waves don't transmit any information.
    Therefore it is not in contradiction with relativity if they have a (phase) velocity larger than the speed of light.
    However, when a photon is detected somewhere, at a positive interference position, it cannot be detected at a time incompatible with the speed of light.

    But it is still a little more complicated than that, I think.
    The heisenberg uncertainties on position and speed is my problem ...

    Michel
     
  8. Jan 29, 2007 #7
    Many things travel faster than speed of light. Has anyone compiled a list?
    One thing doesn't - information.
     
  9. Jan 29, 2007 #8

    mjsd

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    sheaf

    firstly, I can only vaguely see the connection between your title "principle of least action" and the subsequent discussions .. anyway, are you referring to the Path Integral formalism of QM? I think you are asking (tell me if this is not the case): say a particle travels from A to B, now according to Feynman, there are many paths that connects A and B, some longer than others, if the path is too long, then the particle can't get from A to B on time, but is Feynman wrong or special relativity is somehow violated?

    ok, assuming this is your question, let me try to answer it... in a layman way of course (so don't pick on any little techincal errors) . Firstly, the "complex number" you are referring to is probably the amplitude, while the "trajectories" is what I've been calling paths :smile:

    ok, so [tex]A(T)=\sum_i A(T)_i[/tex]
    where
    [tex]A(T)[/tex] is amplitude of a particle going from A and arriving at B in time [tex]T[/tex] while [tex]A(T)_i[/tex] is the amplitude of a particle going from A and arriving at B in time [tex]T[/tex] following a particular path

    now, the crucial thing here is that all paths has the same time [tex]T[/tex]. To find out the actual amplitudes you need QM. Each path is first chopped into segments (then integrated), and then you sum up all paths...etc.
     
  10. Jan 29, 2007 #9
    Wave functions do travel faster than speed of light!
     
  11. Jan 29, 2007 #10
    http://www.bottomlayer.com/bottom/basic_delayed_choice.htm

    Does our choice "change the past"?
    How long can we delay the choice? In Wheeler's original thought experiment, he imagined the phenomenon on a cosmic scale, as follows:

    1. A distant star emits a photon many billions of years ago.

    2. The photon must pass a dense galaxy (or black hole) directly in its path toward earth.

    "Gravitational lensing" predicted by general relativity (and well verified) will make the light bend around the galaxy or black hole. The same photon can, therefore, take either of two paths around the galaxy and still reach earth – it can take the left path and bend back toward earth; or it can take the right path and bend back toward earth. Bending around the left side is the experimental equivalent of going through the left slit of a barrier; bending around the right side is the equivalent of going through the right slit.

    3. The photon continues for a very long time (perhaps a few more billion years) on its way toward earth.

    4. On earth (many billions of years later), an astronomer chooses to use a screen type of light projector, encompassing both sides of the intervening and the surrounding space without focusing or distinguishing among regions. The photon will land somewhere along the field of focus without our astronomer being able to tell which side of the galaxy/black hole the photon passed, left or right. So the distribution pattern of the photon (even of a single photon, but easily recognizable after a lot of photons are collected) will be an interference pattern.
    5. Alternatively, based on what she had for breakfast, our astronomer might choose to use a binocular apparatus, with one side of the binoculars (one telescope) focused exclusively on the left side of the intervening galaxy, and the other side focussed exclusively on the right side of the intervening galaxy. In that case the "pattern" will be a clump of photons at one side, and a clump of photons at the other side.

    Now, for many billions of years the photon is in transit in region 3. Yet we can choose (many billions of years later) which experimental set up to employ – the single wide-focus, or the two narrowly focused instruments.
     
  12. Jan 29, 2007 #11
    Thanks to everyone for replying.
    Yes, I am talking about Feynman's sum over histories. I'll try to explain my question with a diagram. Below I have Minkowski space, and I want to calculate the amplitude for a particle to get from A to B. I'm changing my original question, which took the example of a photon, because it's easier to ask the question with a diagram this way. The diagonal lines show the forward null cone from A. The line on the right hand side outside the null cone is a barrier, so paths can't get through it. My question is, looking at the path integral for the amplitude to get from A to B, the paths that would go through the barrier wouldn't contribute, so would that have an effect on the final answer ? If so, then I could deduce the existence of the barrier, which is not allowed by the principles of relativity, since I am gaining information about something outside the light cone.

    I suspect the answer is no, the barrier won't have an effect on the final amplitude and relativity will live to fight another day, but I've no idea how to perform the integral to prove it. I can see that the off-main-path paths will be very small contributions, but since this question is about principles, even a very small effect is signficant. I know path integrals are extremely difficult to do explicitly, but I thought maybe this one - a free particle in flat space - might be possible.

    Just so people know at what level to pitch their answers - I do have a degree in physics, but I got it 25years ago, so I'm a bit rusty - though I have tried to keep up by reading semi popular stuff.

    Sorry about all the dots, but when I tried spaces, they were automatically concatenated into a single space when I did the preview.

    ................................ x B
    \.......................................................... /
    ...\.................................................. /
    ......\............................................ /
    ........ \ ..................................... /
    ............. \ ............................. / ------------------------------------------------
    .................. \ ....................../
    ......................\ ................/
    ..........................\ ........./
    ...............................x A
     
  13. Jan 29, 2007 #12
    OK I have got mine only 15 years ago.
    Quantum electrodynamics already includes special relativity right? So it can deal with fast moving electrons and stuff like that in flat spacetime.
     
  14. Jan 29, 2007 #13

    pervect

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    That's a terribly misleading thing to say. With ordinary quantum mechanics, for instance, it is for example true (but misleading) to say that Schrodinger's wave equation has solutions that travel faster than 'c'.

    The problem here is with Schrodinger's wave equation, not relativity or quantum mechanics. Schrodinger's wave equation is just an approximation. If you look at the Dirac (for spin 1/2) or Klein Gordon (for spin 0) wave equations, they don't have solutions that travel faster than light.

    http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/tachyons.html

    talks about this a little bit, with some specifics on the Klein-Gordon equation (not very many on the Dirac equation, though)

    Moving onto QED, we have the following:


    The FAQ didn't go into this in enough detail, but I gather that any relativistic quantum field theory will have the above property, which basically says "No observable FTL".
     
  15. Jan 29, 2007 #14
    They don't travel at all.
    :smile:
     
  16. Jan 29, 2007 #15
    I think this is correct. What I meant by "the whole space" is actually the backwards lightcone of B.
     
  17. Jan 29, 2007 #16

    Fredrik

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    Sheaf, you really asked a good question. I've been thinking a lot about this today. These are my thoughts about it.

    A path integral calculation can tell us the probability P(x) that a photon emitted by a laser on Earth will hit a spot x on the moon. (Actually it would be a probability density, but that's just a technicality, so I won't say anything more about that). To find the shape of the function P, we would have to do a path integral calculation for each x we're interested in. We would find that the function P is sharply peaked around the point we're aiming at. It's important to realize that P(x) not zero everywhere else.

    Now if we were to put up an obstacle somewhere near Alpha Centauri, this would eliminate some of the paths from the path integrals we need to evaluate to find the probabilities P(x). But most of the paths that get eliminated would cancel each other out anyway. The effect of those that don't cancel each other out is to change the probabilities P(x) by ridicilously small amounts. This would move the peak of the function a ridicilously small distance from where it "should" be.

    This effect would be instantaneous. It would not take four years, as someone suggested earlier in this thread. So how does this not violate causality? I think I might have the answer, but I haven't done any calculations to verify it.

    The specific point that the peak is moved to depends on the exact size and shape of the obstacle. Suppose that the obstacle is a wall that starts 4 light years to the right of the moon (as seen from the laser's position) and extends 100 m further to the right. Suppose also that we're able extend the wall further to the right. Now we can consider the position of the peak of P as a function of the length of the wall. I believe that if we could make the wall grow continuously and very slowly, the position of the peak would oscillate (pretty rapidly, I think) around the position where it would be if the wall wasn't there. It would however never go very far from that position.

    How far would the peak move? I believe that a calculation would show that the maximum distance it could move is extremely tiny, shorter than the Planck length. This would solve the causality problems, since distances that small aren't measurable anyway.

    However, I suspect that if the distance to the wall is, say, 4 meters instead of 4 light years, the presence of the wall might move the peak a distance that's greater than the Planck length. If that's the case, there's another way that casuality may be saved. It takes time to perform an experiment that determines if the peak has been moved. Sending one photon obviously isn't enough. I suspect that a detailed calculation would show that if the intensity of the laser isn't unreasonably high (perhaps so high that the beam would collapse into a stream of black holes), there's no way to do the experiment in less time that it would take a message traveling at speed c to get from the wall to the point x.

    I might be wrong about some of these details, but something like this must hold if causality is to be preserved.
     
  18. Jan 29, 2007 #17

    mjsd

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    On causalisty:

    Quantum field Theory is the result of combining QM with Spec Rel. It gives an explanation to "action at a distance" that is consistent with Sepcial relativity (that's where propagators come from). It is then not surprising that classical QM may led to paradoxes that seems to violate causality ....anyway, some of you guys may have misinterpreted/misquoted the statements of Feynmann. All paths are paths that get there at the same spacetime coordinate. If they arrive earlier or later, they are a separate event, and if a path is "too long" such that it can't get there in time T unless you somehow violate Spec Rel, then this path is not "a path". ie. doesn't exist. When you are summing up all paths, you ask yourself how many "valid" paths are there to be summed up... not any artibrary paths, eg. paths that arrive there early, late or violate Spec Rel.
     
  19. Jan 30, 2007 #18

    Fredrik

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    I'm pretty sure you got this wrong. Yes, all the paths have the same endpoints, but paths that are "too long" must be included in the path integral to get the correct answer.
     
  20. Jan 30, 2007 #19

    mjsd

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    my definition of a path being "too long" is that they don't have the same end points (same spacial coordinates, but not the same for the time coordinate).
     
  21. Jan 30, 2007 #20
    That's the impression I had too. I remember reading that you have to add all paths, even ones that wander backwards and forwards in time. Thanks for the explanation - I'll have to go and think about it a bit more and draw a few diagrams !

    Given that it's difficult to explicitly calculate even a simple path integral, I was toying with the idea of doing a computer program to build a small 2d lattice and try to approximate the integral as a sum to see what happens. I think the sum still has a ridiculously large number of terms for all but the smallest lattices though.
     
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