Is it possible to derive action principle #2 from QM?

[straycat]

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.

 

[Perturbation]

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

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

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

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.

$$S=\int dt L=\int d^4x \cal{L}$$

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 $\cal{L}$, 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.

$$S=\int d^4x \cal{L}$$$$=\int d^4xR\sqrt{-g}$$

So that when we apply the principle of least action $$\frac{\delta S}{\delta g_{\mu\nu}}=0$$ we must vary the metric determinant as well, to give the familiar field equation.

 

[Physics Monkey]

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.

 

[straycat]

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.

$$S=\int dt L=\int d^4x \cal{L}$$

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: $$\frac{\delta S}{\delta g_{\mu\nu}}=0$$, in a manner analogous to the derivation of Hamilton's action principle?

David

 

[Perturbation]

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.

http://en.wikipedia.org/wiki/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.

 

[straycat]

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?

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.

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.

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.

 

[Perturbation]

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.

 

[abszero]

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.

 

[selfAdjoint]

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.

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.

 

[straycat]

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.

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

 

[straycat]

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.

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

 

[Physics Monkey]

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 $$\int \mathcal{D} g \,e^{i S[g]/\hbar}$$ where $$S[g]$$ 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 $$\delta S = 0$$ 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?

 

[straycat]

The path integral is just so darn elegant, how can you not love the simplicity of the thing?

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!

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.

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 anyone 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."

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.

 

[abszero]

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

If you have a holonomic constraint, then it means that there is a constraint on the problem of the form $$f(\overline{q}, t) = 0$$. 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.

 

[straycat]

If you have a holonomic constraint, then it means that there is a constraint on the problem of the form $$f(\overline{q}, t) = 0$$. 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.

Ya got to 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

 

[selfAdjoint]

The great elementary example of a non-holonomic system is a wheel rolling on a plane. The phase space is not closed.

 

[Perturbation]

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 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).

 

[straycat]

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).

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

 

[Perturbation]

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?

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.

 

[straycat]

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.

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

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

 

[Physics Monkey]

Hi straycat,

As I indicated before, the sort of derivation you are aiming for has actually already been accomplished. Feynman showed how, starting from the notion of a massless spin 2 field, the classical Einstein field equations can be obtained. In the terms of modern effective field theory it is possible to write down the most general action you can imagine (addressing point 1 of my orginal post), quantize the theory in a gauge invariant way using the path integral (point 2 and 3), and then carry out perturbation theory to compute quantum corrections. The theory so defined is not renormalizable in the old fashioned sense, but it is renormalizable in the modern sense so long as you include all (infinite) possible terms in the action.

The theory can be renormalized with a finite number of counterterms at each order in perturbation theory and predictions can be made. It is a satisfying result of the analysis that out of all the possible terms, only those in the classical Einstein field equations are relevant for describing the physics of the low energy world. The whole structure of classical GR can then be seen as emerging from the usual stationary phase approximation to the path integral with the added bonus that it explains the particular form of the Einstein-Hilbert action. This occurs because even though you had to include all possible terms in the quantum action, the terms relevant for low energy physics are precisely the terms in the Einstein-Hilbert action.

A good reference for this is the nice paper by Donoghue http://arxiv.org/abs/gr-qc/9512024 , but be warned that you won't get much past the introduction without some strong grounding in fairly advanced field theory and general relativity. See also the references in this paper for listings of original sources like Feynman's article.

 

[selfAdjoint]

Home truths from the introduction to Donoghue's paper:

We have no reason to suspect that the effects of our present theory are the whole story at the highest energies. Effective field theory allows us to make predictions at present energies without making unwarranted assumptions about what is going on at high energies. In addition, whatever the physics of high energy really is, it will leave residual effects at low energy in the form of highly suppressed non-renormalizable interactions. These can be treated without disrupting the low energy theory.

 

[straycat]

Hey PM and SA,

Thanks for the link to the Donoghue article; sounds like exactly what I was looking for when I launched this thread. I'm printing it out now; we'll see how much I can follow!

I don't know how much either of you have followed my rantings on various other threads, but one of the issues that is of deep interest to me is the notion of replacing the Born rule with outcome counting for determining probabilities. (I know that selfadjoint is in the "democracy of spacetimes?" thread at https://www.physicsforums.com/showthread.php?t=112556 where I post links to the various threads on outcome counting.) As I have stated on those threads and elsewhere [1], I think there are, at the least, "ontological" reasons for preferring outcome counting over the Born rule. In my mind, outcome counting should be raised to the status of a symmetry principle, like the principle of relativity. The question then becomes: can you turn outcome counting into a workable theory that matches experiment?

I am going to assume that Donoghue's derivation either starts with the assumption that the Born rule is true, or starts with assumptions that are logically equivalent to the Born rule. So as I read Donoghue's paper, I will be contemplating the question: if we assume outcome counting instead of the Born rule (or do whatever is the logical equivalent), then can we still derive GR? Maybe, if we're lucky, a couple of other things might even fall into place, and the derivation will be easier!

Of course I have no idea if this is mathematically workable. But to me, the ontological argument is compelling, so it is worth it to pursue this line of reasoning. Of course if it doesn't work, no skin off my nose, I still have my day job!

david / straycat

[1] See eg message #96 at https://www.physicsforums.com/showthread.php?t=109096&page=7 where I try to argue that outcome counting is immune to the following criticism of the Born rule: namely, that the Born rule requires us to assume that most observers in parallel worlds are "mindless hulks."

 

[masudr]

Firstly, it's important to note that the formulation of the dynamics of a system using a Lagrangian, $L$, and a Lagrangian density, $\mathcal{L}$, is entirely equivalent, since we say that

$$L=\int d^3 \vec{x} \mathcal{L}$$

and so we can say that

$$S[\phi]=\int dt L = \int d^4 \vec{x} \mathcal{L}.$$

Then we use the variational principle $\delta S=0$ to derive the field equations. The principle reason I can think of that this approach is better than the previous is that the density approach is manifestly covariant -- the density itself is a tensor density, which means that with appropriate root of $g$ (in this case $\sqrt{g}$) it becomes a tensor (in this case a scalar). And we don't have to do the ugly slices of spacetime into space-like slices.

Secondly, it is important to note that deriving Einstein's field equations from the E-H action is in the realm of classical field theory -- it is akin to deriving the Maxwell equations from the Lagrangian density of the electromagnetic field. A fairly terse but good summary of these results are in Wikipedia (try http://en.wikipedia.org/wiki/Classical_field_theory" ). So the equations of motion for a particle come out of the usual Lagrangian mechanics and the equations of motion for a field (e.g. for the EM field we have Maxwell's equations, for gravitation we have Einstein's equations etc.) come out of Lagrangian field theory. The point is that they both come out of the same variational principle. (In fact the variational principle leads to the Euler-Lagrange equations, which I'm sure you remember, which may or may not be easier to use depending on the situation).

Feynman's path integral method shows that we can get to the classical variational principle of the path of stationary action from the quantum mechanical principle of summing amplitudes from every path.

I hope this clears up some of the misunderstandings in the first post (which may have been cleared up since, and I apologise if I have repeated anyone else's comments).

Masud.

 

[straycat]

Hey Masud,

I hope this clears up some of the misunderstandings in the first post (which may have been cleared up since, and I apologise if I have repeated anyone else's comments).

You're really hitting at the boundary between what I've grokked and what I haven't, so I appreciate your post. This is what makes PF such a great place.

The place where I get hung up is in the step:

$$L=\int d^3 \vec{x} \mathcal{L}$$

I understand the expression on the left as being the lagrangian of a path, which is a 1-dimensional thing, and that L = KE - PE. I even understand that the parameterization of the path matters, because a change of parameter corresponds to a particle moving at different speed at different points along the path (so would result, just to name one thing, in a change in the KE at that point). But I get hung up on how to equate this with the expression on the right. Given a particular path on the left, what is the corresponding 3-d region that we integrate over on the right?

This probably has a simple answer and I'll feel dumb when I finally get it, but it's where I'm stuck!

David

 

[straycat]

Wait a minute, I think I said that wrong. The action is the action of a path, ie the action $$S$$ is a function of the path $$\phi$$, $$S[\phi]$$.

Still, we have the equation:

$$S[\phi]=\int dt L = \int d^4 \vec{x} \mathcal{L}.$$

The integral on the left is the integral over the path $$\phi$$. So what is the integral on the right integrated over? That's where I'm stuck.

 

[masudr]

I think I can see the source of confusion here. Firstly, I'll point out that $\phi$ in what I've written above is the field variable (which is a tensor of some sort), not the path.

In Lagrangian mechanics, where we have the Lagrangian, $L$, the motion of the particle is specified by a path $x(t)$, so we specify the initial and final positions and integrate over time.

In field theory, where we have the Lagrangian density, $\mathcal{L}$, the "motion" (dynamics is a better word here) of the field (which is a tensor of some rank),

$$\phi^{abc...}_{def...}(\vec{x})$$

is specified by the configuration of the field at different points in spacetime.

One important point is that we are demanding Lorentz covariance, so

$$\vec{x}=x^\mu=\left(x^0, x^1, x^2, x^3\right)=\left(ct, x, y, z\right).$$

And the second point is the meaning of the integral. Since I can think of no better way to put it, I quote one of my professors: "At each $t$ between $t_i$ and $t_f$ the field's configuration $\phi \left(\vec{x}\right)$ is chosen such that the integral [in question] through the space-time volume bounded by $t=t_i$ and $t=t_f$ is extremized." Whereas in Lagrangian mechanics we specify the co-ordinates at two times, here we specify the functional dependence of our field variable $\phi$ on $x^i=\left(x, y, z\right)$ at two fixed times.

EDIT: Since I am explicitly using units where $c \ne 1$, I will re-write the integral which gives the action with $c$ explicitly put in:

$$S = \frac{1}{c} \int d^4\vec{x} \mathcal{L} [\phi, \partial\phi, \partial\partial\phi...]$$

 

[straycat]

I think I can see the source of confusion here. Firstly, I'll point out that $\phi$ in what I've written above is the field variable (which is a tensor of some sort), not the path.

Wiki says the same thing here:
http://en.wikipedia.org/wiki/Path_integral_formulation

"The action is referred to technically as a functional of the field: S[φ] where the field φ(xμ) is itself a function of space and time, ..."

OK, I can see that.

In http://en.wikipedia.org/wiki/Lagrangian , two distinct definitions are presented:
$S = \int L dt$
and
$$S[\phi] = \int d^4 \vec{x} \mathcal{L}.$$

and they comment that "Both definitions of the Lagrangian can be seen as special cases of the general form ..."

I still don't see how you can equate the above two actions. The first is the action of a path, and the second is the action of a field. That's pretty much what you say here:

In Lagrangian mechanics, where we have the Lagrangian, $L$, the motion of the particle is specified by a path $x(t)$, so we specify the initial and final positions and integrate over time.

In field theory, where we have the Lagrangian density, $\mathcal{L}$, the "motion" (dynamics is a better word here) of the field (which is a tensor of some rank),

$$\phi^{abc...}_{def...}(\vec{x})$$

is specified by the configuration of the field at different points in spacetime.

That makes sense to me.

One important point is that we are demanding Lorentz covariance, so

$$\vec{x}=x^\mu=\left(x^0, x^1, x^2, x^3\right)=\left(ct, x, y, z\right).$$

Ahh, so this could be a difference between the path integral and QFT methods. In the path integral, you start out from the beginning considering non-classical (non-extremal) paths. From the above statement, if I understand correctly, you start out from the beginning by restricting attention to classical (Lorentz covariant) fields. (?)

This would jive with what I'm about to post in the "outcome counting, the action principle, and GR" thread, ie that in QFT we assume from the outset the action principle, and the action principle implies Einstein's equation, which means we are assuming Lorentz covariance from the beginning.

And the second point is the meaning of the integral. Since I can think of no better way to put it, I quote one of my professors: "At each $t$ between $t_i$ and $t_f$ the field's configuration $\phi \left(\vec{x}\right)$ is chosen such that the integral [in question] through the space-time volume bounded by $t=t_i$ and $t=t_f$ is extremized." Whereas in Lagrangian mechanics we specify the co-ordinates at two times, here we specify the functional dependence of our field variable $\phi$ on $x^i=\left(x, y, z\right)$ at two fixed times.

iiuc, the times $t=t_i$ and $t=t_f$ each define a hypersurface. To make an analogy, the specification of the fields at $t=t_i$ and $t=t_f$ in QFT would be analogous to the specification of the particle position at spacetime locations $a$ and $b$ in the FPI. Does that make sense?

David

 

[straycat]

iiuc, the times $t=t_i$ and $t=t_f$ each define a hypersurface. To make an analogy, the specification of the fields at $t=t_i$ and $t=t_f$ in QFT would be analogous to the specification of the particle position at spacetime locations $a$ and $b$ in the FPI. Does that make sense?

Assuming I've understood the above issue correctly, I have an "ontological" question.

When you do the FPI, you are essentially asking the question: given that I observe a particle at spacetime location $a$, what is the (relative) probability that I will detect it at spacetime location $b$?

So to continue the analogy, in QFT we are (unless I'm mistaken, which I may be) asking the question: given that I observe the field to exist in such-and-such configuration at time $t=t_i$; what is the relative probability that I will observe it to be in such-and-such configuration at time $t=t_f$?

Now I have no objection to the problem, as stated, in the FPI formalism. Classically, there is nothing wrong with assuming that "I observed particle emission at spacetime location $a$." You might object that QM does not allow us to know the exact time and space location of particle emission, but if we define $a$ as a small but not pointlike region, then we evade even that objection.

However, I do propose an objection to the problem, as stated, in QFT. How do you justify ever knowing the exact state of the field along the entire hypersurface defined by $t=t_i$? Most of that (well, all of it except for a point) is outside my past light cone. So how can we even consider the question, as stated?

As usual, I could very well be confused about how QFT works, and I freely admit it ...

David

 

[Perturbation]

The state of the field is given by (for vacuum theory) $a^{s}^{\dagger}_{\mathbf{p}}|0\rangle$, where p is some momentum eigenstate and s a spin polarization. The propogation amplitudes involves integrals over the four-momenta (whose norm is required to be equal to the squared rest mass for particle excitation, corresponding to a pole in the propogator). We don't know the momenta of the field excitations at the end points, we know the positions in Minkowski space-time of the field excitations.

The Feynman transition amplitude is

$$\langle \psi_b|e^{iHT}|\psi_a\rangle =\int \cal{D}$$$$x(t) \exp\left[i\int d^4x \cal{L}\right]$$

Which means that we have two field states, initial and final, defined at $x^{\mu}=(0, \mathbf{x}_a)$ and $x^{\mu}=(T, \mathbf{x}_b)$ and we perform a unitary evolution so as to attain an overlap in the states. Feynman tells us that this overlap is equal to a functional integral, which is weighted by a phase term. There is no dependence of any kind of field configuration other than position of the initial and final states. All we require then is some knowledge of the Lagrangian to construct and quantise our field theory, and we can simply sum over quantities that depend on the intermediate path.

 

[Perturbation]

Wiki says the same thing here:
I still don't see how you can equate the above two actions. The first is the action of a path, and the second is the action of a field. That's pretty much what you say here:

If I take the total mass of a body m, I can express it as the integral over the volume of the body of the mass density. It's the same here: I'm integrating some quantity over the region of space that we're interested into get our Lagrangian. For this reason we call it the Lagrangian density. If I then integrate the mass over some time contour, I'm integrating the mass-density over a four volume. This is an analogous situation to what's causing you problems, but there's nothing wrong with doing it, as the previous example shows.

 

[Perturbation]

that in QFT we assume from the outset the action principle, and the action principle implies Einstein's equation, which means we are assuming Lorentz covariance from the beginning.
David

In elementary formulations of field theory it is indeed assumed. We begin with a classical field, by classical we mean unquantised, and a dynamical equation or a Lagrangian density. In the field theory of the Klein-Gordon and Dirac fields we start with a field equation and then find a Lagrangian density that gives this equation using least action. In general, one begins with a Lagrangian density then constructs the field equations, for example when one builds a general gauge invariant field theory we choose appropriately amicable terms to construct the Lagrangian. We only had the field equations for Dirac and Klein-Gordon first because they both came from the relativistic equation for energy.

In second or cannonical quantisation one would then go about deciding upon a solution for the classical fields of the field equations in terms of operators etc. For this reason the field equations are useful and least action is important. From this method of quantisation propogation amplitudes and all the rules and properties for the field theory follow.

However in the Feynman path integral approach we needn't assume least action to be able to quantise the field theory, all we need do is make the jump from the classical fields to opertors on Hilbert space and have ourselves a Lagrangian. The Feynman rules then follow.