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Natural Science Forum / Physics / Research / April 2004How to Field Theorize Starting from a Lagrangian
I dare say that every student of mathematical physics has encountered the idea that if you wish to create a field theory, you should (a) write down a Lagrangian, perhaps as a sum of terms like L = L_matter + L_interaction + L_field (b) derive from the Lagrangian key ingredients of any field theory, including of course the field equation and (if possible--- perhaps this is a "big if") the equation of motion of "charged" particles, the energy-momentum tensor, etc. Everyone here has probably studied in some detail at least two examples of how this "procedure" works, namely Maxwell's theory of EM and Einstein's theory of gravity (at least in its "linearized" approximation). Many (but not me, alas!) have probably also studied application of an analogous process to quantum field theory examples. And everyone knows (I assume) that Lagrangians (and Hamiltonians) constitute one of the very few things which is immediately applicable to both classical and quantum physics, and (I assume) noone doubts the utility of the Lagrangian approach. Nonetheless, as far as I know, there is currently -no- textbook on "How to Theorize from a Lagrangian"! (But please correct me, with citations, if I am wrong about this.) This gap--- if it indeed exists--- is lamentable, although perhaps not inexplicable. I'd like to try to fill this apparent gap right here in s.p.r.--- not by writing a textbook, of course, but at least by compiling a list of toy theories and a list of the most essential pedagogical points. Such preliminary work might help persuade some physics professor to consider writing the missing (?) textbook. Specifically, I'd like to resurrect an old proposal of mine: to play around here on s.p.r. with a reasonable range of simple "toy theories", in order to explore/learn/teach the "procedure" (by no means entirely algorithmic, as far as I can see) for writing down a Lagrangian and obtaining from it a well-defined field theory (or else recognizing an inconsistency, and reacting intelligently to this misadventure). Because my own background (in math) includes no quantum physics whatever, my participation would have to be largely limited to discussing toy -classical- field theories, but the practicing physicists here could no doubt chime in by mentioning/discussing their favorite toy quantum field theories. BTW, my own knowledge of Lagrangians comes largely from studying the textbooks of Landau/Lifschitz and Ohanian/Ruffini and from playing around with toy examples I have concocted myself--- but I would much prefer to play with toy examples concocted by professional physicists! I propose to start as simply as possible and slowly increase the level of sophistication, e.g. (not neccessarily in this order) 1. first linear, then nonlinear Lagrangians, 2. first Lagrangians of form L(x,u,u_x), then L(x,t,u,u_x,u_t) (more variables!) and L(x,u,u_x, u_(xx)) (higher order derivatives!), 3. first scalar theories, then vector theories, then tensor theories, 4. first classical field theories, then quantum field theories. Issues we might systematically explore in this way could include: 1. how to sensibly select the terms of our Lagrangian, as written in the useful form L = L_matt + L_interaction + L_field 2. how to guess key features of the "free-field" case of the field equation of our theory from looking over L_field, 3. how to write down the Euler-Lagrange equations (allowing for the possibility of second and higher order derivatives), 4. how to verify key features of the (free-field) field equation, e.g. (a) how to determine the group of "point symmetries" and the subgroup of "variational symmetries", (b) how to confirm the existence of wavelike solutions to the (free-field) field equation, (c) how to obtain and study particular solutions (even for nonlinear PDEs) e.g. using Lie's methods, Baecklund symmetries, scattering transform, 5. how to obtain a suitable energy-momentum tensor (and what to do if the "canonical energy-momentum tensor" turns out to be -nonsymmetric-), 6. more generally how to find "conserved currents" (e.g. from "divergence symmetries"), at least in the case where L only contains first order derivatives, 7. how to find suitable "conserved currents" if L contains second order derivatives, 8. how to determine L_interaction from energy-momentum exchange between the field and "particles", if this is possible, or if not, how otherwise to obtain a suitable "equation of motion" for "charged" particles, 9. how to check for at least the most commonly encountered self-inconsistencies which can arise in a field theory, Having explored these issues for a generous range of simple toy theories, we can try to address the question which presumably we would all agree is the question of ultimate interest: how can we write down a Lagrangian which has a reasonable chance of producing a theory which will accomplish stated goals? Let me mention a specific example of a "stated goal". Recall that Maxwell's theory of EM is beset with singularities in the field. Recall also that a few years ago we had some people here who wanted to replace gtr with a theory which would "ameliorate" singularities while preserving the remarkable experimental/observational success of that theory. In principle, as I said at the time, that's not a bad idea, but I claimed then (and repeat now) that this is a -much- more challenging program than (I felt) those people recognized. At that time I proposed to step back and consider first the analogous but easier question for EM, but unfortunately (as I recall the affair, at least) my correspondents were insufficiently willing to do this. I hope and believe that the current student generation is more amenable to such good advice! Actually, I propose to start with scalar theories, e.g. three-dimensional analogues of L = -1/2 [ u_t^2 - u_x^2 - m^2 cos(u)^2 ] (I might have munged a sign or two, but this is supposed to be the Lagrangian which leads to the sine-Gordon equation, a nonlinear wave equation beloved of PDE people), or more generally L = -1/2 [ u_t^2 - u_x^2 ] + some other proposed "perturbing term" which is small for small u (since I doubt the sine-Gordon equation will fully achieve the goal of "ameliorizating" singularities in our scalar field). I am now resurrecting my proposal for at least three reasons: 1. As I said, AFAIK, there is still no textbook on "How to Theorize from a Lagrangian". If this is really the case, this oversight is indeed startling. But anyone who has played around with Lagrangians very long can probably guess a likely explanation: many Lagrangians one is likely to propose even for a "toy theory" lead straight to a -nonlinear- field equation. Now, since Lie's day (1880s) there have been available quite general methods for computing symmetries of differential equations, even nonlinear ones (even systems of coupled nonlinear PDEs), and using then using these symmetries to obtain particular solutions. (By the very nature of Lie's methods, one can specifically look for say wave solutions, which is just what we want for theorizing from a Lagrangian.) Alas, hand computations using these methods can quickly become daunting to humans, so these methods were not much taught even in math departments until rather recently. Fortunately, with the appearance of increasingly powerful packages like Maple, several excellent textbooks explaining Lie's methods have recently appeared, so things are changing for the better! And as it happens, last year I studied (from various books by Peter J. Olver and others) this stuff, and I have even been learning (from books on "soliton theory") a bit about additional useful methods such as Baecklund symmetries. So I at least am willing to attack differential equations with much diminished prejudice in favor of linearity! 2. Steve Carlip (I assume everyone knows he is an occasional but highly valued contributor here) recently finished a nice preprint exploring a class of gravitation theories which allow for possibly differing speeds of EM and gravitational radiation in vacuo. Previously he wrote at least one other preprint addressing what are essentially (I think he would agree) pedagogical issues, so I would -particularly- welcome his contributions. 3. Doug Sweetser recently asked (perhaps in the "Weinberg on GR" thread?--- I forget!) why a -particular- classical field theory (apparently a "vector theory" of gravitation and/or EM) is not workable. I asked, in essence, "precisely what theory do you have in mind"? Something over two weeks ago, Doug responded. Unfortunately, he didn't provide nearly enough information for me to be sure exactly what Lagrangian he had in mind, or how he was trying to obtain a "theory" from it. But I could make a good guess and I quickly generated a long list of comments. Alas, before I could submit my reply, I went down for intersession, and the original message has long since scrolled at our news server. The essence of what I had to say was that what Doug said was so mangled that he should put his paper in a desk drawer, and focus for the next few weeks/months on learning how to posit/use Lagrangians intelligently. Which is of course something I want to do anyway, with wider participation (especially by physics professors). BTW, Doug, if you see this: as I recall, in your reply to my query, you said something to that some unnamed physicist (?) had replied to 40+ emails (!!!) you had sent regarding your Lagrangian, and then you admitted that due to your own careless writing you had wasted much of this effort. Alas, after reading your post I found this story very plausible! You certainly are very lucky to have found such a generous and patient teacher. Be warned that -I- don't have anything close to that amount of patience! But part of what I am trying to get at here is that while I now strongly suspect that you don't really know what you are doing when it comes to Lagrangians, I also suspect that -this isn't your fault-, because of the apparent lack of any textbook which even attempts to -teach- the Lagrangian "procedure". So I hope you will see the wisdom of my advice to step back and study "Lagrangianizing" in more elementary contexts before attacking questions of current interest in the research literature. "T. Essel" (hiding somewhere in cyberspace) > Nonetheless, as far as I know, there is currently -no- textbook on "How to > Theorize from a Lagrangian"! (But please correct me, with citations, if I > am wrong about this.) Erm, how about pretty much every advanced classical mechanics textbook in existence? And pretty much every QFT book for the quantization thereof? If you want to see the really mathy version of this stuff, Deligne and Freed have a set of lecture notes in the IAS volumes on QFT and Strings for mathematicians. Dan also has the following set of lectures on field theory and supersymmetry: <http://www.ma.utexas.edu/users/dafr/pcmi.ps> Aaron > [...] as far as I know, there is currently -no- textbook on "How to > Theorize from a Lagrangian" That's a surprising statement. I would have said that Peskin & Schroeder's "Introduction to QFT", and Weinberg's multi-volume "The Quantum Theory of Fields" together do a good job. P&S introduce the key elements by concentrating on a relatively easy case called "phi^4" theory. They show how to pass from the Lagrangian to a formula for 2-point correlation functions and thence to the S-matrix and practical cross-section calculations. They they do other types of fields and cover more difficult aspects. Weinberg's is a more advanced text. He starts from the premise that the fundamental fields correspond to causal irreducible representations of the Poincare group, and develops the subject from there. However, it's a pre-requisite for P&S that you must already be familiar with basic quantum mechanics. You said that your background includes no quantum physics whatsoever. Without such background, I doubt anyone can teach you QFT, so perhaps a good starting point would be a textbook on introductory quantum mechanics? (E.g: Mandl, or Greiner, or, ummm, well there's lots of them.) > [...] how can we write down a Lagrangian which has a reasonable chance > of producing a theory which will accomplish stated goals? P&S give some intuitive insight into this in one of their early chapters. They basically appeal to the necessity for Poincare invariance, and renormalizability, to restrict the choice of Lagrangians. BTW, over at superstringtheory.com, they have occasional online courses in QM, QFT, and some other stuff. That might be useful in the future if you don't get a satisfying response here on spr. HTH. -- Mike Mowbray Internet: mikem@despammed.com > I dare say that every student of mathematical physics has encountered the > idea that if you wish to create a field theory, you should > > (a) write down a Lagrangian, perhaps as a sum of terms like > > L = L_matter + L_interaction + L_field snip > Nonetheless, as far as I know, there is currently -no- textbook on "How to > Theorize from a Lagrangian"! (But please correct me, with citations, if I > am wrong about this.) As far as I know this matter is well-covered if scattered in the literature. The key paper for the physicist is by Belinfante and concerns the issue of symmetrizing the canonical energy tensor. To this one can add the work of Palatini where he shows how to get GR by varying the connection. For textbooks, one can mention Landau-Lifschitz and Barut. Of course one should have a good understanding of variational calculus going in. For the physicist, one can mention Caratheordory and Gelfand-Fomin. I remember the book by Weinstock as being loaded with excellent problem sets. If you are interested in seeing the process at work in rather new and non-trivial circumstances, drop me an email. -drl On Thu, 1 Apr 2004, Aaron Bergman wrote: > In article <c4hcmb$vqm$1@lfa222122.richmond.edu>, tessel@um.bot wrote: > [quoted text clipped - 4 lines] > Erm, how about pretty much every advanced classical mechanics textbook > in existence? Perhaps I should clarify something: when I referred to an intended audience of "average physics [graduate] student", I did -not- mean "average Princeton physics graduate student"! :-/ (This reminds me of a famous 18th century observation, to the effect that students fall into two classes: a huge class of ineducables and a tiny class of people who effortlessly master all relevant ideas/techniques with only minimal assistance/guidance. When I hang about here for very long, I tend to start to believe this again. Otherwise I sometimes become inspired with the [naive?] idea that somewhere out there some student might need just need a little help.) Be this as it may, my (lack of) background must be showing, because the "classical mechanics" books I have examined (many coauthored by Marsden) completely lack what I was (am?) looking for. Can you (or anyone else) cite a specific textbook (preferably one I am likely to be able to find in my local research library!) which discusses the specific issues I raised for even a half dozen examples? I don't know of anything which even comes close! But I'd be glad to find a book which proves me wrong. > If you want to see the really mathy version of this stuff, Actually, I was probably looking for a lowbrow guide for physics students, suitable for the "average student" I imagined. I take it you feel there is in fact no such "gap" as I imagined? Do other physics faculty here agree with that? If so, I'll gladly cease and desist. "T. Essel" (hiding somewhere in cyberspace) > Actually, I was probably looking for a lowbrow guide for physics students, > suitable for the "average student" I imagined. I take it you feel there > is in fact no such "gap" as I imagined? Do other physics faculty here > agree with that? If so, I'll gladly cease and desist. I am not sure where you see that gap ... surely Landau and Lifshitz do a fairly good job of discussing the action principle in their Classical Mechanics and Classical Field Theory books? Most quantum field theory textbooks also do a quick review of the classical field theory results they are going to use, like the Euler-Lagrange equations, Noether's theorem, Hamiltonian and conserved quantities ... You could complain that Dirac's method of analysis of constraints is not given in more books, but you can't apply it directly to non-Abelian gauge theories anyway --- you need to go to superspace or such to introduce ghosts, and ghosts are better done in the path integral formalism anyway -- so you shouldn't really complain. :-) What else will fill this 'gap' that you are worrying about? -S. Perhaps the thing to do is to just write down the most general lagrangian you can think off that describes the standard model and general relativity. Eg the thing that is experimentally the accepted norm. Palatini and others go into great length to do just that, using the technology of fiber bundles and differential forms. Its messy, and not the best way to learn, but there you have it... THE Lagrangian of physics so to speak! A few general concepts are of course applicable. The first is that, by and large the lagrangian are constructed using symmetry principles. The second is that gravity does indeed behave a little differently, as it involves metric forms as the fundamental thing, as opposed to guage connections. The third, is how to mix quantum mechanics into the picture. In essense, its how you interpret the lagrangian that changes from the move to classical systems to quantum mechanics. Of course, renormalizability then cuts down on the total amount of terms, and justifies why certain things were left out of the lagrangian that a priori could have been there in the first place. > Be this as it may, my (lack of) background must be showing, because the > "classical mechanics" books I have examined (many coauthored by Marsden) [quoted text clipped - 3 lines] > for even a half dozen examples? I don't know of anything which even comes > close! But I'd be glad to find a book which proves me wrong. All the issues you raise are pretty scattered. Goldstein (even though I didn't use it) almost assuredly covers Lagrangians quite extensively. For things like Noether's theorem and nonlinear Lagrangians, you can look in intro to QFT books. Peskin and Schroeder certainly covers much of that stuff. > > If you want to see the really mathy version of this stuff, > > Actually, I was probably looking for a lowbrow guide for physics students, > suitable for the "average student" I imagined. I take it you feel there > is in fact no such "gap" as I imagined? Do other physics faculty here > agree with that? If so, I'll gladly cease and desist. It's not all done in one place in the curriculum always, but I haven't really noticed a gap. The material is generally introduced as needed. Aaron Personally being a student who struggled to get a better grasp on many of these variational/Lagrangian problems myself not to long ago I do agree that there is somewhat of a gap. Landau and Lifschitz while excellent certainly is neither suited for the avarage student, nor does it cover the subject in the generality you proclaim. My personal last brush with variational problems was a (succesfull) attempt to derive the differential equations for the path of a ray of light in a general medium with refractive index n(r) from fermats principle that light always takes the shortes optical path: int n(r)dl. (Kept me from properly studying for my exam as well...) Neither in Landau Lifschitz classical field theory book (I did't have the field theory in matter one available, don't know if it covers this) nor in Jacksons Electrodynamics did I find much help, L&L contained some leads as to which variational principle to use, but that didn't help that much either. (I don't know perhaps it's considered trivial, but if all introduction to variational methods you ever had was Landau Lifschitz classical mechanics and some handwaving it isn't, especially wrt varying the meassure dl) Many fellow students consider variational methods as mysterious and deep and not something to be overly concerned with. I think a ressource that would bring together several simple examples of how variational methods work, some simple Lagrangians for different (toy) applications and which would show how simple variational calculations really are when approached from the right angel would be much appreciated. Cheers, Frank. > I think a ressource that would bring together several simple examples > of how variational methods work, some simple Lagrangians for different > (toy) applications and which would show how simple variational > calculations really are when approached from the right angel would be > much appreciated. I can only offer my (admittedly very short) appendix on the subject in my qft script on my homepage. I hope this helps a little: http://theory.gsi.de/~vanhees/publ/lect.pdf  SignatureHendrik van Hees Cyclotron Institute Phone: +1 979/845-1411 Texas A&M University Fax: +1 979/845-1899 Cyclotron Institute, MS-3366 http://theory.gsi.de/~vanhees/ College Station, TX 77843-3366 On Sun, 4 Apr 2004, Aaron Bergman wrote: > All the issues you raise are pretty scattered. Well, do you at least agree there's some point to gathering the relevant ideas together in one place? > Goldstein (even though I didn't use it) almost assuredly covers > Lagrangians quite extensively. Fine, but how many of the specific "issues for discussion" which I mentioned does this book cover? Also, I think I probably know which book you have in mind, but can you please give a citation? (Sorry, I still haven't gotten around to reskimming Peskin and Shroeder, but I promise to do this.) > It's not all done in one place in the curriculum always, but I haven't > really noticed a gap. Many thanks to Frank Hellman, who commented that he agrees that there -is- something of a gap, and gave another example of the kind of [serious] "Lagranian play" I had in mind. I also eagerly await any comments by Steve Carlip, e.g. regarding searching for an EIH analog. "T. Essel" (hiding somewhere in cyberspace) > On Sun, 4 Apr 2004, Aaron Bergman wrote: > [quoted text clipped - 20 lines] > "Lagranian play" I had in mind. I also eagerly await any comments by > Steve Carlip, e.g. regarding searching for an EIH analog. I agree too, I've never seen anything remotely like what you describe as your ideal. All the introductions to Lagrangians in mechanics textbooks that I have seen, are very skimpy in comparison. Tim > On Sun, 4 Apr 2004, Aaron Bergman wrote: > > > All the issues you raise are pretty scattered. > > Well, do you at least agree there's some point to gathering the relevant > ideas together in one place? This is generally done in math textbooks, although I wouldn't be surprised if there were physics textbooks out there that did the same. > > Goldstein (even though I didn't use it) almost assuredly covers > > Lagrangians quite extensively. > > Fine, but how many of the specific "issues for discussion" which I > mentioned does this book cover? Also, I think I probably know which book > you have in mind, but can you please give a citation? H. Goldstein _Classical Mechanics_. I don't own a copy, so I can't tell you precisely what's in it, but it's one of the standard junior level classical mechanics texts. Aaron > I think a ressource that would bring together several simple examples > of how variational methods work, some simple Lagrangians for different [quoted text clipped - 4 lines] > Cheers, > Frank. There is a book called "Lagrangian Interaction", by Noel A. Doughty. I have it and it's a satisfying survey of the subject. I don't know how much it would help a student, but you could take a look. (Attention please Steve Carlip!!! Do you agree with Aaron and Shagird that the issues I raised are adequately covered in an existing textbook?) On Sun, 4 Apr 2004, Buzurg Shagird wrote: > tessel@tum.bot wrote in message news:<c4kqmp$18q3$1@fiasco.xenopsyche.net>... > [quoted text clipped - 5 lines] > fairly good job of discussing the action principle in their Classical > Mechanics and Classical Field Theory books? Since I mentioned those books as inadequate for students like Doug Sweetser who are trying to formulate and play around with a "Lagrangian field theory" (even a "toy theory"), you may assume that I feel they (and the other books I mentioned) do -not- explain -the "issues for discussion" which I listed- (see below for a repeat of this list). > Most quantum field theory textbooks also do a quick review of the > classical field theory results they are going to use, like the > Euler-Lagrange equations, Noether's theorem, Hamiltonian and conserved > quantities ... Let me clarify: it is true that LL and other books I have studied -do- discuss the following topics: (1) the E-L equations for L containing only first derivatives, (2) a special case of Noether's theorem (physicists may not know the more general statement using variational symmetries or even "divergence symmetries", for which see Olver), (3) Hamiltonian and conservation of energy, (4) EM tensor for a very few and very special cases (e.g. scalar fields, Maxwell, linearized gtr) and conservation of (linear, angular) momentum. Since I've studied those books, I know they discuss these topics and I did not mean to imply that I feel that their discussion of -these- topics is inadequate! My claim is quite different: for purposes of playing around intelligently with Lagrangians, -much more is needed-. Perhaps most obviously, as I pointed out, if you try to perturb your favorite field theory Lagrangian, you will probably encounter a -nonlinear- field equation. For example, the sine-Gordon equation can appear in this way. Now, since an early step in investigating a proposed field theory is to search for "wave solutions", you will need tools for doing this even in the case of a -nonlinear- PDE. Average students who have only studied -linear- PDEs (e.g. solution by separation of variables) are likely to have great difficulty in doing that, but Lie's methods are straightforward and easy to use (the intermediate computations can become tedious, but heck, that's why people invented me). Or again: when I proposed to try to derive the equation of motion of "charged" test particles in a given field, did you and Aaron fail to notice that I was talking about the analogue of the EIH procedure in the case of gtr? (For other readers: Einstein, Infeld, and Hoffmann proposed to remove the assumption that the world lines nonspinning test particles in a vacuum region are timelike geodesics by -deriving- this from the Einstein field equation.) I assume that you and Aaron are both aware that this is -highly nontrivial-; if not, see author = {Joshua N. Goldberg}, title = {The Equations of Motion}, booktitle = {Gravitation: an Introduction to Current Research}, editor = {Louis Witten}, publisher = {Wiley}, pages = {102--129}, year = 1962} Note that I was proposing to explore whether an analogous procedure (for obtaining the equation of motion of "charged" particles from the field equation) can be found for other theories. Another way to think about it: can we determine a suitable (particle-field) interaction term from a proposal for a free-field term in the Lagrangian of our field theory? This is presumably a nontrivial question! > You could complain that Dirac's method of analysis of constraints is not > given in more books, I didn't mention that, probably because I haven't heard of it, at least not under that name. > but you can't apply it directly to non-Abelian gauge theories anyway --- > you need to go to superspace or such to introduce ghosts, and ghosts are > better done in the path integral formalism anyway -- so you shouldn't > really complain. :-) I didn't complain about those issues. Indeed, I forgot to say that one of the "levels of structure" I had in mind was: (5) first Lagrangians admitting U(1) gauge symmetry, then nonabelian gauge symmetries. > What else will fill this 'gap' that you are worrying about? Honestly, at this point I have to wonder if you, Aaron even -read- my post in its entirety! In particular, did either of you read the list of "issues for discussion" that I offered? Did either of you review the contents of the books I mentioned with this list in mind? Did either of you see any of Doug's efforts, or the efforts of other amateurs here over the years, to create some kind of "toy theory" starting from a proposed Lagrangian? I am an amateur myself, but even I could spot zillions of errors committed by these people, some quite elementary and some involving more subtle issues. For that matter, look at the ArXiv--- I have seen quite a few poor quality papers there committing similar errors (but don't ask me for examples, because I've deleted my "file of horrors" from the ArXiv, not to mention that could get ugly--- my interest is not in cutting down people but in preventing others from committing similar fundamental errors, in the ultimate interest of promoting the appearance of better and thus more interesting papers). Thus my suspicion that average students, and perhaps even some folk submitting papers to the ArXiv, need more help than offered by Landau/Lifshitz or Ohanian/Ruffini. For the readers convenience, here again is my list of proposed "issues for discussion": ============== BEGIN SELF-QUOTATION ==================== I propose to start as simply as possible and slowly increase the level of sophistication, e.g. (not neccessarily in this order) 1. first linear, then nonlinear Lagrangians, 2. first Lagrangians of form L(x,u,u_x), then L(x,t,u,u_x,u_t) (more variables!) and L(x,u,u_x, u_(xx)) (higher order derivatives!), 3. first scalar theories, then vector theories, then tensor theories, 4. first classical field theories, then quantum field theories. [5. first Lagrangians admitting U(1) gauge symmetry, then nonabelian gauge symmetries.] Issues we might systematically explore in this way could include: 1. how to sensibly select the terms of our Lagrangian, as written in the useful form L = L_matt + L_interaction + L_field 2. how to guess key features of the "free-field" case of the field equation of our theory from looking over L_field, 3. how to write down the Euler-Lagrange equations (allowing for the possibility of second and higher order derivatives), 4. how to verify key features of the (free-field) field equation, e.g. (a) how to determine the group of "point symmetries" and the subgroup of "variational symmetries", (b) how to confirm the existence of wavelike solutions to the (free-field) field equation, (c) how to obtain and study particular solutions (even for nonlinear PDEs) e.g. using Lie's methods, Baecklund symmetries, scattering transform, 5. how to obtain a suitable energy-momentum tensor (and what to do if the "canonical energy-momentum tensor" turns out to be -nonsymmetric-), 6. more generally how to find "conserved currents" (e.g. from "divergence symmetries"), at least in the case where L only contains first order derivatives, 7. how to find suitable "conserved currents" if L contains second order derivatives, 8. how to determine L_interaction from energy-momentum exchange between the field and "particles", if this is possible, or if not, how otherwise to obtain a suitable "equation of motion" for "charged" particles, 9. how to check for at least the most commonly encountered self-inconsistencies which can arise in a field theory, Having explored these issues for a generous range of simple toy theories, we can try to address the question which presumably we would all agree is the question of ultimate interest: how can we write down a Lagrangian which has a reasonable chance of producing a theory which will accomplish stated goals? Let me mention a specific example of a "stated goal". Recall that Maxwell's theory of EM is beset with singularities in the field. Recall also that a few years ago we had some people here who wanted to replace gtr with a theory which would "ameliorate" singularities while preserving the remarkable experimental/observational success of that theory. In principle, as I said at the time, that's not a bad idea, but I claimed then (and repeat now) that this is a -much- more challenging program than (I felt) those people recognized. At that time I proposed to step back and consider first the analogous but easier question for EM, but unfortunately (as I recall the affair, at least) my correspondents were insufficiently willing to do this. I hope and believe that the current student generation is more amenable to such good advice! Actually, I propose to start with scalar theories, e.g. three-dimensional analogues of L = -1/2 [ u_t^2 - u_x^2 - m^2 cos(u)^2 ] (I might have munged a sign or two, but this is supposed to be the Lagrangian which leads to the sine-Gordon equation, a nonlinear wave equation beloved of PDE people), or more generally L = -1/2 [ u_t^2 - u_x^2 ] + some other proposed "perturbing term" which is small for small u (since I doubt the sine-Gordon equation will fully achieve the goal of "ameliorizating" singularities in our scalar field). ============== END SELF-QUOTATION ======================= I wonder if either Shagird or Aaron really think that average students will be able to attack successfully the example of "stated goal" which I mentioned without help well beyond that offered by Landau/Lifshitz or Ohanian/Rufinni? Or the question of searching for an analogue to the proposal of EIH? Or even searching for "wave solutions" to a nonlinear free-field equation like the sine-Gordon equation? If so, I challenge either of them to show us/them how to do this! Or, much more modestly, to cite chapter and verse from an -existing textbook- where this is done, in enough detail for an average student to follow the discussion. "T. Essel" (hiding somewhere in cyberspace) [lament about lack of good references RE field theory] Two book recommendations 1) Gelfand-Fomin, "Calculus of Variations" 2) Lovelock and Rund, "Tensors, Diff. Forms, and Variational Principles" The latter has a large section at the end in which a combined vector-metric theory is examined in gory detail. The former is the rare math book that really understands physics. We used it in my graduate course in the calculus of variations. I agree that there are a lot of papers on the arxiv that are poorly written noise. For this one can thank the komissar. -drl [..about teaching and learning..] Like many here I learned a lot of things on my own when the time came. Because I was happy reading older books (esp. Klein) I was used to doing things in terms of parameters, including differentiating with respect to them (for determining the envelope). So "differentiating this thing with respect to the amplitude" or the like seemed very natural, and I didn't get hung up on taking derivatives with respect to a velocity. (I think it helps to go directly to the Hamiltonian formulation. Many people get all balled up trying to imagine d/dxdot.) Also I was very comfortable with "first-order thinking" - calculating with differentials, because of the way I learned calculus and DEs. I was accidentally well-prepared. I think I really learned what was going on from Sommerfeld. His little mechanics book is outstanding. See the sections on "Principle of Virtual Work". -drl > Or again: when I proposed to try to derive the equation of motion of > "charged" test particles in a given field, did you and Aaron fail to [quoted text clipped - 4 lines] > Einstein field equation.) I assume that you and Aaron are both aware that > this is -highly nontrivial-; if not, see Unless I misunderstand you, this is a standard exercise in many GR books. It's done in chapter 9 of Stephani, for example. It's not particularly difficult either, so maybe you can elaborate further about what you are referring to? [...] > I wonder if either Shagird or Aaron really think that average students > will be able to attack successfully the example of "stated goal" which I > mentioned without help well beyond that offered by Landau/Lifshitz or > Ohanian/Rufinni? Or the question of searching for an analogue to the > proposal of EIH? Or even searching for "wave solutions" to a nonlinear > free-field equation like the sine-Gordon equation? Depends on the student. It sounds like what you really want is an advanced course in PDEs, in particular, those stemming from lagrangians. I wouldn't call this a gap, nonetheless, because I don't see too many problems stemming from the lack of what you're looking for. I'm not surprised you can point to bad papers on the ArXiv. There are papers with misunderstandings of more things than just PDEs there. There's a lot more things one could add to the curriculum, but aren't there. A good student should be able to pick up a textbook and learn additional stuff as needed. Aaron > On Sun, 4 Apr 2004, Buzurg Shagird wrote: > [quoted text clipped - 7 lines] > the other books I mentioned) do -not- explain -the "issues for discussion" > which I listed- (see below for a repeat of this list). Well, first of all, when I said that I wasn't sure where you see that gap, I did NOT mean ``I know everything there is to know about field theory", NOR ``all possible issues of classical field theory has been discussed in those books." If you got that impression from my post, I am really sorry, that was not my intent at all. I am sure you will agree that no textbook can discuss every possible issue in a given subject, then it will not be a textbook! A textbook is meant for creating a foundation in a given subject, usually by means of a `linearly progressive' discussion from some starting point to some end point, plus occasionally a brief discussion in some tangential topics which the author finds interesting. A good textbook is read by a lot of students, many of whom will not carry on to do research in the subject, but will find an overview quite useful. Your list of ``issues for discussion" includes some things which are covered in the standard textbooks, some things which are covered in the non-standard textbooks, i.e. books which compile existing knowledge, but with a narrow focus or from an uncommon viewpoint, and a few things for which you have look in monographs or research papers. But this kind of situation will exist whenever you want to make a plan for a thorough investigation of some topic, as you have done. I am unable to see the `gap' that you mention because I feel that those books prepare you well enough to do your own investigations into the questions that you have asked -- and as far as I could see from your post, you are indeed well prepared to do that investigation. That is why I was surprised that you see a gap, or that you expect the answers to all your questions in some textbook. > My claim is quite different: for purposes of playing around intelligently > with Lagrangians, -much more is needed-. Only by those who want to `play around' with Lagrangians, in the way that you and a few others on this group may want to do. :-) But that is very much a `special interests' topic. Why would a textbook include all possible ways of playing around with Lagrangians? Instead, what good textbooks do is to point out the _principles_ behind the most common Lagrangians, so that you can make your own physical or mathematical principle, your own action, and your own perturbative quantum field theory (and if you are really lucky, your own non-perturbative qft). :-) > > You could complain that Dirac's method of analysis of constraints is not > > given in more books, > > I didn't mention that, probably because I haven't heard of it, at least > not under that name. You may find it useful to learn about it -- I was being somewhat facetious when I said what's below. You can learn a lot about a field theory by studying the structure of its constraints -- in particular you can identify the physical and unphysical degrees of freedom, and have a fairly good idea of how symmetries are implemented on the quantum states ... > > but you can't apply it directly to non-Abelian gauge theories anyway --- > > you need to go to superspace or such to introduce ghosts, and ghosts are > > better done in the path integral formalism anyway -- so you shouldn't > > really complain. :-) > I wonder if either Shagird or Aaron really think that average students > will be able to attack successfully the example of "stated goal" which I > mentioned without help well beyond that offered by Landau/Lifshitz or I can't speak for Aaron, but I certainly don't expect a student, average or not, to attack any major research problem immediately after reading a standard textbook. I expect them to skim through many books, mongraphs and papers looking for hints and discussions, and above all, to talk to people who are interested in the same sort of problems. -S. Aaron wrote: > This is generally done in math textbooks, Do tell! I have easy access to a good math research library, and my background, as you know, is in math. I have -never- seen either a math or physics book which discusses the issues I listed, so I'd be very interested if you could (third or fourth request, arghgh!) provide a -citation- to such a book. > H. Goldstein _Classical Mechanics_. I don't own a copy, so I can't tell > you precisely what's in it, but it's one of the standard junior level > classical mechanics texts. Thanks, Aaron, this is the book I thought you meant. IIRC it doesn't do the trick, and by now I am quite sick of this whole darned subject (I know--- I brought it up, I just didn't expect a negative response from you in particular), but if I get over this, I'll take another look at this book. > > Or again: when I proposed to try to derive the equation of motion of > > "charged" test particles in a given field, did you and Aaron fail to [quoted text clipped - 9 lines] > particularly difficult either, so maybe you can elaborate further about > what you are referring to? Sorry, I expressed myself very badly. You did misunderstand me. The topic for discussion was -not- how to do EIH for -gtr- but rather how to do something analogous for -other- field theories. This is nontrivial. If you think otherwise, well, by now I am sick of arguing about it. Several other posters suggested further resources--- thanks! Buzurg Shagird wrote: > > Since I mentioned those books as inadequate for students like Doug > > Sweetser who are trying to formulate and play around with a [quoted text clipped - 8 lines] > discussed in those books." If you got that impression from my post, I am > really sorry, that was not my intent at all. I did not think you were saying that. I thought you were saying you think there are already adequate discussions of the issues I raised suitable for -average- students. (I agree that -brilliant- students require much less help.) > Your list of ``issues for discussion" includes some things which are > covered in the standard textbooks, Cites would be appreciated. > some things which are covered in the non-standard textbooks, i.e. books > which compile existing knowledge, but with a narrow focus or from an > uncommon viewpoint, Cites? > and a few things for which you have look in monographs or research > papers. Cites? > I am unable to see the `gap' that you mention because I feel that > those books prepare you well enough to do your own investigations into > the questions that you have asked -- and as far as I could see from > your post, you are indeed well prepared to do that investigation. Yes, I agree that these books prepare -me- and they apparently prepared -you-, and I have no doubt they will adequately prepare creative and highly motivated students with good background and blessed with intelligence, insight, and at least some talent. I am saying that the books I have seen won't prepare students who lack some/all of those things. > > My claim is quite different: for purposes of playing around > > intelligently with Lagrangians, -much more is needed-. [quoted text clipped - 3 lines] > very much a `special interests' topic. Why would a textbook include all > possible ways of playing around with Lagrangians? Would it help if I dropped the word "textbook"? What if I had proposed a book for graduate students of -philosophy of physics-, rather than advanced undergraduate students of physics? And finally, Doug Sweetser wrote: > I was in the position four years ago of having field equations, but no > Lagrangian. What I did was practice with EM, doing the derivation from > the classical EM Lagrange density out to the field equations about a > dozen times. A good way to begin. > In many ways, it is like practicing scales or multiplication tables: > nothing can substitute for practice. After that, it was possible for me > to play with variations. But the summary you provided for me was so full of misconceptions major and minor that I didn't even try to list them! > Before I ramble about anything else, I request you take the above > Lagrange density and generate those field equations. Well, if you were Feynman, you wouldn't need to compute anything to see what will happen here. In a burst of foolish enthusiasm, I promised in effect to try to turn everyone into a new Feynman. Sorry, Doug, I can see I am not helping by dropping out here, but I already told you I don't have sufficient patience to play by any rules but my own, and by now I am sick of the whole topic and am sorry I ever brought it up. So let's just forget it. Thanks to the small number of posters who had a more positive response to my proposed "topics for discussion"! I greatly appreciate the support, but gosh, we were certainly in the minority, arghgh. Alas, I have less time than I anticipated when I offered to lead this discussion. And by now I'm quite sick of arguing tendentiously about what I regard as nonissues, when all I really wanted was to induce people like Aaron to just -play-, darn it all. So I'll withdraw my proposal to discuss these topics. Igor, in principle I -would- still like to try for something less ambitious, namely an exposition of "Lie's methods, Baecklund symmetries, scattering transform". Right now I am too disappointed with the general tenor of the response in this thread to contemplate attempting lengthy expository posts in the near future, but time permitting, perhaps I will try, some time later this month [read "millenium"], to summarize the most useful features of Lie's theory of symmetries of PDEs, including looking for "particular solutions" and for "Baecklund morphisms". "T. Essel" (hiding under a stupid psuedonym) > Aaron wrote: > > > This is generally done in math textbooks, > > Do tell! I have easy access to a good math research library, and my > background, as you know, is in math. I wish I could give you a better answer. Mostly I just remember seeing such things when browsing in a campus bookstore. Typing 'calculus of variations' into Amazon returned a number of hits if that helps. > I have -never- seen either a math or physics book which discusses the > issues I listed, so I'd be very interested if you could (third or fourth > request, arghgh!) provide a -citation- to such a book. I'm not sure you'll ever find everything you want in a single book, especially at advanced levels. One generally needs to read a number of sources to really understand a given subject. [...] > Yes, I agree that these books prepare -me- and they apparently prepared > -you-, and I have no doubt they will adequately prepare creative and > highly motivated students with good background and blessed with > intelligence, insight, and at least some talent. I am saying that the > books I have seen won't prepare students who lack some/all of those > things. This is probably going to come across poorly, but from my perspective, if a student isn't willing to go down to the library and do some research, they probably should consider a different field. Finding a single source that answers all your questions on any given subject is a rare occurrence. On the other hand, I think what you say is a pretty neat idea for a book. If it doesn't exist, it wouldn't be a bad thing if someone wrote it. As a last thought, another reference that occurred to me in the process of writing this is Warren Siegel's book "Fields", hep-th/9912205. I haven't read it, but I know that it begins with classical field theory and then starts quantizing it. You might find it interesting. Aaron Hello: What an odd place to stop a thread. My name appeared several times as the cement-brained Cro-Magnon poster boy (with far gentler comments, but I think that tag is pithier :-) If you'd rather find the ultimate book, good luck to you, but the citations are not piling up as you noted. Me, I'd rather grind. The nice thing about always digging and getting one's hands dirty is that gems can be found on rare occasions. So I was reading a thread (this thread?) recently on SPR, and the author was waxing on how important symmetries in Lagrangians are. Not the first time I have heard this. But I went I looked at the Lagrange density again, this time wanting to dig up a root connected to symmetry. The work took about three minutes tops, located in a different thread. That result may be the key idea driving the proposal because that is how important symmetries in Lagrange densities are. As always, personal issues are irrelevant to me. All I care about are technical critiques of specific equations. doug quaternions.com I should probably stay quiet, but ... > I did not think you were saying that. I thought you were saying you think > there are already adequate discussions of the issues I raised suitable for > -average- students. (I agree that -brilliant- students require much less > help.) I don't know how to make that distinction. I am not being facetious, and I do have some experience with students of physics at all levels. And I mean precisely that I don't have a checklist or a rule of thumb that allows me to assign labels like that to *students*. To professional researchers, yes, but not to students. I believe that any student, if sufficiently motivated, can learn a subject well enough, from the standard sources, to start doing original research in it. Some guidance is often helpful, but I think that it is needed only to keep up the level of motivation. > > Your list of ``issues for discussion" includes some things which are > > covered in the standard textbooks, > > Cites would be appreciated. You already know which ones I am talking about -- Landau and Lifshitz, Goldstein, any introductory book on quantum field theory ... If you want a page and volume reference, you need to ask a more specific question, and hopefully someone who remembers page numbers will be able to help. Okay, maybe I should give an example. Say you want an action which produces a linear eq. of motion. Then you need a Lagrangian quadratic in the fields and their derivatives. Write down all such terms. That is your Lagrangian. Suppose you wish to be a bit more serious and want the action to have some resemblance to physics. Then (all standard texts) write a term quadratic in the time derivative. Add any polynomial in the fields. Suppose you want the theory to be Lorentz invariant. Then make sure your Lagrangian is a Lorentz scalar. Suppose you want the theory to have any other global invariance. Then your Lagrangian must have that invariance as well. That is it, for classical fields. I have ignored Lagrangians for spinors -- you can construct those in pretty much the same way. Everything I said so far is covered in the standard texts. And you know it. So why ask for a cite? > Would it help if I dropped the word "textbook"? What if I had proposed a > book for graduate students of -philosophy of physics-, rather than > advanced undergraduate students of physics? Then I would suggest starting with a paper by Weinberg, ``What is quantum field theory and what did we think it is?" hep-th/9702027. This paper was presented in a conference on Conceptual foundations of quantum field theory, Boston 1996. The proceedings contain several articles which should be interesting to graduate students of philosophy of physics. > Well, if you were Feynman, you wouldn't need to compute anything to see > what will happen here. In a burst of foolish enthusiasm, I promised in > effect to try to turn everyone into a new Feynman. I never met Feynman, but my impression from reading, and talking to, those who had, is that the reason he didn't need to compute anything only because he had already spent a lot of time computing everything he wanted to know. And he could calculate very fast, which helped. :-) (Freeman J. Dyson once said that a QED calculation by Feynman was the fastest piece of physics calculation he had seen.) > Sorry, Doug, I can see I am not helping by dropping out here, but I > already told you I don't have sufficient patience to play by any rules but > my own, and by now I am sick of the whole topic and am sorry I ever > brought it up. So let's just forget it. Why not play by your own rules? Giving up is too easy. :-) -S. On Thu, 15 Apr 2004, Buzurg Shagird wrote: > I should probably stay quiet, but ... To the contrary---- but see my "general comment" below. > > I did not think you were saying that. I thought you were saying you > > think there are already adequate discussions of the issues I raised [quoted text clipped - 10 lines] > is often helpful, but I think that it is needed only to keep up the > level of motivation. Probably this doesn't need saying, but I'll say it anyway: the thing which made s.p.r. worthwhile in the past (and might make it worthwhile in the future) is the presence of (a) eager, able, and well-motivated students at all levels (possibly including "amateurs", "continuing education students", and students not enrolled in formal courses), (b) experienced teachers/researchers willing to spend some unremunerated time helping students. Probably the only thing we disagree about is the degree to which group (a) must be well-prepared by successful solid formal coursework in the requisite math/physics background. But your observations suggest a -general comment- concerning discussions in s.p.r., at which I hope noone will take offense. I stress that this comment is addressed to -all- current s.p.r. participants, and not to you specifically. In my final reply to Doug I pointed out that apparently one thing he and I agree upon is that we'd both like to see more teaching/learning in this group--- specifically more -technical- discussion of -specific- nontrivial issues--- and less -talking- about teaching/learning. I realize that in this thread I have myself been guilty of -talking- about how I'd like to do what I'd like to do, rather than doing it. In fact, feeling myself manuevered into this bad behavior explains why I became despondent! This, together with something beyond my control (lack of access/time/energy), is why I opted out. Before anyone gets defensive over my general comment above, I'd just ask that everyone please remember that in happier days I did follow this practice myself. And I hasten to assure newbies that some posters did express appreciation of my own expository efforts, so such efforts are not -entirely- thankless. And we probably agree there can be other rewards besides gratitude: the combination of good teachers and good students often results in the -teachers- as well as the students learning something valuable! (John Baez has mentioned this as one motivation for his Weeks--- IMO, we would all do well to take this valuable series as a model.) > Okay, maybe I should give an example. Heh, we must both have been thinking the same thing, re my "general comment" :-/ > Say you want an action which produces a linear eq. of motion. Then you need > a Lagrangian quadratic in the fields and their derivatives. Write down all [quoted text clipped - 3 lines] > resemblance to physics. Then (all standard texts) write a term quadratic > in the time derivative. Add any polynomial in the fields. Well, I did mention sine-Gordon as a valuable example of a naturally arising equation (naturally arising in pure math, that is, namely differential geometry!) which -violates- this rule of thumb. But of course I agree that mostly one uses polynomials, and in fact I had that in the back of my mind because it implies that whenever we need to eliminate variables (including derivatives), e.g. by taking further derivatives of a given equation in order to eliminate inessential parameters, we can avail ourselves of Groebner bases for a suitable elimination order. (Actually, trig polynomials are also susceptible to this technique. So are Jacobian elliptic functions, etc.) I haven't yet myself applied elimination orders in quite this way--- at least not in an essential way--- but one thing I was looking forward to was keeping my eye out for opportunities to use this tool (and others) to good effect. > Suppose you want the theory to be Lorentz invariant. Then make sure your > Lagrangian is a Lorentz scalar. ^^^^^^^^^^^^^^^ A function invariant under the standard action on R^4 of the Lorentz group? (For other readers: this would be the notion which is generalized/systematized by the "point symmetry group" of a [system of] differential equation[s]. As an application, we can generalize/systematize the standard Ansatz technique employed in searching for "traveling wave" and "similarity" solutions of PDEs.) > Suppose you want the theory to have any other global invariance. Then your > Lagrangian must have that invariance as well. Agreed! :-) We no doubt agree these particular items are obvious enough (once pointed out!), but this excellent list is indeed exactly the kind of thing I was talking about in my proposal for a kind of "Schaum's Outline" advice/problem book on field theorizing. (If you look at past expository threads by me you'll see that I tried to set exercises which would lead a student to this kind of list of useful observations. Probably we agree about the value of "routine" exercises of this nature.) > Everything I said so far is covered in the standard texts. And you know it. > So why ask for a cite? See my "topics for discussion". IMHO, these issues are certainly -not- "covered" in the standard texts. > I never met Feynman, but my impression from reading, and talking to, > those who had, is that the reason he didn't need to compute anything > only because he had already spent a lot of time computing everything he > wanted to know. Another thing I apparently agree with Doug/you/"everyone" about is the value of hard slog! Although as you say, for some, like Feynman (or Weinberg), slogging is apparently much easier than for most. > Giving up is too easy. :-) I -have- been depressed by the way the thread developed, but the real obstacle is lack of access/time/energy/(?) civility adequate to the task. "T. Essel" (hiding somewhere in cyberspace) Hello T Essel: Let me respond to one point of your post based on my experiences. > In my final reply to Doug I pointed out that apparently one thing he > and I agree upon is that we'd both like to see more teaching/learning > in this group--- specifically more -technical- discussion of > -specific- nontrivial issues--- and less -talking- about > teaching/learning. This is difficult to do either in a newsgroup or in a technical book. The reason is typography. With a pencil, it is easy to write a great variety of symbols. With ASCII, it is a greater challenge. Even with LaTeX, people write far fewer equations than are down with pencil. I have done ASCII derivations to this newsgroup. It is my observation that few reply to such posts. The monospaced letters don't translate efficiently to how math by pencils look, so most hit a few rapid spacebars and move on. I do that myself! For a series of talks I gave, I decided to put in the effort of recreating with LaTeX what I would do with a pencil. That required a BIG block of time (I was unemployed at the time). I used texmacs which was designed to speed things up. I had a policy of always stating exactly where I was starting from. Usually in under 8 steps I got to the end. No steps were "left for the reader" :-) To make it available in both pdf and html also was a time sink. I enjoyed this time unemployed. How things look matters to communication efficiency, doug quaternions.com On Thu, 15 Apr 2004, Doug Sweetser wrote: > What an odd place to stop a thread. My name appeared several times as > the cement-brained Cro-Magnon poster boy (with far gentler comments, but > I think that tag is pithier :-) > > Me, I'd rather grind. The nice thing about always digging and getting > one's hands dirty is that gems can be found on rare occasions. Although I did repeatedly cite you as an exemplar of a seriously confused student, you certainly are not the only one I had in mind! I just couldn't remember anyone else's name off the top of my head. And if I really thought you were "cement-brained", of course I wouldn't have tried to engage you as a "Lagrangian playmate"! But in any case, as it turns out--- as I said a few weeks ago--- I don't after all have sufficient access/energy/time/civility(?) to pursue the ambitious thread I impulsively tried--- with good intentions--- to initiate. Because my intention was certainly not to discourage you from playing, I will attempt one last time to clarify what I -was- trying to urge you to do. First, let me say that I -entirely agree- with you concerning the value of playing with specific Lagrangians! And of course I entirely approve of the search for "gems"! > As always, personal issues are irrelevant to me. All I care about are > technical critiques of specific equations. Actually, I -do- want you (and me, and other physics fans) to -continue- to "get your hands dirty" by playing with specific models. My suggestion was certainly not to stop playing! Rather, I said that I think that you are in over your head (wrt your current level of insight/sophistication) regarding -the specific Lagrangian- you have been trying to study. I suggested that you back off from that -specific- proposal for the time being. I proposed that instead you > start as simply as possible and slowly increase the level of > sophistication, studying > (not neccessarily in this order) > [quoted text clipped - 6 lines] > > 4. first classical field theories, then quantum field theories, I suggested that for various specific Lagrangians you systematically explore, with our help, the following issues (this was not neccessarily an exhaustive list!): > 1. how to sensibly select the terms of our Lagrangian, as written [when appropriate] > in the useful form > [quoted text clipped - 34 lines] > 9. how to check for at least the most commonly encountered > self-inconsistencies which can arise in a field theory, I suggested that this kind of systematic exploration would help you/others to develop a -highly valuable skill-: the ability to write down an intelligently chosen Lagrangian leading to a field theory which has a "reasonable chance" of achieving a stated goal (e.g. behaving like Maxwell's theory under common laboratory conditions, while achieving something Maxwell's theory does not, such as ameliorating point charge singularities). If you followed this advice, I am confident that you would also acquire the ability to quickly see, in many cases, what is wrong with an "obviously bad" choice of Lagrangian--- such as the one you proposed. You would also be more likely, I think, to avoid wasting the time/energy of the amazingly generous email correspondent you mentioned and of various posters here. Certainly you would be much more likely to obtain here the kind of -focused- feedback which -immediately improves your understanding- of some specific issue. (The "wastage" which I and others have cited is clearly due to multiple misunderstandings concerning numerous issues at various levels of sophistication. This kind of mutual multiple misunderstanding is -much- more likely to arise if any participant is in over his head, and once it has happened, IMO, the only real fix is to erase the whole discussion and to start all over again at a lower level.) And, while I have no idea what would turn up if you followed my advice, in my experience systematic exploration along the lines I suggested is the best way to uncover "gems". To be sure, I would expect that any "gems" uncovered might not be directly relevant to the original goal (developing the skill of intelligent field theorizing), but might involve some completely unexpected insight into who-knows-what. But that's probably half the fun of finding a mathematical gem--- typically, the lucky explorer finds not just one stone but a veritable Kimberly. And to keep up the metaphor, the stones aren't diamonds or rubies but something -completely new-, yet of comparable beauty. So I also regret that this thread never developed (as was my goal) into the specific critique of specific proposed Lagrangians. And I vigorously applaud your desire for this kind of focused discussion on technical issues! I think our only disagreement was over -which- Lagrangians to discuss -when-. And if you reconsider following my advice, others here are certainly capable of helping you even if I cannot after all participate myself. (I haven't -completely- wimped out--- I do still hope to exposit here, when I have more access/time/energy, the computation/applications of symmetry groups of partial differential equations and perhaps additional techniques such as scattering theory and Baecklund morphisms.) Wishing you the best of luck with your future Lagrangian play, "T. Essel" (hiding somewhere in cyberspace) > Issues we might systematically explore in this way could include: > > 1. how to sensibly select the terms of our Lagrangian, as written in the > useful form > > L = L_matt + L_interaction + L_field I think this format for the Lagrangian is found in gauge theories. this is a restricted class of theories, but an important one. For gauge theories, it is easy to write the Lagrangian in the way you ask, you can find it in any book on gauge theory. But for more general theories, the Lagrangian won't take this form. or at least, there isn't a distinction between the "matter field" and the "field field". you can always seperate the linear terms from the nonlinear terms and call the former L_fields and the latter L_interaction. > 2. how to guess key features of the "free-field" case of the field > equation of our theory from looking over L_field, Which key features about the free-field version of your theory do you want to guess? In general, i don't think free fields have all that many features. It seems to me like you can know just about everything you might want to know about a free field if you know its mass and spin. > 3. how to write down the Euler-Lagrange equations (allowing for the > possibility of second and higher order derivatives), I recall this homework question in Goldstein. It is easy to write down the Euler-Lagrange equation for Lagrangians with higher derivatives, and not particularly enlightening. One thing I wouldn't mind knowing more about: as i understand it, the reason we do not use Lagrangians with higher order derivatives is that they would violate locality. This is, for example, the criterion by which we throw out the square root of the Klein-Gordon equation (d\phi/dt=sqrt{d^2/dx^2+m^2}\phi) as a useful theory in some field theory textbooks. it contains derivatives of all orders, and so is nonlocal. if anyone wants to say why higher order derivatives violate locality, i would love to hear it. > 5. how to obtain a suitable energy-momentum tensor (and what to do if the > "canonical energy-momentum tensor" turns out to be -nonsymmetric-), For a really nice treatment of the Belinfante procedure, and other discussions of the stress tensor, see Di Francesco, Conformal Field Theory. > 6. more generally how to find "conserved currents" (e.g. from "divergence > symmetries"), at least in the case where L only contains first order > derivatives, I don't know what a divergence symmetry is. Does Noether's procedure not work here? > 7. how to find suitable "conserved currents" if L contains second order > derivatives, I don't see why we shouldn't be able to do Noether's procedure on a Lagrangian with higher order derivatives.... I complained: > > [...] as far as I know, there is currently -no- textbook on "How to > > Theorize from a Lagrangian" On Fri, 2 Apr 2004, MM replied: > I would have said that Peskin & Schroeder's "Introduction to QFT", and > Weinberg's multi-volume "The Quantum Theory of Fields" together do a > good job. OK, I've made a mental note. But AFAIK neither of these two study the questions I asked regarding -classical- field theories. Does anyone else agree that those questions are by no means trivial? I am beginning to wonder... I trust people who have the books I mentioned, namely 1. Landau/Lifschitz, Mechanics, 2. Landau/Lifschitz, Classical Theory of Fields), 3. Ohanian/Ruffini, Gravitation and Spacetime, 4. Weinberg, Gravitation and Cosmology will agree that the questions I raised are largely unaddressed in these books, although from playing around I do understand why LL and OR make certain comments which only apply in examples beyond the two examples these books discuss. I expect (perhaps) naively that my questions should also be relevant in QFT, e.g. because I am vaguely aware of mathematical analogies between the usual suspects (PDEs arising in classical field theory and in quantum theory). Is this expectation also wrong? "T. Essel" (hiding somewhere in cyberspace) I complained: > > [...] as far as I know, there is currently -no- textbook on "How to > > Theorize from a Lagrangian" On Fri, 2 Apr 2004, MM replied: > I would have said that Peskin & Schroeder's "Introduction to QFT", and > Weinberg's multi-volume "The Quantum Theory of Fields" together do a > good job. OK, I've made a mental note. But AFAIK neither of these two study the questions I asked regarding -classical- field theories. Does anyone else agree that those questions are by no means trivial? I am beginning to wonder... I trust people who have the books I mentioned, namely 1. Landau/Lifschitz, Mechanics, 2. Landau/Lifschitz, Classical Theory of Fields), 3. Ohanian/Ruffini, Gravitation and Spacetime, 4. Weinberg, Gravitation and Cosmology will agree that the questions I raised are largely unaddressed in these books, although from playing around I do understand why LL and OR make certain comments which only apply in examples beyond the two examples these books discuss. I expect (perhaps) naively that my questions should also be relevant in QFT, e.g. because I am vaguely aware of mathematical analogies between the usual suspects (PDEs arising in classical field theory and in quantum theory). Is this expectation also wrong? "T. Essel" (hiding somewhere in cyberspace) On Fri, 2 Apr 2004, Danny Ross Lunsford wrote: > tessel@um.bot wrote: > [quoted text clipped - 5 lines] > literature. The key paper for the physicist is by Belinfante and > concerns the issue of symmetrizing the canonical energy tensor. Oh good, Ohanian/Ruffini make the requisite remark but fail to give any citation. Can you or anyone else give me a bit more than just an author name? > To this one can add the work of Palatini where he shows how to get GR by > varying the connection. > > For textbooks, one can mention Landau-Lifschitz and Barut. ^^^^^^^^^^^^^^^^ Sigh... by now it seems clear that my original post must have been very badly worded, and I apologize! I did specifically mention (but I should have given a list) LL, Mechanics, LL, Classical Theory of Fields, Ohanian/Ruffini, Gravitation and Spacetime, Weinberg, Gravitation and Cosmology as textbooks I myself have studied. I regard LL as superb on the topics covered; my claim was that this coverage is -inadequate- the purpose I have in mind. (I think I understand LL, but another subsidiary point here is that in my experience, the average student will not. As evidence that I may not be the only one who thinks LL/Weinberg/MTW aim too high for many students here, I note the steady decrease in the mathematical sophistication assumed by authors from the "first generation" classics, LL/Weinberg/MTW, through the excellent "second generation" textbooks by Schutz/D'Inverno, to the recent "third generation" textbooks by Carroll and Hartle. (*Heh---since Carroll apparently sometimes reads this group, maybe he will comment on this alleged downdumbing? BTW, despite this comment, I think this is an excellent textbook; see RWWW! Downdumbing isn't neccessarily a bad thing--- in a way, what I am doing in this thread is proposing to expand some side remarks by LL/OR into a sketch of the textbook length discussion I feel these issues deserve, precisely because, while subtle, they seem to frequently arise in the -practice- of field theorizing.) What purpose do I mean? Well, to repeat my list of "topics for discussion" and my example of one possible "stated goal" for a field theory defined in terms of a Lagrangian (and possibly additional assumptions, e.g. if needed, a force law): ======= BEGIN SELF QUOTATION ============ I propose to start as simply as possible and slowly increase the level of sophistication, e.g. (not neccessarily in this order) 1. first linear, then nonlinear Lagrangians, 2. first Lagrangians of form L(x,u,u_x), then L(x,t,u,u_x,u_t) (more variables!) and L(x,u,u_x, u_(xx)) (higher order derivatives!), 3. first scalar theories, then vector theories, then tensor theories, 4. first classical field theories, then quantum field theories. [5. first theories admitting U(1) gauge symmetries, then nonabelian gauge symmetries.] Issues we might systematically explore in this way could include: 1. how to sensibly select the terms of our Lagrangian, as written in the useful form L = L_matt + L_interaction + L_field 2. how to guess key features of the "free-field" case of the field equation of our theory from looking over L_field, 3. how to write down the Euler-Lagrange equations (allowing for the possibility of second and higher order derivatives), 4. how to verify key features of the (free-field) field equation, e.g. (a) how to determine the group of "point symmetries" and the subgroup of "variational symmetries", (b) how to confirm the existence of wavelike solutions to the (free-field) field equation, (c) how to obtain and study particular solutions (even for nonlinear PDEs) e.g. using Lie's methods, Baecklund symmetries, scattering transform, 5. how to obtain a suitable energy-momentum tensor (and what to do if the "canonical energy-momentum tensor" turns out to be -nonsymmetric-), 6. more generally how to find "conserved currents" (e.g. from "divergence symmetries"), at least in the case where L only contains first order derivatives, 7. how to find suitable "conserved currents" if L contains second order derivatives, 8. how to determine L_interaction from energy-momentum exchange between the field and "particles", if this is possible, or if not, how otherwise to obtain a suitable "equation of motion" for "charged" particles, 9. how to check for at least the most commonly encountered self-inconsistencies which can arise in a field theory, Having explored these issues for a generous range of simple toy theories, we can try to address the question which presumably we would all agree is the question of ultimate interest: how can we write down a Lagrangian which has a reasonable chance of producing a theory which will accomplish stated goals? Let me mention a specific example of a "stated goal". Recall that Maxwell's theory of EM is beset with singularities in the field. Recall also that a few years ago we had some people here who wanted to replace gtr with a theory which would "ameliorate" singularities while preserving the remarkable experimental/observational success of that theory. In principle, as I said at the time, that's not a bad idea, but I claimed then (and repeat now) that this is a -much- more challenging program than (I felt) those people recognized. At that time I proposed to step back and consider first the analogous but easier question for EM, but unfortunately (as I recall the affair, at least) my correspondents were insufficiently willing to do this. I hope and believe that the current student generation is more amenable to such good advice! Actually, I propose to start with scalar theories, e.g. three-dimensional analogues of L = -1/2 [ u_t^2 - u_x^2 - m^2 cos(u)^2 ] (I might have munged a sign or two, but this is supposed to be the Lagrangian which leads to the sine-Gordon equation, a nonlinear wave equation beloved of PDE people), or more generally L = -1/2 [ u_t^2 - u_x^2 ] + some other proposed "perturbing term" which is small for small u (since I doubt the sine-Gordon equation will fully achieve the goal of "ameliorizating" singularities in our scalar field). ======= END SELF QUOTATION ============= I doubt that anyone familiar with the books I listed just above (at least not anyone who has actually tried to play with Lagrangians) would say that they are adequate for working toward the "stated goal" I mentioned as one possible example! Of course, an excellent student should in principle be able to figure out what is needed on his/her own. But most students (obviously!) are average, not excellent. I was in effect suggesting that the better students--- I immodestly implicitly counted myself in that group!--- help out the less able by illustrating "intelligent field theorizing", by working through the detailed analysis of some interesting toy theories. Again, I was not saying that existing textbooks do not cover Lagrangians, but rather than for intelligently positing/analyzing even toy theories, considerably more background material/insight is needed. Apparently this background material has not yet been presented in a single textbook, which, I said, I find very odd, even though I mentioned a plausible reason why this might be the case. Please note that I was also saying that part of the point here would be to guide average students in the very first step, the -intelligent- choice of a Lagrangian to analyze. And the last step, how to modify that initial guess with the benefit of the experience gained by analyzing the alleged field theory it "defines" (perhaps with additional stuff about "force law"--- see my comment about the nontriviality of EIH). A major point of my proposal to fill an alleged gap was this: only by playing with a goodly number of toy theories can one begin to appreciate what is likely to be a "bad" choice of Lagrangian. C.f. Doug's proposal in the "Weinberg on GR" thread, which I regard as "obviously bad". (Doug---sorry if I seem to picking on you; I'm not; I already noted that while I think you munged it in your choice of Lagrangian and what you did thereafer, but I said that I don't think this is your fault, since existing textbooks give so little guidance for the -practice- of field theorizing) Recall also that as far as I can tell, Doug has studied pretty much the same books that I have. > Of course one should have a good understanding of variational calculus > going in. For the physicist, one can mention Caratheordory and > Gelfand-Fomin. I remember the book by Weinstock as being loaded with > excellent problem sets. IIRC, Weinstock does mention the more general form of the Euler-Lagrange equation. Unfortunately, I don't have Gelfand-Fomin, but Caratheodory is a classic! > If you are interested in seeing the process at work in rather new and > non-trivial circumstances, drop me an email. Go for it! (I.e., an expository post to this n.g.--- but recall my own regretable but undeniable lack of background in quantum physics.) "T. Essel" (hiding somewhere in cyberspace) Hello T. Essel: I was in the position four years ago of having field equations, but no Lagrangian. What I did was practice with EM, doing the derivation from the classical EM Lagrange density out to the field equations about a dozen times. In many ways, it is like practicing scales or multiplication tables: nothing can substitute for practice. After that, it was possible for me to play with variations. Working out the units gave me a great insight. A Lagrange density is all the mass (aka energy) in a volume. This is a scalar that has everything that can happen in a box. The action integrates over the volume and time. If that integral does not change as something is varied over an arbitrary amount of time, then a symmetry is found. [In a personal language observation, it is why I prefer the phrase "Lagrange density" to "Lagrangian" because it reminds me of all the energy in a volume.] The field I was trying to find a Lagrange density for was this: Jq^u - Jm^u = A^u;v_;v which looks quite similar to EM: Jq^u = A^u,v_,v Therefore it was appropriate to try variations, once simple repetition lead to some practical, not linguistic, mastery. The Lagrange density I study is this one: L = -(Jq^u - Jm^u) A_u - 1/2 A^u;v A_u;v The first time I wrote this down, I had Jm^u with the same sign as Jq^u. That leads to fields were all like charges repel. The signs between the coupling term (J^u A_u) and the field strength term (A^u;v A_u;v) determine if like charges attract or repel. It would have prevented some embarrassment for me if I had read that first :-) I also had quite a bit of fear about calculating the stress energy tensor. Again, practice with EM removed that inhibition. I was able to do that for this Lagrange density. Before I ramble about anything else, I request you take the above Lagrange density and generate those field equations. Practice is good! If one is skilled with an action, then it turns out to be a one liner. The way I did it, I wrote out all the terms without indices, then started to take derivatives. That process takes a few pages. doug > (c) how to obtain and study particular solutions (even for nonlinear PDEs) > e.g. using Lie's methods, Baecklund symmetries, scattering transform, Independent of all else, I'd be happy to follow your exposition of this topic. Igor |
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