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Natural Science Forum / Physics / Research / September 2004Quantization....
Hi everyone, Skimming over the latest entry in the String Coffee Table (not understanding any of it) I was plunged into some of the discussions on the relaxed quantization conditions with some people arguing that that alone was basis enough not to take QLG serious. This triggered some questions I had built up from about two years ago when I was first confronted with this strange and subtle procedure. I am of course aware that there are many many technicalities (ordering, pathe integral meassures) and different methods of quantization which are equivalent (are they? Mathematically? Physically?) I further no very very little about them so whack me and send me back to the text books if neccesary. What i really wonder about though is why people believe in quantization this strongly! It appears to me that quantization is a way to retain as much physics as possible (particularly several observable/symmetry algebras) while introducing the superposition principle, which forces us to use a vector space as physical state space (and we of course want a hilbert space to actually have a rich enough structure to work with). The superposition principle further requires that our physics be linear and thus our observables (or at least the one defining the physical flow) be linear operators. Now this approach is at least in though rooted in the idea of an observable as something subject to experiment. A fundamental theory should of course be able to reproduce the process of experimenting itself (I don't want to venture into whether or not unitary physics is capable of doing so here, there are enough threads with this question around to keep me reading for the better part of the still young century). The observables are however related to symmetries, so retaining the pbservable algebra can be thought of as retaining the physical symmetries as well (and indeed in Weinberg's QFT book for example (which I still haven't gotten around to read past chapter 3), they appear as generators of the appropriate groups first). At the moment that is the way I like to think about this (QM wise, not talking about QFT or any such monstrosities yet). Cannoncial quantization is a convenient method to construct quantum theories obeying certain symmetries. (it furthermore is a process that ensures we have a nice semi classical limit and some sort of locality apparently) It is blatantly apparant looking around that many physicist think of quantization as something a lot stronger. Not a convenient method of constructing a physical theory but a physical ingredient in itself. Thus the question does anybody attempt to contruct quantum mechanics not from a quantization but from scratch? (I suspect AQFT would fall into this category) Or even try to develop a method of systematically constructing such theories? (Like the Lagrangian formalism for classical physics) What is the basis of this strong believe in quantization. What are the particular properties of cannoncially quantized theories that are considered physically neccesary? What is in particular, this locality condition. In QFT it means the commutator of space like seperated fields vanish. Is that what is meant? Are there analogous elements in Diracs electron? I personally always thought cannonical quantization was rather ad hoc and purely formal (draw hats on the p and q and make the curly brackets into cornered ones and you have succesfully quantized and passed the exam!). Less so with path integrals, but still. I always felt that really QM should stand on it's own. It is after all the theory supposed to be underlying classical mechanics and it seems odd and limiting that our only way to construct such theories is to construct it's classical limit first! This purely formal operation, used carefully to constuct QM and then quickly forgotten about (Don't you dare to use the wrong coordinates when quantizing! p and q it is! And put them in the right order) was then suddenly applied in a much more general manner in QFT without really worrying about anything! And now it seems that "quantizing gravity" and "constructing a quantum theory of gravity" are used as (almost) equivalent! I remember it was even claimed that the former was a purely mathematical problem, while the second is of course manifestly physical. So... What's the general opinion on quantization? -- f Right on, Frank. I couldn't agree more. The more you think about it, the less sense it makes. Creating a theory (quantum mechanics) from a limiting case of it (classical mechanics) is circular. Much better to concentrate on symmetries, i.e. examine the implications of having in QM/QFT a representation of the group of spacetime displacements and rotations (& if relativistic, boosts as well). Quantization does not in any case always work. If one chooses 1/x as the co-ordinate instead of x it will not work. Gauge symmetry in quantum fields blows it away also. One might ask why people take such pains to get it to work. It could be that it is because it is formal, and formality is reassuring. Or maybe because the great Dirac sanctioned it. Whatever ... but to me the procedure is no more than just a curiosity. > The more you think about it, the less sense it makes. Creating a theory > (quantum mechanics) from a limiting case of it (classical mechanics) is > circular. "Quantization" is just timil lacissalc eht -- "the classical limit" written backwards; which is just the Correspondence Principle, which in turn is not only valid but a prerequisite for any new physics. There's nothing circular about the fact that the world, in the large, is classical. Quantum theory has to reduce to classical physics in the limit. > > The more you think about it, the less sense it makes. Creating a theory > > (quantum mechanics) from a limiting case of it (classical mechanics) is [quoted text clipped - 7 lines] > large, is classical. Quantum theory has to reduce to classical > physics in the limit. See my discussion with Arnold Neumaier next to this, it is possibly a bit "intrackable" therefore I point this out here: "The classical limit" of quantum theories constructed from quantizing classical field theories is "per definition" the classical field theory itself. This classical field theory does not need to coincide with the coarse grained, large scale, "classical" behaviour of the quantum theory, which is of course what the correspondence principle is about. There is confusion over to different kinds of "classic" one technical and one physical. I guess my original questions could be rephrased in the language Arnold Neumaier (and I presume most physicists?) uses: Has anyone investigated the possibility of constructing quantum theories which don't have a classical limit but neithertheless have classical physics as an effective theory on a coarse grained large scale. What particular properties that are ordinarily imposed through the process of quantization would be required for such a theory to be viable as a physical "quantum theory". --- frank The only classical limit that makes any kind of sense to me is the latter one, i.e. the "coarse-grained, large-scale" behaviour. The requirements of quantum theory AFAIC are just that the physical states are a Hilbert space upon which the groups believed to be symmetries of nature, including the Poincaré group, are represented non-trivially. This makes the representations of the groups unitary. The irreducible reps of the Poincaré group have been classified by Wigner, and the normalisation of the vectors in these is then determined apart from a scaling. One can then apply the argument in Weinberg QFT, Vol. 1, Ch. 4 to get to the algebra of annihilation and creation operators, and form Fourier transforms of the latter to get quantum fields, i.e. operators associated with spacetime points. The commutator functions are then determined. If we use canonical quantization we get to the same place for spin zero, apart from a factor of (2\pi)^3, but we cannot do half-integer spin at all because, classically, there is no such thing as a fermionic field. For integer spins above zero we get nonsense unless we tamper with our definition of quantization. See if you can guess which method I think is the best. On the matter of the classical limit, of course massless spin one irreps in position space look a lot like Maxwell's equations, and it seems highly likely that the expectation values of the quantum operators representing a photon field are just the classical EM field. Maybe, but it is not necessarily obvious, and one must be careful of treating classical equations as just expectations of quantum operator equations because of the basic fact that <AB> is not the same as <A><B>. Why get hung up on formalism, anyway? The currently-popular approaches to quantum gravity, i.e. loops and strings are based on formal rather than pragmatic notions, and maybe that is their undoing. Neither has provided a classical limit (according to the "sensible" definition), and if they cannot even do that then how do we know that they have anything to do with gravity at all? > Has anyone investigated the possibility of constructing quantum > theories which don't have a classical limit but neithertheless have [quoted text clipped - 4 lines] > process of quantization would be required for such a theory to be > viable as a physical "quantum theory". All depends on how ambitious you are... The Dirac equation has no classical limit. In general, fermions are hm... not "so classical". But you don't need Dirac equation to reason/speculate about, say, neutron stars. An effective many-fermion theories are alive and well. Everything is stable, the fermionic repulsive potential is a decent model from a practical viewpoint. With bosons the situation may be more clumsy. A system which undergoes a condensation is not very classical either. But there are reasonable collective degrees of freedom - sometimes... I am not sure (I am not a physicist), but do you need a full quantum theory to describe the flow, the thermodynamics, vortices, etc. in liquid helium? Jerzy Karczmarczuk Computer science, Univ. Caen, France >>Has anyone investigated the possibility of constructing quantum >>theories which don't have a classical limit but neithertheless have [quoted text clipped - 19 lines] > a physicist), but do you need a full quantum theory to describe the > flow, the thermodynamics, vortices, etc. in liquid helium? On the fluid flow level (Navier-Stokes etc.), everything is already classical. The underlying kinetic equation (Boltzmann etc.) are described by quantum kinetic equations, which are essentially classical dynamical equations but with a collision term revealing the underlying quantum structure. The transition from quantum to classical is thus a gradual thing, as your models get coarser and coarser (i.e., larger and larger time and space scales), they lose more and more of their quantum features, and only some remnants remain. Arnold Neumaier >>Has anyone investigated the possibility of constructing quantum >>theories which don't have a classical limit but neithertheless have [quoted text clipped - 19 lines] > a physicist), but do you need a full quantum theory to describe the > flow, the thermodynamics, vortices, etc. in liquid helium? On the fluid flow level (Navier-Stokes etc.), everything is already classical. The underlying kinetic equation (Boltzmann etc.) are described by quantum kinetic equations, which are essentially classical dynamical equations but with a collision term revealing the underlying quantum structure. The transition from quantum to classical is thus a gradual thing, as your models get coarser and coarser (i.e., larger and larger time and space scales), they lose more and more of their quantum features, and only some remnants remain. Arnold Neumaier > I guess my original questions could be rephrased in the language > Arnold Neumaier (and I presume most physicists?) uses: [quoted text clipped - 3 lines] > classical physics as an effective theory on a coarse grained large > scale. Spin is generally considered a nonclassical effect. (Though there are ways to model classically spin 1/2 systems, these models came long after spin was discovered as a QM phenomenon.) Thus Pauli's extension of Schroedinger's N-particle equations do not come from a classical theory (whi is how you defined what you want a classical limit to mean) but its large-N behavior is of course spin-independent and hence gives classical mechanics. > What particular properties that are ordinarily imposed through the > process of quantization would be required for such a theory to be > viable as a physical "quantum theory". 1. A Hilbert space. 2. An algebra of linear operators on the Hilbert space whose eigenvalues or expectations can be related to experiment. 3. A definition of state which induces well-defined expectations with nonnegative probabilities. 4. A special operator H whose expectation defines the energy of the system, and which acts as the generator for time translations. (Actually, a whole group of such generators, Poincare times Gauge.) This is the ideal wish list. In practice, QFT already compromises; not even QED has be phrased in a way such that 1-4 holds... Thus no one knows what are the exact requirements, and people are free to try things out. The requirements for success are better known, since they just mean: find a consistent framework in which all the known stuff comes out convincingly and uniquely after taking the appropriate limits. So whatever changes in the basic framework are done, they will be viewed with suspicion as long as you didn't succeed, and they will be the new standard once they succeeded. Arnold Neumaier > I personally always thought cannonical quantization was rather ad hoc > and purely formal (draw hats on the p and q and make the curly [quoted text clipped - 4 lines] > and limiting that our only way to construct such theories is to > construct it's classical limit first! Quantization just means constructing a quantum theory for a theory of which we already know what is the classical limit (as it happens for general relativity or N-particle mechanics). No matter how you do it, you must come up with a Hilbert space and a dynamics, of which you can prove that it reduces in the classical limit to what is known and already well tested. Of course, once one _has_ the quantum theory, one no longer needs the classical theory (but can get it as a limit) - but to _discover_ the right theory needs som - from the classical theory. Canonical quantization and path integrals are just two more or less systematic ways of doing that. They are much less ad hoc than you might think. > This purely formal operation, used carefully to constuct QM and then > quickly forgotten about (Don't you dare to use the wrong coordinates [quoted text clipped - 3 lines] > And now it seems that "quantizing gravity" and "constructing a quantum > theory of gravity" are used as (almost) equivalent! Two names for exactly the same problem. Arnold Neumaier > Quantization just means constructing a quantum theory for a theory > of which we already know what is the classical limit Implicitly assuming that the process of taking the limit is exactly the one used in NRQM. An assumption that fails in almost everything beyond NRQM, already in QED it's not straightforward to recover the field picture, the dirac field that serves to quantize the electron has no classical meaning and in non abelian gauge field theories the classical theories we construct them from are not a good description of their classical limit (asymptotic freedom and all). Especially in the latter case we are using quantization mostly to implement symmetries I think. (as it happens for > general relativity or N-particle mechanics). Given that this doesn't happen to be the case for the above examples and given the conceptual differences between gr and newtonian mechanics this appears a questionable motivation at best. Nevermind the fact that straightforward path integral quantization of the Einstein-Hilbert action is apparently very very ill defined. > They are much less ad hoc > than you might think. Perhaps ad hoc was the wrong word, purely formal might be more appropriate. Insufficiently general as a method for constructing quantum theories is what I really mean. Consider the various generalizations of QM considered and how they appear naturally in a different approach: gr-qc/9706069 > > And now it seems that "quantizing gravity" and "constructing a quantum > > theory of gravity" are used as (almost) equivalent! > > Two names for exactly the same problem. Yet implying different approaches! "Quantizing gravity" implies starting with GR whereas "constructing a quantum theory of gravity" means ending up with GR in some limit! That this is considered equivalent is precisely what I question! For reference I'll link you to the discussion that irked me: Thomas Thiemann started of this particular branch of the discussion with this comment: http://golem.ph.utexas.edu/string/archives/000299.html#c000588 (Here is one of the responses: http://golem.ph.utexas.edu/~distler/blog/archives/000307.html as far as I can see this has been settled ) Consequently several different formal steps are discussed which apparently lead to radically different results. In particular it is ambigious when to promote classical variables to operators. Since this is a purely formal step it seems that every prescription as to when to do it can only be anwsered "ad hoc" and justified by later success, and not have a sound basis in theory. If there is a theory and studies that look at this, that look at precisely what we are doing and why we need to do what we do when and where we do it to get physical results please point me towards it, the standard text books contain precious little material or references on this! In particular there is a lot of stuff going around there I do not understand about anomalities and whatnot. It appears, however, that Thiemann's quantization creates theories that satisfy the right symmetries and the super position principle. If they are then not equivalent to standard quantization, then standard quantization is a more strict mathematical framework then these physical conditions, and what I ask is what additional conditions and properties it introduces and how they can be interpretated physically (locality is shouted a lot, hu?) --- frank >>Quantization just means constructing a quantum theory for a theory >>of which we already know what is the classical limit > > Implicitly assuming that the process of taking the limit is exactly > the one used in NRQM. An assumption that fails in almost everything > beyond NRQM, Haven't you heard of NRQED, NRQCD, etc? All relativistic field theories have reasonable nonrelativistic limits, though, at present, the limit cannot be taken in a mathematical sense since there is no mathematical formulation of QED etc. > already in QED it's not straightforward to recover the > field picture, the dirac field that serves to quantize the electron > has no classical meaning It only needs to have a qunatum meaning. > and in non abelian gauge field theories the > classical theories we construct them from are not a good description > of their classical limit (asymptotic freedom and all). They are. The usual 1-loop results are low order expansions around the classical limit. > the fact that straightforward path integral quantization of the > Einstein-Hilbert action is apparently very very ill defined. Straightforward path integral quantization of the anharmonic oscillator is already ill-defined. What counts are the final results with which one works numerically, not the intermediate formal detours. To 1-loop order (which is all one needs for all applications of quantum general relativity in the near future), there is no problem with quantum gravity. At higher loops, one simply needs to take account of a few more coupling constants for each order. There is nothing intrinsically wrong here, as long as one doesn't want to have a full theory. The common problem to all relativistic QFTs is that they lack a firm mathematical basis; but this has nothing to do with the problem of quantization. > Consider the various generalizations of QM considered and how they > appear naturally in a different approach: > gr-qc/9706069 This looks like an interesting paper. The symplectic formulation of QM might well be the one that generalizes to a full theory of quantum gravity. But even thatr would involve 'quantizing gravity'! >>>And now it seems that "quantizing gravity" and "constructing a quantum >>>theory of gravity" are used as (almost) equivalent! [quoted text clipped - 5 lines] > means ending up with GR in some limit! That this is considered > equivalent is precisely what I question! To construct the theory for the first time (and to motivate it later for each new generation of physics) one _must_ go the first way, since how can we guess what the right theory should look like if we didn't. To verify that the theory is indeed quantum gravity one needs to take the second path. Thus both paths are complementary, and give the same result in case of success. That none of the current approaches to quantum gravity is close to any sort of success is a different matter. Arnold Neumaier Quote from physicsforum/Haelfix: "What I'm looking for is somewhat similar to the questions the original poster asked.. Namely, why should this be a unique prescription mathematically. Are there theorems that guarentee uniqueness, at least up to some level of rigid assumptions. In fact, even a good review of what motivated the details of the prescription historically would be useful. I'm curious as to why the LQG people feel they are justified in using the relaxed poisson bracket formalism.. Surely there was a wide range of literature premedidating the result, at least to justify it in toy problems.." Haelfix, that's about the question I'm asking, though my preconception on the last bit is rather: Why shouldn't they? I'm not just concerend with uniquness as in Stone von Neumann, (which incidently I hadn't heared about, part of what I was asking for in my original post! Thanks for the pointer!) but the physical basis of this all (which of course should be hidden in the formalism, if that formalism were exhaustive). I would rather ask why not to quantize in a different way if it preserves symmetries and superposition principle. Arnold Neumaier, NRQED and NRQCD are not exactly relevant here are they, as I'm asking for classical limits, not non-relativistic limits. >>>Quantization just means constructing a quantum theory for a theory >>>of which we already know what is the classical limit >> the dirac field that serves to quantize the electron >> has no classical meaning > It only needs to have a qunatum meaning. Could you explain this apparent change of mind? We take a classical theory, a non abelian gauge field, or the dirac field. We quantize it, and the result is a quantum theory which classical limit looks nothing like the classical theory we started out with. And I'm not interessted in pertubation theory. As a matter of fact the to first order mentality made me sick during my QFT lectures. There is physics out there that is not scattering and is highly non perturbative, especially in QCD! Quark quark scattering is not very exciting, confinement is. Can you get confinement from a correction to the classical non abelian gauge field theories? I don't know, I can't check, but I've been told no, repeatedly. What then is our reason then for using the initial theory if not it's symmetries? (And isn't there a bad case of considering those things important we can calculate and just ignoring the rest we can't deal with in QFT? The fact that one of the questions routinely asked for success of QG is graviton graviton scattering kind of says so to me, that's the only thing we can calculate in QFT so it's all we ask for anymore. Most of physics is bound states not scattering! Or is there a theorem that says that the full dynamics of a theory can be recovered from the SMatrix?) > To construct the theory for the first time (and to motivate it later > for each new generation of physics) one LaTeX graphic is being > generated. Reload this page in a moment. the first way, > since how can we guess what the right theory should look like if > we didn't. Well we could try considering why the classical theory looks how it looks, what principles underly quantum mechanics and combine the symmetries and physical principles of each theory to give a new one. Blindly using a prescription because it worked so far never looks like good physics (or even good science) to me. There is a historical example as well. Einstein didn't arrive at GR by Special-Relativizing Newtonian Gravity, but by looking at the equivalence principle and at the Lorentz group and putting them together. In particular going back to the LQG qunatization, rephrasing the question again what I wonder is, if it fails normal qunatization of NRQM, why so? What physics does it miss? --- frank > NRQED and NRQCD are not exactly relevant here are they, as I'm asking > for classical limits, not non-relativistic limits. Oh, sorry, I didn't read carefully enough. The classical limit is the theory defined by taking the Lagrangian occuring in the functional formalism and making the corresponding action stationary. Note that a functional integral is an integral in which all fields have classical meaning. The quantum interpretation comes from taking the functional integral as a generating functional for S-matrix elements, while the classical interpretation comes from taking a saddle point approximation. Thus the classical limit of the standard model is a mathematically well-defined theory, while the quantum version is only perturbatively defined, which means, it is mathematically undefined - even for QED. Nevertheless, the renormalization prescription make at least the coefficients of the asymptotoc series in hbar well-defined, which is all particle physics use to extract approximate physical information. In this relaxed sense, the quantum standard model is also well-defined. > There is physics out there that is not scattering and is highly non > perturbative, especially in QCD! Quark quark scattering is not very > exciting, confinement is. Can you get confinement from a correction to > the classical non abelian gauge field theories? I don't know, I can't > check, but I've been told no, repeatedly. Not at present. But this is likely due to our lack of understanding, and not to the wrong action. > What then is our reason then for using the initial theory if not it's > symmetries? It is the fact that it correctly describes the high energy scattering. We would not use it if that were in disagreement with experiment. > (And isn't there a bad case of considering those things important we > can calculate and just ignoring the rest we can't deal with in QFT? We can only do the things we understand. That's why they are important. If we could do the rest, it would be very important, too, but at present, it just means that there are very important open problems. If you find a way to _not_ ignore the rest we can't deal with in QFT and get experimentally verifiable predictions, you'll become famous... > The fact that one of the questions routinely asked for success of QG > is graviton graviton scattering kind of says so to me, that's the only > thing we can calculate in QFT so it's all we ask for anymore. Yes. We can ask for more, but no one know how possibly to get an answer. > Most of > physics is bound states not scattering! Or is there a theorem that > says that the full dynamics of a theory can be recovered from the > SMatrix?) There are hints, no theory. Bound states are supposed to be poles of the S-matrix, and Bethe-Salpeter equations for the bound state dynamics can be obtained approximately from resumming infinite familis of Feynman diagrams. See Chapter 14 of Weinberg's QFT I and the section on 'Bound states in relativistic QFT' in my theoretical physics FAQ at http://www.mat.univie.ac.at/~neum/physics-faq.txt > Well we could try considering why the classical theory looks how it > looks, what principles underly quantum mechanics and combine the > symmetries and physical principles of each theory to give a new one. Doing exactly this lead to the standard model. > Blindly using a prescription because it worked so far never looks like > good physics (or even good science) to me. People successfully matching theory and reality never use prescriptions blindly. > There is a historical example as well. Einstein didn't arrive at GR by > Special-Relativizing Newtonian Gravity, but by looking at the > equivalence principle and at the Lorentz group and putting them > together. Well, this is how one puts it today. But in fact, to arrive at the latter, Einstein must first have done the former. In MTW, it is shown how this naturally leads to standard general relativity... Arnld Neumaier > > > NRQED and NRQCD are not exactly relevant here are they, as I'm asking [quoted text clipped - 8 lines] > while the classical interpretation comes from taking a saddle point > approximation. This is not correct AFAIK. The classical limit of QED is NOT the QED Lagrangian. Classically there is no Dirac field! Apparently things become worse in Non Abelian GFT. AFAIK the classical quark field does not form protons and so on. So going back to my original point, constructing a quantum theory with the correct classical limit is a valid argument for the NRQM but fails for QFTs where it simply isn't true anymore. In fact it was an open question for some time whether Non Abelian Gauge Field Theories described physics until confinement was discovered which is manifestly important in the correct classical limit, but not present in the classical gauge field theories. >From what I understand asymptotic freedom implies that paths far from the classical saddle point contribute heavily to the path integral. > Not at present. But this is likely due to our lack of understanding, > and not to the wrong action. I'm not implying wrong action. That's not my argument/question. My question is really this: What physical reasons are there to believe that "standard" quantization is the right method to construct Quantum theories? The usual argument is the classical limit which is not accurate anymore as far as I can see. > > (And isn't there a bad case of considering those things important we > > can calculate and just ignoring the rest we can't deal with in QFT? > > We can only do the things we understand. That's why they are important. Hmmm... I would disagree. That's the kind of thinking that delayed the development of non linear physics. What is important is decided by the physics, not our mathematical difficulties. > > The fact that one of the questions routinely asked for success of QG > > is graviton graviton scattering kind of says so to me, that's the only > > thing we can calculate in QFT so it's all we ask for anymore. > > Yes. We can ask for more, but no one know how possibly to get an answer. Or just for something different really. Why always scattering (asked the disgruntled undergrad student ;) > 'Bound states in relativistic QFT' in my theoretical physics FAQ at > http://www.mat.univie.ac.at/~neum/physics-faq.txt Thanks for the pointer. That was a bit of a tangent, not real the core point of my question here. > > Well we could try considering why the classical theory looks how it > > looks, what principles underly quantum mechanics and combine the > > symmetries and physical principles of each theory to give a new one. > > Doing exactly this lead to the standard model. At least from my limited POV not. The principles underlying QM are never examined. We just "do quantization" again. Now that would probably be a fair roundabout way to summarize my problem: What is the pysical significance of standard quantization? If Thiemann's quantization method is indeed different, what are the differences in physics implied? > > Blindly using a prescription because it worked so far never looks like > > good physics (or even good science) to me. > > People successfully matching theory and reality never use prescriptions > blindly. Yet Thiemann got critizised heavily for deviating from standard quantization (at the String Coffee Table, check my link in my earlier post), and suggesting that it was up to experiment to decide what method was right. > > There is a historical example as well. Einstein didn't arrive at GR by > > Special-Relativizing Newtonian Gravity, but by looking at the [quoted text clipped - 3 lines] > Well, this is how one puts it today. But in fact, to arrive at the latter, > Einstein must first have done the former. Must he? And even if that was the way he proceeded technically wasn't he mostly guided by an understanding, or an idea, that the equivalence principle should be the defining element of Newtonian Gravity to carry over? And that is where I find things lacking at least from my POV today. Everybody is trying to make an established algorithm work without looking at the physical principles of this algorithm! Instead of trying to make the standard quantization algorithm work with the equations of GR we could for example try to create (from scratch or from wherever you like) a theory where the superposition principle holds and we have diffeomorphism invariance/general covariance. These are both physical principles as opposed to formal algorithms. Only we have no clue how to go about creating such a theory, do we? Possibly because we never actually studied how to construct quantum theories except for the standard quantization? With the few pointers I got here and at the physics forum I already have a clearer picture on why quantization is considered to be without alternative by so many. So the picture is clearing up a lot already... cheers, frank >>The classical limit is the >>theory defined by taking the Lagrangian occuring in the functional [quoted text clipped - 6 lines] >> > This is not correct AFAIK. You can check it directly for any nonrelativistic field theory. But it works in general. > The classical limit of QED is NOT the QED Lagrangian. > Classically there is no Dirac field! Classically there is an electron density field W(x,p) given by the Wigner transform of Psi(x)Psi(y), where Psi(x) is the classical Grassmann field occuring in the Lagrangian, satisfying a Dirac equation with an electromagnetic interaction added. This field is measurable and plays a role in semiconductor modeling. Psi(x) itself has no classical meaning as an anticommuting field, but has the numerical advantage that it is a field in 3 instead of 6 variables. One could write the classical action directly in terms of W(x,p), but then one would have to use a nonstandard variational principle since not all variations of W(x,p) are allowed. > Apparently things become worse in Non Abelian GFT. AFAIK the classical > quark field does not form protons and so on. This doesn't mean much. Quantum properties need not be preserved in the classical limit. For example, the classical hydrogen atom doesn't have a discrete part in its spectrum. But bound states are just what comes out of discrete spectra... > So going back to my > original point, constructing a quantum theory with the correct > classical limit is a valid argument for the NRQM but fails for QFTs > where it simply isn't true anymore. But this is not how things are done. There was never a classical theory of strong interactions, so there was never a point quantizing the given classical theory. The theory of strong interactions was modeled as a quantum theory; but for computational reasons one can only work close to the classical limit. Thus one looks at the quantum theory and takes its classical limit by letting hbar go to zero, and one obtains exactly what I told you: Since the k loop contributions scale with hbar^k, they disappear in the classical limit, so only the tree diagrams are left, which correspond to the saddle point approximation in the functional integral. > In fact it was an open question for some time whether Non Abelian > Gauge Field Theories described physics until confinement was > discovered which is manifestly important in the correct classical > limit, but not present in the classical gauge field theories. If you'd follow this reasoning for nonrelativistic QM, the classical limit of the N-particle Schr"odinger equation would not be the N-particle Newton dynamics, but a complicated dynamics in terms of bound states. This is _not_ what is commonly called the classical limit. What your intuition amounts to is rather a coarse-grained limit, giving an effective classical theory. > I'm not implying wrong action. That's not my argument/question. My > question is really this: What physical reasons are there to believe > that "standard" quantization is the right method to construct Quantum > theories? The usual argument is the classical limit which is not > accurate anymore as far as I can see. The classical limit was never a strong argument; it was just the starting point in the early days of QM. The correct arguments are given in Weinberg's QFT book, Volume I, Chapters 3-7. One needs a classical action to be able to implement unitarity of the S-matrix and the cluster decomposition. The first is essential for a correct probabilistic interpretation of QFT, since it amounts to preservation of probability, and the second is necessary to account for the fact that all our experiments are done locally, and what is far away does not contribute significantly except theough effectively classical far fields. >>We can only do the things we understand. That's why they are important. > > Hmmm... I would disagree. That's the kind of thinking that delayed the > development of non linear physics. What is important is decided by the > physics, not our mathematical difficulties. This is not true. Celestial mechanics was always nonlinear. Nonlinearities in field theories were known since Euler. What delayed the development of nonlinear physics was only the lack of numerically accessible techniques to exploit the nonlinearity. So one had to be content with linearizing everything. This changed as soon as computers became available on a significantly wide scale. > Or just for something different really. Why always scattering (asked > the disgruntled undergrad student ;) Because scattering is what is macroscopically left of microscopic processes. The only other thing is bound states - and this is indeed poorly understood in QFT, though people are working on it using whatever approximation tricks they can come up with. None of them is neat, so it is unlikely that what they do is the last word. >>>Well we could try considering why the classical theory looks how it >>>looks, what principles underly quantum mechanics and combine the [quoted text clipped - 4 lines] > At least from my limited POV not. The principles underlying QM are > never examined. We just "do quantization" again. This is like saying, 'The principles underlying mathematics are never examined', upon the observation thateveryone uses them uncritically. Both statements are false - these principles are examined by a minority, as it should be. The majority can be content with the fact that the principles work well, and just use them. >>People successfully matching theory and reality never use prescriptions >>blindly. [quoted text clipped - 3 lines] > post), and suggesting that it was up to experiment to decide what > method was right. Well, he is not yet successful. As long as ideas are tentative and not validated by experiment, they are always hard to defend. Success comes late - either with a triumphal experimental verification, or if people realize that a new way is significantly simpler than the tradition. If neither happens, people will stick to the tradition, except for a minority who lives from exploring the consequences of the idea. Innovative research is always a risky business - one must be prepared to continue one's work no matter how much it is criticied, but one must also learn as much as possible from one's critics. Then - if it is indeed the right track - success will come sooner or later. But who knows beforehand what will turn out to be the right track? So people have a right to be critical... >>>There is a historical example as well. Einstein didn't arrive at GR by >>>Special-Relativizing Newtonian Gravity, but by looking at the [quoted text clipped - 8 lines] > principle should be the defining element of Newtonian Gravity to carry > over? Well, it would be the task of a historian to tell a coherent and realistic story of how he really got his understanding. Judging from my own experience, understanding is not something that springs into one's head without preparation, but is the result of walking attentively and openminded along many blind alleys, until one sees one which smells like being the real thing. Then one starts grinding away in this direction, and in this process discovers what should have been the guiding principle that would have avoided all the dead ends, bringing one directly to the goal. Then, and only then, the right understanding governs the remainder of the search. This is not only my personal experience but seems to be the general pattern: See G. Polya, Mathematical Discovery, John Wiley and Sons, New York, 1962. So I am guessing that Einstein just took the standard pedestrian way, and somewhere along it discovered that everything becomes neat and simple if he makes the equivalence principle the guideline of his reasoning. > And that is where I find things lacking at least from my POV today. > Everybody is trying to make an established algorithm work without > looking at the physical principles of this algorithm! Well, if you are right, and are the only one seeing the right way to approach the problem, why don't you go ahead and do it better? Acording to your assessment you have essentially no competition, and all the answers lie in store for him who finds the key... > Instead of trying to make the standard quantization algorithm work > with the equations of GR we could for example try to create (from > scratch or from wherever you like) a theory where the superposition > principle holds and we have diffeomorphism invariance/general > covariance. These are both physical principles as opposed to formal > algorithms. There is a big difference between trying to do something and achieving it. What you want to do is like seeing all the problems of modern society and saying, 'Why not just build a new society from scratch?' people have tried and ended up no better, usually worse. Real improvements come from patiently building upon the best of what already exists, being open-minded but critical about new possibilities, and trying to integrate what looks most promising. Arnold Neumaier > Classically there is an electron density field W(x,p)... I think the major problem here is that we have different conceptions of the classical limit. I think with the way you consider the classical limit the statement that the Lagrangean is indeed the classical limit is just tautological. To me it appears that the classical limit of QED would be a point charge particle together with maxwells equations as described by the simple action: S = Int m dtau + Int F^mu^nu F_mu_nu + Int J^mu A_mu This is the classical theory which one aims to quantize with QED. (After all there are "two quantizations" from here to QED via the Dirac field, many argue that there is no such thing as two quantizations and I tend to find their arguments convincing but then we do indeed not start with the classical limit!) After all we are looking at transition propabibilities between particle states. Quantization implies the big conceptual step of rendering fields into particles of differing properties, which is of course why we quantize the electron using the dirac field and end up with a quantum particle again. (I am aware that this sentence is perhaps not entirely meaningfull since of course in QM we don't have strictly seperated concepts of field and particle to begin with). I think non perturbative quantum effects certainly do contribute to the classical, decoherent, newtonian, particle behaviour of the theory. > The correct arguments are given in Weinberg's QFT book, Volume I, > Chapters 3-7. One needs a classical action to be able to implement > unitarity of the S-matrix and the cluster decomposition. I know, I am still studying this book, and it is precisely the kind of thing I ask for. If there is more detailed work in this direction, and so on. > >>We can only do the things we understand. That's why they are important. > > [quoted text clipped - 8 lines] > So one had to be content with linearizing everything. This changed as > soon as computers became available on a significantly wide scale. That's a rather beneficial view of history. There are numerous quotes that reflect the opinion widely held until the middle of the 20th century that the non linear case was just an aberration of the linear case. That by having understood the linear case we had captured the essence of physics. I would rather interpretate the numerical side of things the other way around: Until people were directly confronted by it through numerical simulations, and by it's relevance for weather systems for example, no one bothered to study it in depth. Selective perception, what we can do is what is important until proven otherwise. > Both statements are false - these principles are examined by a minority, > as it should be. The majority can be content with the fact that > the principles work well, and just use them. Well then, that goes way back to my initial question! Please point me towards this minority, their results and text books (if existent)! That's basically what I was asking for! I have already, through this and related discussion and some other things, discovered a host of things never mentioned in any of my QM or QFT text books, particularly the Stone von Neumann Theorem and Diracs book "Lectures on QM". Besides the absence of this questions in standard literature, they also seem absent in some discussions I have seen around here and on the String Coffe Table. I have seen there the argument that modifying the standard QFT quantization procedure (as done in QLG) basically amounts to throwing away QM, or at least a deep modification of QM, without any mention as to what physical principles are encoded in the process of quantization and which of them are violated by the relaxed quantization. QLG people argue (implicitly) that a lot less of the details of quantization is physical and, due to the experimental success of QM/QFT, neccesary then String theorists who argue that we have to keep it intact. In generally it is accepted that the different quantization leads to a different physics, but how does the physics differ? I find no reference to that in the discussion. > Innovative research is always a risky business - one must be prepared > to continue one's work no matter how much it is criticied, but one must > also learn as much as possible from one's critics. But that is precisely what I'm asking for. See above. Are they right in saying that this is a mayor radical change in the theory? Or is it rather an unphysical technicality? This is a valid question even in the absence of a decisive anwser, for it can well influence the standings of the theory, the amount of work invested in it, and so on. If the criticism is valid it needs to be addressed of course. > So I am guessing that Einstein just took the standard pedestrian > way, and somewhere along it discovered that everything becomes neat and > simple if he makes the equivalence principle the guideline of his > reasoning. Reasonable enough. The question I'm asking is if we start our blind journey from the right point, or if we fail so far because we already took a wrong turn a while ago. (Here comes the obligatory Einstein quote...) "The mere formulation of a problem is far more essential than its solution, which may be merely a matter of mathematical or experimental skills. To raise new questions, new possibilities, to regard old problems from a new angle requires creative imagination and marks real advances in science." - Albert Einstein That's kind of what eats me when we keep asking for an SMatrix, perhaps that's the wwrong question, perhaps new questions need to be asked... > > And that is where I find things lacking at least from my POV today. > > Everybody is trying to make an established algorithm work without [quoted text clipped - 4 lines] > Acording to your assessment you have essentially no competition, > and all the answers lie in store for him who finds the key... I plan to, I am, however, well aware of my place at the moment. That place is about a couple of years away from having learned enough to do actuall research. Nevertheless along the way to the frontier things irk me left and right and about things people have done before me and continue doing which I don't understand, or which, to me, seem questionable. Then I ask. Usually presuming that someone very clever in the past has had the same objections and questions I raise now and solved them, and that there are people out there who have all the experience I lack and know about this and can tell me about it, and thus I advance a step further on the path to the frontier. Thanks for taking the time to anwser and discuss all this stuff :) --- thanks, Frank Hellmann >>Classically there is an electron density field W(x,p)... > > I think the major problem here is that we have different conceptions > of the classical limit. Yes. I take a classical limit to be what it says, namely a limit hbar to 0. > I think with the way you consider the > classical limit the statement that the Lagrangean is indeed the > classical limit is just tautological. No. It is, e.g., the nontrivial statement that in a path integral of exp(-iS/hbar), only the path of least action contributes as hbar goes to zero. Or a number of other statements like that which recover something classical from something quantum when hbar to 0, e.g. ERhrenfest's theorem. > To me it appears that the classical limit of QED would be a point > charge particle together with maxwells equations as described by the > simple action: > S = Int m dtau + Int F^mu^nu F_mu_nu + Int J^mu A_mu > This is the classical theory which one aims to quantize with QED. But this is not the classical limit, as you can easily convince yourself. QED is _not_ classical electrodynamics canonically quantized!!! (Classical electrodynamics with point particles is not even a consistent theory.) >>The correct arguments are given in Weinberg's QFT book, Volume I, >>Chapters 3-7. One needs a classical action to be able to implement [quoted text clipped - 3 lines] > thing I ask for. If there is more detailed work in this direction, and > so on. I haven't seen more on this that would be of any substantial value for the question at hand. >>Celestial mechanics was always nonlinear. >>Nonlinearities in field theories were known since Euler. [quoted text clipped - 8 lines] > case. That by having understood the linear case we had captured the > essence of physics. Look at Poincare's work to convince yourself of the opposite. If some people have the opinion you state, this does not yet mean that it was consensus in physics. Progress in physics is always the work of a minority. >>Both statements are false - these principles are examined by a minority, >>as it should be. The majority can be content with the fact that [quoted text clipped - 3 lines] > towards this minority, their results and text books (if existent)! > That's basically what I was asking for! Please restate your initial question, then, in the light of the current discussion. I don't remember what it was... > I have already, through this and related discussion and some other > things, discovered a host of things never mentioned in any of my QM or > QFT text books, particularly the Stone von Neumann Theorem and Diracs > book "Lectures on QM". Text books are supposed to give an easy road into modern physics, and not a thorough discussion of the fundamentals. For special questions one must go to special sources... > I have seen there the argument that modifying the standard QFT > quantization procedure (as done in QLG) basically amounts to throwing [quoted text clipped - 8 lines] > different physics, but how does the physics differ? I find no > reference to that in the discussion. I can't comment on that since I avoid going deeply into work that has no contact with experiment. From what I have seen (at a somewhat superficial level of understanding) I trust neither string theory nor LQG to be close to the truth. >>Innovative research is always a risky business - one must be prepared >>to continue one's work no matter how much it is criticied, but one must [quoted text clipped - 3 lines] > in saying that this is a mayor radical change in the theory? Or is it > rather an unphysical technicality? Critics usually just present a statement, or point to an incoherence in an opponent's statement. To learn from it is a nontrivial task, since it means that one has to find out a) how to make the criticism strongest, in a constructive sense, and b) how to defend the original statement. Finding this out is learning from it. >>So I am guessing that Einstein just took the standard pedestrian >>way, and somewhere along it discovered that everything becomes neat and [quoted text clipped - 4 lines] > journey from the right point, or if we fail so far because we already > took a wrong turn a while ago. Everyone starts their journey from where they are, in the direction they find most promising. The others observe what they do and have to make up their own mind. If people knew what is the right start and the right direction, QG were solved by now. > (Here comes the obligatory Einstein quote...) > > "The mere formulation of a problem is far more essential than its > solution, which may be merely a matter of mathematical or experimental > skills. Here he means 'numerical solution', while with 'formulation' he means 'theory'. > To raise new questions, new possibilities, to regard old > problems from a new angle requires creative imagination and marks real [quoted text clipped - 4 lines] > perhaps that's the wrong question, perhaps new questions need to be > asked... Flat Minkowski QFT gives no more than the S-matrix, so if you want to ask for more, maybe you'd first ask it on the level of QED. For example, what is the QED dynamics at finite times? The S-matrix only tells you what happens at time t=+inf, given knowledge about t=-inf. Realizing this and knowing of the big bang, it is perhaps understandable why the quest for a quantum gravity S-matrix must have failed... >>Well, if you are right, and are the only one seeing the right way to >>approach the problem, why don't you go ahead and do it better? [quoted text clipped - 10 lines] > solved them, and that there are people out there who have all the > experience I lack and know about this and can tell me about it, The frontier is the frontier because there is no clear understanding of what is beyond. Those who had the questions and found real answers published it and andvanced the state of the art. The others can only share their experience and their chart of the uncharted territory. As you can see from the conflicting opinions, these charts are not reliable. You need to learn to see with your own eyes, take your own risks, and find out for yourself what can be trusted. There are no guides beyond a certain point. Arnold Neumaier > No. It is, e.g., the nontrivial statement that in a path integral of > exp(-iS/hbar), only the path of least action contributes > as hbar goes to zero. Ah but I wonder if that would be true for a non abelian Y-M Field? I would find it surprising if such a theorem would hold, much more if it were proven. > > To me it appears that the classical limit of QED would be a point > > charge particle together with maxwells equations as described by the [quoted text clipped - 8 lines] > (Classical electrodynamics with point particles is not even a > consistent theory.) If I may quote your initial reply: "Quantization just means constructing a quantum theory for a theory of which we already know what is the classical limit (as it happens for general relativity or N-particle mechanics). No matter how you do it, you must come up with a Hilbert space and a dynamics, of which you can prove that it reduces in the classical limit to what is known and already well tested." Well tested implies relation to experiment. As you stated yourself QED is not the quantization of well tested classical electromagnetism but of an unphysical classical limit involving the un observable thereby untestable Dirac field. And this of course goes to the heart of my questions. > Please restate your initial question, then, in the light of the > current discussion. I don't remember what it was... I think what has become clear to me is that there is no simple question to ask here, except the very general inquiry about the physical(mathematical) meaning of quantization, and why people have so much faith in it. > Text books are supposed to give an easy road into modern physics, > and not a thorough discussion of the fundamentals. For special questions > one must go to special sources... That's one of the things I asked for, where to find them... (a short cut of "Go to the library and go through the back catalogue of the usual journals...") > > That's kind of what eats me when we keep asking for an SMatrix, > > perhaps that's the wrong question, perhaps new questions need to be [quoted text clipped - 3 lines] > ask for more, maybe you'd first ask it on the level of QED. For example, > what is the QED dynamics at finite times? I can't remember where I read it but the quote went something like: "If a theoretical physicist can't solve a problem he will invariably attempt the next more difficult one." - some mathematician. ;) > The S-matrix only tells you what happens at time t=+inf, given knowledge > about t=-inf. Realizing this and knowing of the big bang, it is perhaps > understandable why the quest for a quantum gravity S-matrix must have > failed... Indeed! > You need to learn to see with your own eyes, take your own risks, > and find out for yourself what can be trusted. There are no guides > beyond a certain point. There's much to learn before I get to that point. But reaching it is my main motivation. --- frank >>No. It is, e.g., the nontrivial statement that in a path integral of >>exp(-iS/hbar), only the path of least action contributes [quoted text clipped - 3 lines] > would find it surprising if such a theorem would hold, much more if it > were proven. My statement is simply saddle point integration, well-known in asymptotic analysis. It is independent of the form of the action, and holds in general. >>>To me it appears that the classical limit of QED would be a point >>>charge particle together with maxwells equations as described by the [quoted text clipped - 24 lines] > untestable Dirac field. > And this of course goes to the heart of my questions. Quantixation of a classical theory should recover the classical theory as a classical limit. But the classical limit can exist even without an underlying well-tested classical theory. This is the case for QED. Here the classical theory which is second-quantized is _not_ classical electrodynamics but the theory whose classical action is given by the QED Lagrangian. To arrive from QED at classical electrodynamics one has to do more than just the classical limit. One also needs to get rid of antiparticles and Fermions by performing a nonrelativistic limit of the matter part. Then one gets indeed classical electrodynamics. > I think what has become clear to me is that there is no simple > question to ask here, except the very general inquiry about the > physical(mathematical) meaning of quantization, and why people have so > much faith in it. The faith comes from the success of the standard model. So people believe that the structure of the standard model and its subtheories is somehow important to quantum mechanics. But quantization itself is a very vague concept, and needs to be qualified to have a definite meaning - and this can be done in different ways. >> For special questions >>one must go to special sources... >> > That's one of the things I asked for, where to find them... (a short > cut of "Go to the library and go through the back catalogue of the > usual journals...") If your questions are specific enough, you can hope to get specific answers. >>>That's kind of what eats me when we keep asking for an SMatrix, >>>perhaps that's the wrong question, perhaps new questions need to be [quoted text clipped - 9 lines] > - some mathematician. > ;) You wanted new questions to be asked, that don't center around the S-matrix. I gave an important example which should not be more difficult than quantum gravity. Though probably not much easier either... Arnold Neumaier > >>>Quantization just means constructing a quantum theory for a theory > >>>of which we already know what is the classical limit > > >> the dirac field that serves to quantize the electron > >> has no classical meaning > > > It only needs to have a qunatum meaning. > Could you explain this apparent change of mind? > We take a classical theory, a non abelian gauge field, or the dirac > field. We quantize it, and the result is a quantum theory which > classical limit looks nothing like the classical theory we started out > with. Sorry to "barge in" but I have been asking myself questions very similar to the ones Mr Hellmann is asking so this thread struck me. First of all, I think his last question to Mr Neumaier above (Could you explain this apparent change of mind) his a good one which, afaik has not been answered yet. I have been bothered by the usual QFT approach of quantizing classical fields for a long time now (see the recent thread of over 90 posts on physicsforums.com!) and Weinberg's book has been a gift from the gods to me (well, from *a* god :-)! I am talking here about the KG field, the Dirac field, the Y-M fields, etc. All those weird fields that were postulated to exist (they certainly had not been observed as classical fields experimentally!) and which were used as the *starting point* of QFT. Now, I understood the analogy to E&M, but it all seemed like a huge leap of faith. But Weinberg has clarified important points. He does it the way I have always wanted to see it done: he starts only from the need for a theory which allows the number of particles to vary and then he imposes the obvious conditions (cluster decomposition, unitarity, locality, Lorentz invariance, etc). And everything that is usually taken as postulated starting point, the concept of fields, the Lagrangian and the wave equations pop up as consequences. A field is only introduced as a way to write linear combinations of creation/annihilation operators that have specific transformation properties. There is no need whatsoever for a classical field in that approach! And it makes much more sense to me. For years, I have been asking "but what are those crazy classical fields that we start with?" And people have basically answered that they were "obviously" needed and that this was the "obvious" way to introduce an infinite number of degrees of freedom (incorrect, since one can write combination of any number of creation/annihilation operators without introducing fields...the correct explanation is a question of Lorentz invariance, not degrees of freedom). So I now think of the quantum fields as simply an efficient way to group creation/annihilation operators in groupings that have specific properties under Lorentz transformations. That these groupings have the same form as classicla fields whose amplitudes would have been turned into operators is an aside, the way I see it now. It does not imply that the corresponding fields with operators replaced by c-numbers are meaningful. I am still open to the idea that they *might* be real in some sense (in a coherent state sense) or that at least they are *useful* (like to calculate solitons type configurations like instantons, etc) but for now they sound more like simple book keeping devices to regroup > I have been bothered by the usual QFT approach of quantizing classical > fields for a long time now (see the recent thread of over 90 posts on [quoted text clipped - 20 lines] > For years, I have been asking "but what are those crazy classical > fields that we start with?" They are classical in the sense that they describe something in terms of classical mechanics theory, but not in terms of classical physics. Classical physics is the physics of processes slowly varying in space and time; of course elementary particles do not belong there. But classical mechanics can also be considered as an abstract mathematical framework for dynamics in phase spaces that has much wider applicability. The classical fields that make up the fields going into the path integral belong in this sense to classical mechanics. Arnold Neumaier > So... What's the general opinion on quantization? Ah I just found a most interessting quote by John Baez in his introductory category text: "There is a famous saying about quantization due to Edward Nelson: "First quantization is a mystery, but second quantization is a functor!" No one is a true mathematical physicist unless they can explain that remark. So, let me explain that remark! First quantization is a mystery. It is the attempt to get from a classical description of a physical system to a quantum description of the "same" system. Now it doesn't seem to be true that God created a classical universe on the first day and then quantized it on the second day. So it's unnatural to try to get from classical to quantum mechanics. Nonetheless we are inclined to do so since we understand classical mechanics better. So we'd like to find a way to start with a classical mechanics problem -- that is, a phase space and a Hamiltonian function on it -- and cook up a quantum mechanics problem -- that is, a Hilbert space with a Hamiltonian operator on it. It has become clear that there is no utterly general systematic procedure for doing so. Mathematically, if quantization were "natural" it would be a functor from the category whose objects are symplectic manifolds (= phase spaces) and whose morphisms are symplectic maps (= canonical transformations) to the category whose objects are Hilbert spaces and whose morphisms are unitary operators. Alas, there is no such nice functor. So quantization is always an ad hoc and problematic thing to attempt. A lot is known about it, but more isn't. That's why first quantization is a mystery." --- frank > Mathematically, if quantization were "natural" it would be a functor > from the category whose objects are symplectic manifolds (= phase [quoted text clipped - 4 lines] > attempt. A lot is known about it, but more isn't. That's why first > quantization is a mystery." Do don't really need a symplectic manifold, do we? We can do with a Poisson structure alone (and especially every symplectic structure induces a Poisson structure) and carry out deformation quantization for instance. Ah, btw. is this really more general than we need in practise or are there examples of classical phase spaces coming along with a Poisson-structure we wnat to deform but not with a symplectic one in physics? Seems odd to me 'cause the symplectic structure is central to classical physics, isn't it? So, was Poisson geometry and deformation quantization invented for the sake of itself resp. for it's mathematical beauty or was there some physical drive behind this too? If I make any errors caused by conceptual misunderstanding and this question is dumb, so please let me know too. Best regards, Florian. >>Mathematically, if quantization were "natural" it would be a functor >>from the category whose objects are symplectic manifolds (= phase [quoted text clipped - 9 lines] > induces a Poisson structure) and carry out deformation quantization > for instance. This is not a morphism, since the multiplication of operators is also deformed. > Ah, btw. is this really more general than we need in > practise or are there examples of classical phase spaces coming along > with a Poisson-structure we wnat to deform but not with a symplectic > one in physics? One could want to quantize a spinning top - which has no symplectic description. > Seems odd to me 'cause the symplectic structure is > central to classical physics, isn't it? Poisson structures are important in classical physics. There is a nice book by Marsden and Ratiu about classical mechanics in terms of Poisson structures. Arnold Neumaier > >>Mathematically, if quantization were "natural" it would be a functor > >>from the category whose objects are symplectic manifolds (= phase [quoted text clipped - 12 lines] > This is not a morphism, since the multiplication of > operators is also deformed. Hm, good point. Wouldn't is be "physically unreasonable" if it would be a morphism? As symmetries garantee the existence of automorphisms besides the identity and ASSUMED these automorphisms are "mapped" to automorphims between the quantizised model different from the identity, anomalies and other effects concerning symmetries important to qunatum physics could never occure, could they? (Although there is the possibility to sidestep this problem if the assumed functor beween morphisms in the classical theory and the quantum theory is "degenerate" enough to make the correspondence between the classical and the quantum theory break down, this should not occure in general, I think, but I have not enough knowledge to rule out this possibility). However, the main line of my reasoning here is that the fact that quantization is NOT a functor should not be seen as a bad mathematical accident but has a physical meaning. (In deformation quantization the quantization of morphisms between Poisson-manifolds can not be performed in general and the situation in the prescence of symmetries is not fully understood yet, as far as I know.) > Poisson structures are important in classical physics. > There is a nice book by Marsden and Ratiu about classical mechanics > in terms of Poisson structures. Thank you for the hint. Florian. The question of whether it is possible to construct several differing quantum theories having the same classical limit is interesting. If you attempt to quantize gravity by considering a path integral over all possible space-time geometries, you must specify which such geometries to allow. Should it be possible to define a causal structure on them? Can baby-universes branch off as "time" progresses? If so, does the fact that the curvature becomes singular at the point where they branch off matter, or does this mean that such geometries should not be included in the path integral? In some sense, we may consider geometries, which do not really have a classical counterpart - should these be included? And so on. So even if we just consider one way of quantizing a classical theory, i e the path integral, there are ambiguities, which can lead to radically different quantum theories with the same classical limit. It may simply be that there is not enough information in a classical theory to allow us to extract a unique quantum theory from it. > Ah I just found a most interessting quote by John Baez in his > introductory category text: [quoted text clipped - 3 lines] > functor!" No one is a true mathematical physicist unless they can > explain that remark. For those who want to understand this without being mathematical physicists: mystery = there is no definite algorithm but only intuition to guide one how to do it functor = there is a unique way of proceeding that works once you defined where you start. Second quantization is a well-defined prescription for going from a single-particle Hilbert space and operators to an indefinite-number-of-particles Hilbert space and operators, with nice properites. First quantization, in contrast, is a set of didactical cues to go from classical physics to quantum physics, a bridge for the newcomer without underlying substance. Unfortunately, second quantization is only a tool in finding the right quantum theories, and needs to be complemented by other stuff (primarily renormalization) before a workable QFT arises... Arnold Neumaier |
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