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A democracy of spacetimes?

  1. Feb 28, 2006 #1
    Hello all,

    I have been thinking about ways that classical mechanics can arise in the classical limit of QM, and I'm wondering how this might occur in quantum gravity. In particular: in the Feynman path integral technique, we start with what Feynman calls the "democracy of paths," according to which all paths, even non-classical ones, are attributed equal amplitude (equal absolute value, that is). In the classical limit, it is easily shown that the classical path of least action is "more probable" (speaking loosely) than all the rest, and in this way, Hamilton's action principle, and thus Newtonian mechanics, may be understood to be valid in the classical limit of QM. I state this a little more carefully in the thread:
    https://www.physicsforums.com/showthread.php?t=112257

    So here's what I'm wondering. Within any of the various quantum gravity programmes, is any attempt made to accomplish a similar derivation that yields, not (merely) Newtonian mechanics, but (more generally) GR? Here's what I'm envisioning: instead of a "democracy of paths," we have a "democracy of spacetimes" (DOS). Let me complete the analogy: in the FPI, we start with "all possible paths," which includes non-classical paths. In the "democracy of spacetimes," we start with "all possible spacetimes," which means we include spacetimes that do not obey Einstein's equation. In the FPI, we attribute an amplitude to each path, each with equivalent absolute value. In "DOS," we figure some way to assign probabilities -- I suppose there are various ways to do this. In the FPI, the classical paths turn out to be "more probable" than non classical paths. In DOS, we hope to show that classical spacetimes (ie, those that obey Einstein's equation) are "more probable" than non-classical spacetimes.

    So I'm wondering: does LQG or any other theory of quantum gravity fit the above description?

    David the Amateur Physicist :blushing:
     
  2. jcsd
  3. Mar 1, 2006 #2
    Hi David.

    Thanks for this post. I am sure someone qualified to think about this is going to answer soon, but in the meanwhile, I should like to take a shot, subject to corrections of course. First, I admire the post as a nice piece of thinking.

    Now, it occurs to me that there is a difference between what is possible and what is probable. We might place very strong limits on what is possible. We might want to say that there is no spacetime in which Einstein's equation is violated. BTW maybe you could tell me which of Einstein's equations you have in mind here. I guess GR since you mention it earlier, but just checking.

    Seems to me then that when Feynman says all possible paths must be considered, he is not saying that impossible paths should be added in also. So, if I have interpreted correctly, the implication is that there are many paths we consider impossible under classical rules, but which are not really impossible at all, merely improbable, under quantum rules.

    Let's say GR is a local theory, by which I mean that there is strictly no action at a distance. Then entanglement might seem to be impossible under GR. However, we have some experimental evidence that suggests entanglement does occur. Now we have two ways to approach this difficulty. In one way, our notion of GR must be wrong somehow, even though we can find no error in any test of GR as it stands. But in the other way, there may be an error somehow in our notion of locality.

    Locality is not so well defined as is GR so it seems a good candidate for investigation. If spacetime is flat, for example, like a sheet of paper, might we somehow bend the sheet of paper, even twist it, in a way that two regions which are not local when the paper is lying on the table might be brought close together, locally? Well sure, it works for the two dimensional paper surface when we work in three dimensions.

    So maybe we might ask if some objects which we see in three dimensions of space are really embedded in higher dimensions, so that what we see in three dimensions is really only part of the higher dimensional object. Then maybe the higher dimensional object could undergoe some kind of folding or twisting which would make the seemingly impossible happen. Two parts of the object which are strictly far apart in three dimensions may be brought close together in four dimensions, and thus interact without violation of extended rules of locality.

    I may be mistaken and my study is far from complete, but it seems to me now that many of the approaches to quantum gravity have taken the path outlined above. I could stick my neck further out of my shell like a turtle trying to get a good view of his tail, and try to connect this idea directly to LQG or M or something, but it is ash wednesday on the Christian calender and we are invited to consider the probability of our own humiliation. So I won't.

    Thanks,

    Richard
     
  4. Mar 1, 2006 #3
    how about an ecology of spacetimes?

    Darwinism seems to be a ubiquitous process arising from parallel schemes working independently in non-compromise which emerges as a complex whole
     
    Last edited: Mar 1, 2006
  5. Mar 1, 2006 #4
    This is a great discussion, creating and exploring new ideas is very important in physics but, we need to make sure that our ideas or theories are scientific.

    First you need to develop the idea sufficiently so that it becomes a theory, then you need to develop the formalism necessary to express the theory mathematically. All the while you need to make sure your theory takes into account all experimental facts and prior theories, also make sure the theory is self-consistent. Then to make your theory scientific you need to differentiate it from all other competing theories by making testable, falsifiable predictions.

    I think you see why it is so difficult to develop a viable theory of spacetime, the conditions it needs to satisfy are quite strict. Anyway if you believe in the power of your idea then follow it but remember, it will take time to develop the theory and there is no guarantee the predictions will accord with experiment.

    John G.
     
  6. Mar 4, 2006 #5
    I agree with everything you said above. I do like my idea enough to have pursued it in some further detail, albeit in a slightly different direction -- see, eg, post #72 on this thread:

    https://www.physicsforums.com/showthread.php?t=109096&page=5

    in which I am trying to understand some of the various "ontological" issues surrounding QM and quantum gravity.

    You will note that I call it a "toy model" -- I am not so presumptuos as to think that it has progressed to the status of a "theory." :blushing: Call it what you will, I *am* pursuing it as far as I can take it. All in good fun!

    David
     
  7. Mar 4, 2006 #6
    Hey Richard,

    I suppose I am thinking of GR.

    Well, by my understanding, when Feynman talks about "all possible paths," he means any path that connects a and b. It must be continuous, although (iiuc) it is not necessarily differentiable. So this includes classical paths, of course, but it also includes paths that are, as you say, "impossible" classically, eg paths that zig and zag all over the place. And as you say, under quantum rules these classically impossible paths are quite possible!

    Now, defining what exactly constitutes the set of "all possible paths" is not exactly straightforward. You have to dig into the formalism of the calculus of variations to figure out what exactly constitutes a "path." So this is an example of an idea that is not terribly difficult to state intuitively, even though its implementation is difficult. My understanding is that the people who worked this out in the first place were guided by some extent by trial and error in their definition of what constitutes a "possible path," and they simply picked the definition(s) that "worked" -- ie, yielded the Schrodinger equation as the end result. In fact, the exact definition may differ from one problem to the next. You could call that cheating, but hey, you do what you gotta do ...

    According to the path that you outline above, you would be imagining that two seemingly dissconnected regions of spacetime are, in fact, joined together by what I suppose we would call a wormhole. A more general statement would be that spacetime is multiply connected. In fact, some people have suggested that spacetime is multiple connected at the small scale, although not (necessarily) at the large scale, and that the small scale multiple-connectedness is what we mean by "quantum foam" -- see, eg, the illustration at this link:

    http://universe-review.ca/I01-16-quantumfoam.jpg

    Note that you do not in fact need any extra dimensions (like a 4th spatial dimension) to do this.

    So this might be one way to explain quantum entanglement. Note, however, that QM places no limit on the distance over which "entanglement" may be manifest. (The Gisin group, for instance, has demonstrated entanglement of photons miles apart.) So we would have to modify the quantum foam picture above to allow "wormholes" that join regions of spacetime that are miles apart (ie, miles apart if you go from a to b outside the wormhole, but really close if you go from a to b via the wormhole).

    There is one reason that I do not favor the above as a means of explaining entanglement. The reason has to do with the fact that "spooky action at a distance" is a very different beast than classical causal influences. The difference is that classical causal influences can be used to transmit information -- think of a signal being propagated at speed c via fluctuations in the EM field. The collapse of the wavefunction, otoh, cannot be used to transmit a signal.

    So if we imagine that entanglement is explained somehow by the fact that two seemingly separated particles are in fact joined by a wormhole, then it would seem to follow that it IS (or should be) possible to transmit information "faster than light," ie across the wormhole from one particle to its entangled partner. This would not violate GR because it would be founded on the postulate that the wormhole places, let's say, particle a in the future light cone of particle b via the wormhole. So then how do we explain the fact that quantum entanglement cannot be used to transmit signals? We need a different model of entanglement, imho -- one that explains the EPR correlations, but which does NOT allow FTL signal propagation. To see an example of how I would envision this, see the "toy model" I presented in message #72 of the PF thread with url in my previous post.

    David
     
  8. Mar 4, 2006 #7
    I have read a tiny bit about Smolin's idea to apply Darwinism to physics. This is a good example of "intellectual cross-training." I actually have a PhD in cell biology, so for me, thinking about physics is my own exercise in "cross-training." :wink:

    The issue that springs directly to my mind when we try to apply Darwin to quantum garvity is this: how do we define the measure over the space of possibilities? In evolution, you simply count up the number of organisms at a specific instant of time. eg, if you start out with one microbe, and it splits into 100, and one of them has a mutation that out-competes (say by multiplying faster) the other 99, then over the long haul the ratio of microbes with to microbes without the mutation approaches infinity, even though it started out at 1:99.

    So in quantum theory, instead of microbes, we have, say, universes, and each universe can (somehow) create lots of baby universes that looks a lot like its mother universe, perhaps modified by some stepwise mutation. And the idea is to determine which characteristics would result in the "greatest number" of universes. So here's the question: how do we count universes? Do we simply count them up to get a number N_i, and take the ratio over the total number N_tot of universes, and say that the probability that we find ourselves in the i^th set of universes is equal to N_i / N_tot? iow, do we define a probability P_i = N_i / N_tot ?

    I actually think that the above method of calculating probabilities is, more or less, exactly how we should go about it. This method is more or less the same thing as "outcome counting" that I have discussed in several other threads:

    thread: "My paper on the Born rule:"
    https://www.physicsforums.com/showthread.php?t=95585
    started by Patrick Van Esch to discuss outcome counting (which he calls the APP) as an alternative to the Born rule.

    thread: "Are world counts incoherent?"
    https://www.physicsforums.com/showthread.php?t=101339
    started by Robin Hanson to discuss whether it is possible to define probabilities using a the outcome counting method (which he calls world counts) in place of the Born rule.

    thread: "Attempts to make the born rule emerge explicitly from outcome counting"
    https://www.physicsforums.com/showthread.php?t=101982
    started by me to discuss, well you can figure it out from the title :).

    The difficulty that I think Smolin will encounter is that, after 80 years of QM, people are so familiar with the Born rule that they cannot imagine it being replaced by anything so mundane as outcome counting for the calculation of probabilities. Now, it may be that I am misinterpreting the whole Darwinian thing. But I can't think of what else new is brought to the physics table by thinking about evolution, other than how we think about probabilities ...

    david
     
  9. Mar 4, 2006 #8

    selfAdjoint

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    Stray, the question of the measure, in the technical sense, over the space of possible universes is very much on the minds of both the LQG and the string people. Particle physicists have a long history with measure questions, going back to the historic days of Weyl and von Neumann. Modern measure theory and quantum field theory more or less came to maturity in parallel, and there was a lot of interaction; notoriously Schwartz's definition of distributions (1945) saved Dirac's "delta function" from mathematical contempt.
     
  10. Mar 5, 2006 #9
    In his Dublin paper Hawking describes his approach to the information problem as "taking the path integral over metrics of all possible topologies". You might gain some insight from his paper.
     
  11. Mar 5, 2006 #10
    Is this Hawking's Euclidean path integral method? I have butted heads against it several times. The notion of considering all possible topologies as existing in superposition with one another, analogous to the notion that Feynman paths exist in superposition to one another, is one that I like a lot. There are still some aspects of his scheme that I have not figured out, like where he says in his Dublin talk [1]:

    "I adopt the Euclidean approach, the only sane way to do quantum
    gravity non-perturbative. In this, the time evolution of an initial
    state, is given by a Path integral over all positive definite metrics,
    that go between two surfaces, that are a distance T apart at
    infinity."

    How do you define (or choose) these two surfaces? I haven't figured that out yet.

    Also, I haven't figured out how probability enters Hawking's scheme. Does he assume a standard quantum mechanical "probability rule," ie does he assign an amplitude to, say, each possible topology (plus metric)?

    Right now I'm trying to figure out how QFT works. I understand the lagrangian as used in the FPI, and trying to understand the definition of the lagrange density -- this is where I'm at in studying QFT. My next step will be to try to implement outcome counting within the context of QFT, which I think is related to the issue of measure that selfAdjoint mentioned earlier.

    David

    [1] http://pancake.uchicago.edu/~carroll/hawkingdublin.txt
     
  12. Mar 7, 2006 #11
    http://universe-review.ca/I01-16-quantumfoam.jpg

    Link to quantum foam picture, David, post 6 this thread.


    Hi David

    The idea of a ‘quantum foam’ can be illustrated in three dimensions or even in two dimensions, as in the flat picture on the screen of my computer when I call up the image you linked in the other thread….

    http://universe-review.ca/I01-16-quantumfoam.jpg

    However I am not certain that all the effects implied in the illustration can be fully revealed in a two dimensional image. We are invited to mentally add a third spatial dimension by means of our knowledge of parallax, which we do in part by adding an imaginary time dimension to the two dimensional image, a dimension in which we figuratively “look from side to side,” a figurative motion which includes one step parallel to one of the axis in the 2d picture and one step orthogonal to the picture in the imaginary time dimension. We are also invited to add a further imaginary time dimension in which we think of the foam in motion parallel to our own sense of time duration, such as can be illustrated by a film loop, playing the same bit of time over and over in many bits of our own time history. So when we look at the two dimensional image in the link we also imagine what it might have looked like in an imaginary previous instant (imaginary because it is only a drawing and there is no “real” previous instant) and also in an imaginary future instant, so giving the image the semblance of motion, in a purely imaginary time. We do this automatically and we learned to do it when we were very young, so we almost never think of the required mental steps that we use, but I think if you go through your sequence of thoughts, you will have to agree that the simple two dimensions presented do in fact rely on several ‘hidden’ dimensions to make the image a viable model of quantum foam.

    You see that the number of required dimensions is a kind of mental trick, a perceptual tool we use to compare our knowledge of the world as it was in the past with our sensation of the world as it is in this present instant. If you are looking at a 2d drawing, you do the mental tricks necessary to see the concept of the drawing in more than two dimensions. The real question is not how many dimensions are available, but how many dimensions are necessary to convert the image we are presented into the concept which we are invited to consider. Of course we want to present images in the smallest number of dimensions necessary to illustrate the idea, for reasons of economy, clarity, and elegance. Humans are not used to thinking or imagining in more than three or four dimensions, but contrary to popularly held beliefs, there is really no good reason to say that we cannot see higher dimensional systems.

    We can and commonly do act in ways that can be recorded on and analysed in three spatial and one time dimension, as is the classical insistance.

    What is spatial distance? Degree of contact? A metric involving square terms and co-instantaneous poles of separation? How many poles does it take to change a lightbulb? Is an infinite sequence of regressions a line in time? Any reasonable set of such regresions becomes almost instantaneously complicated. We are reduced to the "contact" area of parallel exposure across time dimensions which can be seen as spatially void. The contact areas in parallel have a chance or probability of coming close together in some possible sequence, so we account for the immediate probability by limiting our vision to some set of real things in our environment.



    So I didn’t mean as strong a linkage between the idea of the wormhole and the idea of entanglement as you have taken. They are related ideas, both involving a sort of bending of flat spacetime, but the bend does not necessarily correspond. One may be a timelike curve, the other a spacelike curve, or more likely they are each combinations of partial differentials of timelike and spacelike elements. Related ideas, certainly, but related like the idea of going to Paris is related to the idea of going to Singapore. Many of the required steps are the same, even identical, but the result is not the same at all. Anyway you are probably already aware of the work that has been done showing that wormholes are not likely candidates for ftl signaling devices either, due to additive feedback loops which violate conservation of energy and make the entire system ‘blow up’ to infinities that are clearly on the other side of the line which divides us from what is not possible. Wormholes could only exist if there were some damping mechanism which could keep the two ends from being causally local to each other.

    In short, I think higher dimensions are something very routine in human existence, but something we have not thought about clearly. It isn’t hocus-pocus, nor even action at a distance. It is just that we need a better idea of spacetime geometry to account for experimental results.

    Much more to say, but have resolved to try to become crisp.

    R.
     
    Last edited: Mar 7, 2006
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