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QM and action principles

  1. Feb 26, 2006 #1
    For a long time, I have appreciated the fact that Hamilton's principle of least action can be derived in a straightforward manner via the Feynman path integral (FPI) technique. (This is explained very nicely in the classic textbook: [1].) Since Hamilton's principle of least action leads to Newtonian mechanics, then we can say that Newtonian mechanics may be derived from quantum mechanics.

    Recently, I have learned that Einstein's equations of general relativity may be derived from a similar action principle, in a manner analagous to the derivation of Newtonian mechanics from Hamilton's principle. (This is discussed in MTW, section 17.5, box 17.2, section 2.) Here is the difference: in Hamilton's principle, the action is defined as the one-dimensional integral of the lagrangian corresponding to a particle path; but for the derivation of GR, the action is defined iiuc as a 4-dimensional volume integral of a lagrange density. Call the first "action principle #1" (Hamilton's principle), and the second "action principle #2". We have:

    QM (FPI) ==> action principle #1 ==> Newtonian mechanics

    as well as

    action principle #2 ==> GR

    So here's my question: is it possible to derive "action principle #2" from QM? If so, could we conclude that GR follows necessarily from QM? If not -- is there any other way to derive action principle #2?

    David

    [1] R.P Feynman and A.R. Hibbs. Quantum Mechanics and Path Integrals (McGraw-Hill, Boston, MA, 1965)

    PS: I should perhaps point out that Hamilton's principle of least action should actually be called the principle of extremal action. Mathematically, the classical action is a stationary point -- not necessarily a minimum.
     
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  3. Feb 26, 2006 #2
    The Feynman path integral is itself not a derivation of the principle of least action. The principle of least action is more of a tool used in elementary quantum field theory and a point for justifying the path integral approach, which came after the initiation of field theory. I believe Dirac and Feynman worked on and took inspiration from the idea of stationary phase from Stocastic analysis, though I've not studied this itself. The most prominant use for the functional formalism is for deriving Feynman rules for field theories. It also removes the necessity for a Hamiltonian function for quantising the field theory, allowing the Lagrangian density to take priority.

    The Feynman transition amplitude is

    [tex]\langle \psi_b|e^{-iHT}|\psi_a\rangle =\int\cal{D}\psi[/tex][tex]\exp\left[ i\int d^4x\cal{L}\right][/tex]

    The limits of the time integral on the right hand side being 0 and T.

    The functional approach too takes an action of the form of a four-dimensional integral in Minkowski space-time over the Lagrangian DENSITY (see above). This is entirely equivalent to a one-dimensional time integral of the Lagrangian.

    [tex]S=\int dt L=\int d^4x \cal{L}[/tex]

    The condition satisfied by the field in order for the action to be stationary is equivalent in both cases when the field equation involves the Lagrangian and the Lagrangian density. In quantum field theory, and indeed modern physics in general, it is the Lagrangian density that is the more favourable quantity.

    As for the Einstein-Hilbert action of general relativity, the action still takes the form of an ordinary four-dimensional integration in Minkowski coordinates over the Lagrangian density [itex]\cal{L}[/itex], but the specific Lagrangian density introduced by Hilbert includes the square root of the metric determinant that ensures we're performing a coordinate invariant integral.

    [tex]S=\int d^4x \cal{L}[/tex][tex]=\int d^4xR\sqrt{-g}[/tex]

    So that when we apply the principle of least action [tex]\frac{\delta S}{\delta g_{\mu\nu}}=0[/tex] we must vary the metric determinant as well, to give the familiar field equation.
     
    Last edited: Feb 26, 2006
  4. Feb 26, 2006 #3

    Physics Monkey

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    I would argue that one can use the Feynman path integral to "derive" the action principle. This is how Feynman himself presented it. What I mean is that if you regard the Feynman path integral as the fundamental description, then the least action principle necessarily flows from this description. The historical route by which the path integral was arrived at doesn't invalidate the statement that the classical limit of the Feynman path integral is the action principle. In other words, just because we noticed the path integral later doesn't mean we can't use it to justify or understand the action principle (which is fairly mysterious within classical mechanics).

    Of course none of this invalidates what perturbation said, which is something like the historical view of things.
     
    Last edited: Feb 26, 2006
  5. Feb 26, 2006 #4
    Hey Perturbation,

    Thanks for the reply. The conceptual transition from Lagrangian to Lagrangian density has always been a bit perplexing to me. Just so you know where I'm coming from, I had some exposure to the calculus of variations in college (> 10 years ago), and at one point I could actually do some simple calculations with it. So at the least, I do have an intuitive picture in my mind of what it means to vary a path.

    I am having trouble, though, understanding how you can equate the above two integrals. The one on the left is a one-d path integral, and the one on the right is over a 4-d volume, right? So should I envision the 4-d volume on the right as being like a long thin 4-d "tube" surrounding a 1-d path? iow, the volume integral is (I am asking) performed over a region defined as those points whose shortest distance to the 1-d path is less than or equal to some (infinitessimal) length. If this is correct, then I suppose that the method of varying the path is the same no matter which integral we use to define the action. On the other hand, perhaps the variation of the volume is more complicated than varying a path -- ie we could vary the boundary of the volume in more complicated ways than we could vary the path.

    btw, I agree with what Physics Monkey said about the derivation of the action principle from the FPI. So I am still wondering whether it would be possible to derive "action principle #2" from the FPI. Now I have never heard anyone say that it is possible to derive GR from QM, so I'm guessing the answer to my question is "no" -- but I would like to understand why. Would it be possible, I wonder, to formulate the FPI using the lagrangian density (instead of the lagrangian), as perturbation indicated above, and use it to derive "action principle #2," ie the statement: [tex]\frac{\delta S}{\delta g_{\mu\nu}}=0[/tex], in a manner analogous to the derivation of Hamilton's action principle?

    David
     
  6. Feb 27, 2006 #5
    I would agree with you, by the way, that the Feynman path integral can be seen as a derivation or justification of the principle of least action, I was just giving an historical account in my first post.

    The functional formalism using a Lagrangian density and Lagrangian are equivalent, as the generic Feynman transition amplitude formula involves the exponentiation of the action, which, as I indicated above, is equivalent to both a time integral and a four-volume integral over the Lagrangian and Lagrangian density respectively. Whilst I was learning field theory it was the Lagrangian density that was used for deriving the equations of motion etc., so I'm used to using it over the regular Lagrangian, but as I've said the two methods are identical.

    Wikipedea-Lagrangian Density

    The variation of the four-volume integral isn't much more difficult than that of the time integral, we can use Gauss' theorem to reduce one of the terms to zero by making the variations vanish at the boundary of the integration domain.

    What exactly is the derivation of the principle of least action using path integrals? Is it just that in the classical limit, the paths that give small contributions to the amplitude become negligable and tends to the classical path, that satisfied by least action? Since this is just a general attempt at justifying stationary action in the classical limit, it will suffice as a derivation for that used in GR, even though the ideas behind the path integral are quantum mechanical.

    There may well be a more rigorous derivation, but I've not studied the path integral outside of the chapter or so on it in Schroeder and Peskin. When I get some money I'll probably buy a book on it.
     
    Last edited: Feb 27, 2006
  7. Feb 27, 2006 #6
    Basically. In the classical limit, if you consider a collection of paths grouped around a stationary point, then they all have approximately the same phase, which means they interfere constructively. If you take a similar collection of paths that are grouped around a non-stationary point, then they will tend to have wildly varying phases wrt one another so they will interfere destructively.

    Imagine placing a bunch of barriers that block any path that is not near to a stationary point, so the only remaining paths are the ones very close to the classical path. Classically, we expect that these barriers have no effect because the particle doesn't travel those paths. The FPI tells us that these barriers have little or no effect, but for a different reason: all of the blocked paths interfered destructively, so blocking them off has little or no effect on the overall amplitude.

    The Feynman and Hibbs book is a good one to have on your shelf, imho. It's where I first encountered the above derivation, and no one is better than Feynman at explaining this sort of thing.
     
  8. Feb 28, 2006 #7
    Ah, yeah, that was the justification I was referring to. Cheers for the reference, that was the book I was after actually. Though the only copy I've found on Amazon is a tad expensive.
     
    Last edited: Feb 28, 2006
  9. Feb 28, 2006 #8
    I think it's probably important to note that Hamilton's principle does not lead to Newton's equations, it's the other way around. To see an example of the failings of Hamilton's principle to obtain Newton's equations, I recommend reading "Nonholonomic Constraints: A Test Case" in the Am. Jour. of Physics by Ian Gatland, and "The enigma of nonholonomic constraints" also in Am. Jour. of Physics by Raymond Flannery.

    The key point is that Newton's laws handle nonholonomic constraints with no problem, whereas Hamilton's principle requires some major modifications just to get to a form where it can handle a very small class of them.
     
  10. Feb 28, 2006 #9

    selfAdjoint

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    HAH! Lucky me! I was able to buy it decades ago for a stiff but reasonable technical textbook price.

    Feymann's argument as given by Straycat is a modern updating of Huygens' original construction of Snell's Rule, which I think was Feynmann's inspiration.
     
  11. Feb 28, 2006 #10
    If that is true, it seems to me a pretty significant finding, since it would mean I think that GR is a necessary consequence of QM.

    When you use the lagrangian density action principle in QM, you are looking for the volume that yields a stationary action wrt variations in the boundary of the volume. the lagrangian density, as I understand it, is not varied. But when you derive GR, you are varying the metric, not the boundary -- which means we are varying the lagrangian density, since it is a function of the metric (see the expression given by Perturbation above.) So when we compare these two action principles, are we comparing apples and oranges here? It would be cool if the answer is yes, but I'm still not sure.

    In addition: is it OK to take in the expression for the lagrangian density used in the GR derivation (given by Perturbation above) and plug it into the equation when we employ the QM (FPI) method? I would suppose that would be OK.

    David / straycat
     
  12. Feb 28, 2006 #11
    I've never heard of a nonholonomic constraint; what is that?

    I do know that I've read from a lot of sources that Hamilton's principle is generally accepted as being equivalent to Newton's laws, in the sense that either one may be used to derive the other: Hamilton <==> Newton
     
  13. Feb 28, 2006 #12

    Physics Monkey

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    Hi straycat,

    In some sense all you have to do to derive GR from quantum theory is simply to say you want the classical limit of the path integral [tex] \int \mathcal{D} g \,e^{i S[g]/\hbar} [/tex] where [tex] S[g] [/tex] is the usual Hilbert action for GR. At this point we can feel free to wave our hands, chant "stationary phase" several times, and conclude that the classical "path" is the one for which [tex] \delta S = 0 [/tex] just like usual.

    There are abundant problems with this "justification". I'll mention a few in no particular order. First of all, its not clear why I should use the usual Hilbert action in the path integral. In fact, if you do the quantum field theory of GR as an effective theory then you find that quantum fluctuations actually change the Lagrangian; new terms appear that weren't there in the first place. Second, its not at all clear that the path integral I've "defined" is anything like a reasonable quantity even in the physicist's generalized sense of reasonable. There is all kinds of redundancy in the path integral I've written since many different metrics represent the same spacetime geometry (if they are related by a coordinate transformation). Third, it's not even clear what sort of spacetimes I should sum over in the path integral. Should they be asymptotically flat or maybe closed? I can go on like this for quite a while. Some of these problems have been adressed, and some of the more interesting attempts at quantum gravity, in my opinion, are those that try to define and calculate the path integral i.e. causal dynamical triangulations.

    The path integral is just so darn elegant, how can you not love the simplicity of the thing?
     
    Last edited: Feb 28, 2006
  14. Feb 28, 2006 #13
    It is, and I do! :!!) But that just makes it all the more attractive a target for deriving it from a different principle. iow if we have two wonderful/elegant/simple/powerful principles A and B, and we can marry them by showing Principle A <==> Principle B, then now instead of two, we have one super principle! :biggrin:

    This is the sort of objection that makes my mind reel and my eyes glaze over, because what looked like a tractable problem -- integrate over a (fixed) lagrangian -- now becomes: integrate over a fluctuating lagrangian. What the hell does that even mean???? fluctuations, with respect to what variable??

    In an effort to unglaze my eyes, I have tried to come up with a way at least to define what we are trying to do. If you take a gander at message #72 in this thread:
    https://www.physicsforums.com/showthread.php?t=109096&page=5
    you'll see a "toy model" that I presented to ttn (for the sake of discussing some ontological issues). One of the reasons I like it is that it gives me a handle, at least conceptually, on how to approach the above issue: the lagrangian fluctuates from one spacetime to the next, but remains well-defined within any one spacetime. From the little I know about the various quantum gravity programs (like LQG, causal dynamical trangulations), my simple toy model could be made compatible with at least some of them.

    Your other objections are well-taken, and could be summarized: "math is hard." :grumpy:

    I wonder whether, in any of the quantum gravity programs, the following approach might not work: start out with a very general spacetime (or I suppose an ensemble of "all possible spacetimes") in which we do NOT assume GR. Then plug in some sort of probabilistic assumption, akin to Feynman's "democracy of paths" -- but in this case we have a "democracy of spacetimes" or some such thing. You could think of the FPI as showing that, in the classical limit, the classical path is simply "more probable" than other paths. (This is an uncarefully worded statement, but hopefully you see what I mean.) So perhaps it could be shown, in similar manner, that in some classical limit, spacetimes that obey GR are "more probable" than spacetimes that don't? This is the essential idea that motivated me to start this thread.
     
  15. Feb 28, 2006 #14
    If you have a holonomic constraint, then it means that there is a constraint on the problem of the form [tex]f(\overline{q}, t) = 0[/tex]. Nonholonomic constraints are the rest. The papers I cited mention a few examples. The point is that the calculus of variations coupled with the usual treatment given holonomic constraints given in texts like the Goldstein don't hold up for nonholonomic constraints, but Newton's laws do (this is the subject of Gatland's paper). Therefore, the problems that Hamilton's principle can handle are a subset of the problems Newton's laws can handle, so in that sense Newton's laws are more general.
     
  16. Feb 28, 2006 #15
    Ya gotta love google. Took me about 10 seconds to download the Flannery paper.

    Let's see, skimming through, I'm trying to figure out the physical significance of a "nonholonomic system" ...

    In the conclusion:
    "The displacements dqj in nonholonomic systems violate this rule because they cause nonzero changes dgkÞ0 in the constraint conditions and the displaced paths are not geometrically possible."

    So a nonholonomic system means that we are dealing (in mathematical terms) with paths that are not geometrically possible? Does that mean that the variational method "rips" a discontinuity into the paths, or some such thing, thus rendering the hamilton method unfeasible .... ?

    The paper argues for replacing the Hamilton principle with "the more basic principle of D’Alembert." Sounds like, for the purpose of the thread, the underlying idea is (probably) still valid: you can use QM to derive classical (Newtonian) mechanics. Maybe you need to use the "principle of D"Alembert" instead of Hamilton's principle, but I'm still betting that the core concept remains intact.

    btw I rephrased the question that I asked in message #1 to ask whether any of the attempts at quantum gravity try to derive GR via some sort of an action principle, and posted to the "Beyond the standard model" section:
    https://www.physicsforums.com/showthread.php?t=112556
     
  17. Mar 1, 2006 #16

    selfAdjoint

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    The great elementary example of a non-holonomic system is a wheel rolling on a plane. The phase space is not closed.
     
  18. Mar 1, 2006 #17
    In QFT you're varying the fields that are the generalised trajectories of regular Lagrangian field theory, not the volume. Since the Lagrangian density is a functional of such fields, if you vary the fields then the Lagrangian density is varied (though obviously we want to find the field equations satisfied such that this variation vanishes, which is the whole point of the calculus of variations).
     
    Last edited: Mar 1, 2006
  19. Mar 4, 2006 #18
    Hmm. You are tantalizing me with the possibility that the answer to my original question is "yes." But dammit, if it could be done, why hasn't anyone touted the achievement? I suppose it's because of the issues raised by physics monkey in post #12. Which makes me continue to think that a derivation of the action principle ("#2") is something we should be looking for in a theory of quantum gravity -- see my thread in the "beyond the standard model" section:

    https://www.physicsforums.com/showthread.php?t=112556

    I'll need to make myself as familiar with the variational method of QFT (integrating over 4-volumes) as I am with the method of the FPI (using 1-d integrals). Do you know of a good short introduction / overview to the method? I have included a pdf attachment of my own 1 page introduction / overview of the FPI. Something like that would be helpful for me to establish a handle on the variational method of QFT. What's your favorite QFT text, btw, once I decide to dig in really deep?

    btw, here is a news article I ran across in another forum that mentions nonholonomic constraints:

    http://www.sciencedaily.com/releases/2006/03/060301094931.htm

    Thanks to a-zero for mentioning this topic.

    David
     

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  20. Mar 4, 2006 #19
    Have look through the various Wiki' articles on Lagrangians and things as to why the density is more favourable. The variational method doesn't really differ. I've not been studying QFT long, I've actually been self-studying (as I have to do with anything above the dizying heights of A level) from Schroeder and Peskin and only started about new year time (read straight through to chapter 9, I'll probably go back through and read some of the bits I didn't get before I go onto the next chapter). When I get some money [gah, need a job] I was going to buy Wald's book on QFT in curved space-times and black hole thermodynamics, so I'll get back to you on that one.
     
  21. Mar 4, 2006 #20
    I did look through the wiki article on lagrangians, and it is helpful. I also just sent off for S&P as well as Zee from amazon. Although perhaps Wald is what I should get, since my whole point has been to study the variational technique in curved spacetime ... if you ever decide Wald is a good read, send me a line :smile:

    And while you're working on studying QFT, perhaps you could think about the problem of deriving GR from QFT? I'm sure you could get a few PRL papers out of that! (I'm going to be trying the same thing. But I have the distinct impression you're more adept at math than I am, so you could probably beat me. ;-) )

    D
     
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