Three Mysteries

There’s a well-known list of high-profile problems in fundamental theoretical physics that have gotten most of the attention of the field during the past few decades (examples would be the problems of quantizing gravity, solving QCD, explaining dark energy, finding a model of dark matter, breaking supersymmetry and connecting it to experiment, etc.). Progress on these problems has been rather minimal, and in reaction one recent trend has organizations such as FQXI promoting research into questions that are much more “philosophical” (for instance, they are now asking for grant proposals to study “The Nature of Time”). In this posting I’d like to discuss a different class of problems, ones which I believe haven’t gotten anywhere near the attention they deserve, for an interesting reason.

The three problems share the characteristic of being apparently of a purely technical nature. The argument against paying much attention to them is that, in each case, even if one were to find a satisfactory solution, it might not be very interesting. It’s possible that all one would discover is that the conventional wisdom about these problems, that they’re just “technical” and thus not of much significance, is correct. The argument for paying more attention is that the technical problem may be an indication that we’re doing something wrong, that there is something of significance about the Standard Model that we haven’t yet understood. Achieving this understanding may lead us to the insight needed to successfully get beyond the Standard Model. At the moment all eyes are on the LHC, with the hope that experiment will lead to new insight. Whether this will work out is still to be seen, but in any case it looks like it’s going to take a few years. Perhaps theorists with nothing better to do but wait will want to consider thinking about these problems.

Non-Perturbative BRST

The BRST method used to deal with the gauge symmetry of perturbative Yang-Mills theory does not appear to generalize to the full non-perturbative theory, for a rather fundamental reason. This was first pointed out by Neuberger back in 1986 (Phys. Lett. B, 183 (1987), p337-40.), who argued that, non-perturbatively, the phenomenon of Gribov copies implies that expectation values of gauge-invariant observables will vanish. I’ve written elsewhere about a different approach to BRST that I’m working on (see here), which is still at a stage where I only fully understand what is going on in some toy quantum-mechanical models. My own point of view is that there’s still a lot of very non-trivial things to be understood about gauge symmetry in QFT and that the BRST sort of homological techniques for dealing with it are of deep significance. Others will disagree, arguing that gauge symmetry is just an un-physical redundancy in our description of nature, and how one treats it is a technical problem that is not of a physically significant nature.

One reaction to this question is to just give up on BRST outside of perturbation theory as something unnecessary. In lattice gauge theory computations, one doesn’t fix a gauge or need to invoke BRST. However, one can only get away with this in vector-like theories, not chiral gauge theories like the Standard Model. Non-perturbative chiral gauge theories have their own problems…

Non-perturbative Chiral Gauge Theory

Since the early days of lattice gauge theory, it became apparent that chiral symmetry was problematic on the lattice. One way of seeing this is that naively there should be no chiral anomaly on the lattice. The problem was made more precise by a well-known argument of Nielsen-Ninomiya. More recently, it has become clear that one can consistently introduce chiral symmetry on the lattice, at the cost of using fermion fields that take values in an infinite dimensional space. One such construction is known as “overlap fermions”, which have the crucial property of satisfying relations first written down by Ginsparg and Wilson. This kind of construction solves the problem of dealing with the global chiral symmetry in theories like QCD, but it still leaves unsolved the problem of how to deal with a gauged chiral symmetry, such as the gauge symmetry of the Standard Model.

Poppitz and Shang have recently written a nice review of the problem, entitled Chiral Lattice Gauge Theories Via Mirror-Fermion Decoupling: A Mission (im)Possible? They comment about the significance of the problem as follows:

Apart from interest in physics of the Standard Model — which, at low energies, is a weakly-coupled spontaneously broken chiral gauge theory that does not obviously call for a lattice study — interest in strong chiral gauge dynamics has both intensified and abated during the past few decades. From the overview in the next Section, it should be clear that while there exist potential applications of strong chiral gauge dynamics to particle physics, at the moment it appears difficult to identify “the” chiral theory most relevant to particle physics model-building (apart from the weakly-coupled Standard Model, of course). Thus, the problem of a lattice formulation of chiral gauge theories is currently largely of theoretical interest. This may or may not change after the LHC data is understood. Regardless, we find the problem sufficiently intriguing to devote some effort to its study.

In a footnote they compare two points of view on this: Creutz who argues that the question is important since otherwise we don’t know if the Standard Model makes sense, and Kaplan who points out that if there is some complicated and un-enlightening solution to the problem, it won’t be worth the effort to implement.

You can read more about the problem in the references given in the Poppitz-Shang article.

Euclideanized Fermions

Another peculiarity of chiral theories arises when one tries to understand how they behave under Wick rotation. Non-perturbative QFT calculations are well-defined not in Minkowski space, but in Euclidean space, with physical observables recovered by analytic continuation. But the behavior of spinors in Minkowski and Euclidean space is quite different, leading to a very confusing situation. Despite several attempts over the years to sort this out for myself, I remain confused, and can’t help suspecting that there is more to this than a purely technical problem. One natural mathematical setting for trying to think about this is the twistor formalism, where complexified, compactified Minkowski space is the Grassmanian of complex 2-planes in complex 4-space. The problem though is that thinking this way requires taking as basic variables holomorphic quantities, and how this fits into the standard QFT formalism is unclear. Perhaps the current vogue for twistor methods to study gauge-theory amplitudes will shed some light on this.

On the general problem of Wick rotation, about the deepest thinking that I’ve seen has been that of Graeme Segal, who deals with the issue in the 2d context in his famous manuscript “The Definition of Conformal Field Theory”. I saw recently that he’s given some talks in Europe on “Wick Rotation in Quantum Field Theory”, which makes me quite curious about what he had to say on the topic.

For some indication of why this confusion over Minkowski versus Euclidean spinors remains and doesn’t get cleared up, you can take a look at what happened recently when Jacques Distler raised it in related form on his blog here (he was asking about it in the context of the pure spinor formulation of the superstring). I’m not convinced by his claim that the thing to do is to go to Euclidean space-time variables, while keeping Minkowski spinors. Neither is Lubos, and he and Jacques manage to have an argument about this that sheds more heat than light. It ends up with Lubos accusing Jacques of behaving like Peter Woit, which lead to him being banned from commenting on the blog. While this all is, as Jacques describes it “teh funny”, it would be interesting to see a serious discussion of the issue. Since it in some sense is all about how one treats time, perhaps one could get FQXI funding to study this subject.

Update: Lubos Motl has immediately come up with a long posting explaining why these are all non-problems, of concern only to those like myself who are “hopeless students”, “confused by many rudimentary technicalities that prevented him from thinking about serious, genuinely physical topics.” If I would just understand AdS/CFT and Matrix theory I would realize that gauge symmetry is an irrelevance. Few in the theoretical physics community are as far gone as Lubos, but unfortunately he’s not the only one that thinks that concern with these “technicalities” is evidence that someone just doesn’t understand the basics of the subject.

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76 Responses to Three Mysteries

  1. Interested says:

    “Others will disagree, arguing that gauge symmetry is just an un-physical redundancy in our description of nature, and how one treats it is a technical problem that is not of a physically significant nature.”

    Interesting. Please elaborate on that argument.

  2. Trent says:

    Peter, you write,

    “More recently, it has become clear that one can consistently introduce chiral symmetry on the lattice, at the cost of using fermion fields that take values in an infinite dimensional space.”

    This is complete rubbish. Fermions in the overlap formulation have exactly the same number of components as for example fermions in the Wilson formulation. At each space time point they have 4 Dirac indices and 3 color indices.

    What you might be confusing with overlap fermions are domain wall fermions. However domain wall fermions do not solve the chirality problem exactly, there is a residual breaking. In the limit when the residual breaking goes to zero one recovers overlap fermions, which have, as stated above, the same number of components as conventional Wilson fermions. No surprise there, because overlap fermions are based on Wilson fermions, only the Dirac operator is not the Wilson matrix, but a function of it (related to the sign function, if someone was curious).

    Trent

  3. Trent says:

    You mentioned Distler’s blog, where I also commented and pointed out that the initial assertion of Distler is incorrect. He also admits this, so this issue is very clear and there is no confusion. All of this happened before Lubos entered the scene though 🙂

    Basically, Distler was claiming that in Minkowski space \eta_\mu^\mu = d – 2 whereas in Euclidean space \eta_\mu^\mu = d so he thought there was a problem somewhere.

    But actually there is no problem at all, as I pointed it out to him and he admitted to being wrong, since also in Minkowski space we have \eta_\mu^\mu = d and there is no minus 2 anywhere.

    This is basically elementary linear algebra or vector calculus or whatever you want to call it, at any rate part of QED 101 material. I was surprised to see Distler getting confused about such simple things, but good thing is he quickly admitted to being wrong. Gentlemanly, one could say.

    This is also my take on your issue. What is exactly wrong with Wick rotation and analytical continuation once it’s done properly (with positivity constraints)? In lattice gauge theory the constraint is called “reflection positivity”. Please state a specific problem or specific quantity one can not calculate in a Euclidean setting. Things like “oh, it’s so mysterious” doesn’t count 🙂

    Trent

  4. Peter Woit says:

    Trent,

    Apologies for the imprecise statement about the overlap formalism and the other ideas about how to deal with chirality that it grew out of. For a detailed explanation by an expert of the point I was referring to, see Neuberger’s hep-lat/9808036, where his summary of the situation goes:

    “At the base of this progress lies a world consisting of an infinite number of fermions, all but one having very large masses. This infinity is fully under control and consequently can be completely eliminated.”

    As for the Wick rotation question, I should make it clear that this “mystery” is of a different nature than the two others. It’s not that there’s a well-defined problem of non-existence of a needed non-perturbative construction (at least in flat space-time). It’s rather that I just don’t see a clear explanation of the relationship between the QFTs in Minkowski signature and Euclidean signature, especially when dealing with chiral fermions. What I’m looking for is something that treats the spinor geometry in a coordinate-invariant way, so would make sense even in a curved background. Maybe such a thing exists and I’m just unaware of it, but the literature I’ve looked at doesn’t seem to contain such a thing. The discussion between Lubos and Jacques indicates that maybe I’m not the only one confused by this topic.

  5. Peter Woit says:

    Interested,

    This is a complicated topic. Two quick points about it are:

    1. In principle, you are supposed to be able to get rid of the problem of gauge symmetry by working just with gauge-invariant variables. The problem with with this argument is that the (gauge-dependent) space of connections (gauge fields) is a linear space that we know how to quantize, and with coordinates that are local and transform simply under Lorentz transformations. If you try to eliminate the gauge degrees of freedom, go to the “reduced space” and quantize this instead, you lose all these nice properties and encounter intractable “technical” problems.

    2. In a Hamiltonian formalism, you should be able to deal with gauge symmetry by just looking at states invariant under the gauge group. However, something more subtle is going on when you actually try and implement this. Complexifying, there are no gauge-invariant states, so the standard Gupta-Bleuler trick is to impose invariance only under half the gauge symmetry. BRST deals with this issue in a trickier and very interesting way.

  6. Trent says:

    Peter, you write

    “What I’m looking for is something that treats the spinor geometry in a coordinate-invariant way, so would make sense even in a curved background.”

    Such a thing certainly exists and is well-known, it’s called the spinor bundle of a Riemannian manifold. The metric can be curved or flat, doesn’t matter. The spinor bundle is defined as long as a certain Stiefel-Whitney class is zero and the various choices are parametrized by another Stiefel-Whitney class. So there is no mystery at all, in my opinion, around defining spinors for a curved manifold.

    Trent

  7. Peter Woit says:

    Trent,

    Of course I know how to define the spinor bundle on a curved manifold. The problem is Wick rotation or analytic continuation.

    To have the usual notion of analytic continuation, relating sections of spinor bundles over a Riemannian signature space and over a Minkowski signature space, it seems to me you need to represent the two spaces as subspaces of a complex manifold of twice the (real) dimension, and the spinor bundles as restrictions of a holomorphic bundle. This is why I mentioned the Grassmanian and twistor theory. There you have exactly this, and even better, the (half)-spinor bundle is just the tautological bundle over the Grassmannian.

    This is all just a story about classical fields, and what you really want is a quantum field theory. That’s where I start to get confused….

  8. anonymous says:

    Just to point out a recent progress towards resolving the Neuberger problem done in http://arxiv.org/pdf/0710.2410 and http://arxiv.org/pdf/0812.2992 . Though it is true that the Neuberger problem is a technical problem (just for the lattice), they want to resolve it to do proper simulations then – the same motivation as to have chiral gauge theory on the lattice: to do non-perturbative computations for QCD in a practical way.

  9. Interested says:

    Thanks Peter.

    My impression of your post was that you were explaining that guage symmetry doesn’t have a physical basis.

    On something a little unrelated, I would just like to let you know my impression is that, outside particle physics, nobody cares about a theory that doesn’t make experimental predictions. As such shouldn’t it be called the String Hypothesis (hypotheses??)?

    So keep fighting the good fight.

  10. Interested says:

    Sorry

    My impression is that *others* believe guage symmetry doesn’t have a physics basis. I am aware (If I remember your book/blogs correctly) that you think that symmetry is the key to making progress.

  11. Peter Woit says:

    Interested,

    I don’t think the term “has a physics basis” is very well defined, but it’s true that I think properly understanding the implications of gauge symmetry will give deep insight into fundamental theory. Others think it’s a technical device that can in principle be ignored.

    Yes, calling string theory a “theory” has led to all sorts of confusion and trouble. But, that’s far off-topic, please try and keep this posting string-free unless strings have something to do with one of these three problems.

  12. CKK says:

    What Lubos describes is in fact one of the great achievements of modern physics – the realization that problems too hard for conventional methods may always be solved by psychology – all one has to do is convince enough peers that the problem is a non-issue, a technicality, a shortcoming of mathematics which has nothing to do with real physics, and the problem will simply go away.

    This clever method has been very successful recently, particularly in string theory, where it is by far the most popular technique with a large number of devoted fans.

  13. ijc says:

    I sometimes appreciate what Lubos has to say, but his reaction to these issues betrays his disappointingly un-mathematical character. The issues are technical matters, but that does not make them less important, unless we are to be satisfied with doing only calculations according to symbolic replacement rules, without any definition a meaning for the formal objects.

    At no point did you (Peter) even attack the un-rigorous establishment in theoretical physics, perhaps you accept it as necessary ever since Newton introduced calculus without defining it, and yet Lubos’ reaction is quite defensive. He really does seem to prefer physicists who are so intoxicated by their own egomania that they just consider their mastery of the replacement rules to suffice for understanding:

    LM: ” More experienced and powerful physicists have no trouble with the formalism and they can instantly penetrate it and get to the actual physics.”

    That’s right, LM and his powerful friends have no concern over writing nonsense such as complex measures on function space; not only do they have no concern, but they judge anyone who wants to properly contextualize these “analytic continuations” into a mathematical space as a bad student. More powerful physicist have no trouble with replacement rules like “t – > i T”, and they KNOW that these kinds of rules are as complete and total of an understanding as there could ever be.

  14. Peter Woit says:

    ijc,

    Historically I think there is good reason for the standard prejudice amongst physicists to just charge ahead and not worry about what appear to be technicalities. You may get somewhere this way that you wouldn’t have gotten to if you had got hung up thinking about something that didn’t turn out to be really necessary.

    After 30 some years though of charging ahead and getting nowhere (or farther and farther from anywhere one would like to be), this may be an unusual historical moment in which it would be a good idea to go back and take a closer look at those things one doesn’t completely understand, even if someone has a plausible sounding argument about why this is likely to be a waste of time.

  15. mathphys says:

    When I try to read Motl’s post, my browser comes up with a warning message that says “To protect your security, Internet Explorer blocked this site from downloading files to your computer. Click here for options…”.

    I think he puts so many ads and so forth on his web pages it’s impossible to read now.

  16. Peter Woit says:

    mathphys,

    For the past couple days, Lubos’s blog has been hanging and crashing one of the browsers (the gnome browser, epiphany) on the Linux machine I use at work. In this he joins Jacques, whose blog for a year or so has been giving an error message when you try to connect to it with epiphany. Firefox does still work on both of them.

    A week or so ago, I noticed a loud noise at home periodically, finally realized that it was occurring whenever I had a browser window open on Lubos’s blog. Turns out it was the processor fan on my Windows machine, which started cranking at top speed to keep the processor cool as it tried to deal with Lubos’s web-pages…

  17. N. Nakanishi says:

    Peter,
    I would like to point out that non-perturbative covariant BRS-quantization of gauge field theory was formulated in a satisfactory way more than thirty years ago. For reviews, see

    T. Kugo and I. Ojima, Prog. Theor. Phys. Suppl. 66 (1979) 1.
    N. Nakanishi and I. Ojima, Covariant Operator Formalism of Gauge Theories and Quantum Gravity (World Scientific, 1990).

    The trouble of Gribov copies is nothing more than a defect of the path-integral approach, just like the doubling problem of fermion poles in the lattice theory. The path-integral approach cannot give a complete quantum field theory, because in its own framework one cannot prove the unitarity of the physical S-matrix. The path-integral approach induced such inadequate notions as instantons, axions, etc. It is generally inconsistent with field equations, and therefore violates the Noether theorem, because of the use of T*-product instead of T-product. This fact induced incorrect appearance of various anomalies such as gravitational anomaly, FP ghost-number anomaly, etc.

  18. Peter Orland says:

    N. Nakanishi

    Many people have noticed that the Gribov problem occurs in Hamiltonian quantization of non-Abelian gauge theories – not just in path integrals. The problem of copies means there is an essential difficulty in solving Gauss’ law, for most gauge conditions. Gribov copies are not an artifact of any particular formalism. The only way to avoid the problem is not fixing the gauge. This is fine for some purposes (strong-coupling lattice, Monte-Carlo), of course.

  19. Spivak says:

    Lubos wrote an article regarding three mysteries you suggested, I am very much interested to hear your counterargument.

  20. A. says:

    Agree with Peter W. that nonperturbative BRST is an important problem, esp. in regard to Gribov and the difficulty of quantising on the reduced space of gauge fields. As already illustrated by the opposing opinions, here, of whether Gribov is relevant nonperturbatively, this is a hard topic to make progress in. Also agree with Peter O. that the copies are in no way attached to a particular formulism. It’s easy to find examples in the literature which make no reference to the path integral approach.

    One issue which makes this area difficult to persue is the reluctance of large parts of the community to care about it — I’ve found that the average reviewer is typically very hostile to any paper which tries to make a point of the importance of Gribov. Often the reaction is just “go to the lattice” and everything will ultimately be fine (even when the reviewer admits that there are still copies on the lattice).

  21. abc says:

    Hi,
    As rightly pointed out by A., Gribov copies do exist on the lattice too (e.g. Recently, Gribov copies are fully classified on the lattice though for a ‘toy’ model http://arxiv.org/pdf/0912.0450). Their effects on the gluon propagators on the lattice is an active research area mainly being done by several groups in Germany, Australia and Brazil. And many issues need to be settled down there.
    Best,
    abc

  22. mathphys said: When I try to read Motl’s post, my browser comes up with a warning message that says “To protect your security, Internet Explorer blocked this site from downloading files to your computer. Click here for options…”.

    I have similar problem with Mac. For several times I have got the warning that the loading of the contents of the page puts my computer to a non-responsive state.

  23. Thomas Larsson says:

    Firefox with NoScript, Adblock Plus and WOT (not Woit): Lubos’ page loads rapidly and without problems on both Ubuntu and Windows, even on a old machine.

  24. Peter Woit says:

    Spivak,

    Responding to Lubos in detail would be a full-time job, but here are a few other comments:

    1. On the Wick rotation issue he’s mostly arguing with Distler, not me. I think we agree that this is a crucial part of QFT, with QFTs not well-defined in Minkowski space. Exactly how this works for chiral spinors, non-perturbatively, in non-flat backgrounds, seems to me to be something still not completely understood. Lubos just asserts that it works without providing a reference to where one can find the details.

    2. He claims that these problems are just lattice artifacts, that some other non-perturbative formulation of gauge theory won’t have these issues. The problem is that there is no such thing as a completely well-defined non-perturbative definition of chiral gauge theory outside of the lattice framework. The lesson of many years of work on understanding the problem of chirality on the lattice is that it’s not a lattice artifact, but something inherent in the use of a cut-off, some version of which is needed to make sense of the QFT. His comments about the problems of supersymmetry on the lattice are irrelevant, this is a question about a non-supersymmetric theory.

    3. The general criticism he and others make of anyone trying to work on these issues, that they’re just second-raters working on uninteresting technicalities, unable to, like real men, work on AdS/CFT, I think I’ve dealt with in the posting and earlier comments.

  25. Chris Austin says:

    Hi Peter,

    What is your opinion on Peter Hirschfeld’s paper “Strong evidence that Gribov copying does not affect the gauge theory functional integral,” Nuclear Physics B 157 (1979) 37-44, which he wrote whilst an undergraduate at Princeton, with advice from Curtis Callan, if I remember correctly?

    Best,
    Chris

  26. Peter Woit says:

    Hi Chris,

    I’d never seen that paper, just took a look at it. Peter O. is more of an expert on this question, and maybe he knows if there is a problem with this Peter H. paper, and if so what. From my quick look, there are several things to worry about in his argument (non-compactness of the gauge group, existence of the kind of coordinates he wants, whether the intersection number he gets is zero…). In any case, if his claim is that one can just ignore the contributions of Gribov copies, that seems to conflict with the large amount of later work on the subject.

  27. F says:

    “Non-perturbative QFT calculations are well-defined not in Minkowski space, but in Euclidean space, with physical observables recovered by analytic continuation.”

    Peter, your comment above is not quite correct. Euclideanization is ONE way to handle non-perturbative calculations in QFT, but not the only way. Light cone quantization of quantum field theories is a standard tool that can be used to obtain non-perturbative solutions without a Wick rotation of the time coordinate. In this way, one can avoid a lot of the subtle issues you mention arising from changing the space-time signature of a theory.

  28. anon. says:

    Just to be precise about the issues of BRST and Gribov problems — they don’t affect the nonperturbative definition of the theory since, as you say, the lattice can always calculate gauge-invariant quantities. They are relevant when people want to investigate, e.g., the “nonperturbative gluon propagator” in some particular gauge. But there’s no deep reason to think such things are good physical quantities to study.

    Similarly, the Gribov problem exists for many gauges in the continuum (e.g. Landau gauge), but not for others (Weinberg’s book, for instance, skirts the problem by using axial gauge).

    So I think really the only problem is that one cannot write down an action that is simultaneously manifestly local, manifestly Lorentz-invariant, and gauge fixed without running into Gribov problems. (Although Zwanziger has a formalism attempting to do this by adding more auxiliary fields, which I’ve never fully digested.)

    Whether one things these problems are interesting is a matter of taste — I find them somewhat interesting, but I doubt they hold the key to a more solid nonperturbative understanding of QCD, since any such understanding should be expressible in terms of only gauge-invariant quantities.

  29. Peter Orland says:

    anon.

    The reason Gribov copies are interesting is the following. The physical configuration space of Yang-Mills theories are gauge orbits, not gauge fields. A gauge orbit is an equivalence class of gauge connections, the equivalence relation being gauge-equivalent.

    If you don’t want to find coordinates on the space of orbits, then you don’t have to worry about the Gribov problem. If you do, then you do have to worry.

    So whether or not it is important depends on how you want to study gauge theories. In perturbation theory, you can ignore copies, which correspond to large gauge transformations. If you study gauge-invariant functionals (like Wilson loops) and somehow have a calculational scheme avoiding gauge fixing, you can bypass the issue altogether. Unfortunately, no one has such a scheme for QCD.

    The Gribov problem is present in axial gauges as well, though open boundary conditions (instead of periodic boundary conditions) ameliorate the situation considerably.

  30. N. Nakanishi says:

    Peter Orland,
    Please give me a reference, if any, which shows that the Gribov problem is relevant to the operator formalism of the
    BRS-quantized gauge theory. I note that in the operator formalism, of the BRS-quantized gauge theory, c-number solutions to the gauge fixing condition are totally unnecessary; the canonical quantization can be carried out exactly in the same way as in the non-gauge theory.

  31. Peter Orland says:

    N. Nakanishi,

    Respectfully, I don’t have to give you a reference. The Gribov issue is very general, and it isn’t my problem whether the methods you advocate are right or wrong. That is for you to work out.

    Anyway, Peter W. insists this is not a physics discussion board, so I will just say this:

    You need to show that large gauge transformations are properly taking into account in your formalism (that is how copies arise). Everything I have seen on BRS (except Herbert Neuberger’s work) deals with infinitesimal gauge transformations. For example, how do you handle the Gribov horizon, where the Fadeev-Popov determinant vanishes?

  32. N. Nakanishi says:

    anon,
    You claim “one cannot write down an action that is simultaneously manifestly local, manifestly Lorentz-invariant, and gauge fixed without running into Gribov problems”.

    Such a theory does exist! It is the Kugo-Ojima theory (see,
    Prog. Theor. Phys. Suppl. 66 (1979) 1-130).

  33. N. Nakanishi says:

    Peter Orland,

    It seems to me that you do not understand the fundamental nature of quantum field theory. In quantum field theory, symmetries are described by Lie algebra but not Lie group; that is, I mean that only infinitesimal transformations are
    relevant. To recognize this point is particularly important in supersymmetry, because there is no “supergroup”. The same is true for the local gauge symmetry: finite-size local gauge transformations are irrelevant to the quantum gauge theory. The c-number gauge trasformation is a purely classical concept; in quantum gauge theory, there exist only the BRS transformations. So, it is totally unnecesary to take account of the Gribov copies in the proper framework of quantum field theory. The FP determinant is a consequence of the path-integral formalism; it is a misleading notion based on the classical gauge theory.

  34. Peter Orland says:

    N. Nakanishi,

    Yes, call me a dummy. Other people have said worse.

    Only infinitesmal transformations are relevant? Not so. Global properties of symmetry groups are important, even in the theory of spin.

    The Fadeev-Popov determinant, can be derived from the Haar measure of a lattice theory. There is nothing misleading about it. Is the lattice theory just a classical concept to you (or a misleading notion)?

  35. Peter Orland says:

    Anyway, I will not comment further on this thread. I am sure Peter W. is tired of insults flying around.

  36. N. Nakanishi says:

    Peter Woit,

    Since Peter Orland does not wish to continue the discussion, I would like to have your opinion. Do you still assert that the non-perturbative BRS-quantization of gauge theory is not yet formulated?

    Peter Orland’s comments provide no logical basis for claiming that the Gribov problem is relevant to the operator formalism of the BRS-quantized gauge theory. What he commented are nothing more than his feelings. (Reply to his question: The lattice gauge theory is essentially based on the path integral. There is no BRS-formulated lattice theory.)

  37. Z says:

    Hello, I find the subject of gauge symmetries very interesting but also very confusing, I tried asking Lubos about some of the details of his argument about gouge symmetries being just technicalities but without luck so far, perhaps someone here could be more helpful, here is what is bothering me:

    On one hand Lubos argues that gouge fields represent unphysical degrees of freedom and it agrees with what I’ve read generally. But on the other hand it is also said that gouge bosons mediate interactions, for example a photon is a gouge boson associated with U(1) symmetry. Isn’t that a contradiction? Photons are certainly physically real, so why are they modeled by gouge fields which should only describe unphysical states?

    Lubos also claims and Peter seems to (at least partly) agree that one can reformulate the theory without any gouge degrees of freedom but if that is indeed the case that what will happen to the photon and other gouge bosons in such a description? What will mediate forces?

    Finally the gouge symmetry associated with photons – U(1) represents quantum phase and while there may be no way to directly observe this phase is not really unphysical as it does affect experimental outcomes (position of interference fringes in in double slit experiments or Aharonov Bohm variants for example).

    So are gouge degrees of freedom completely unphysical or do some of them describe physically real phenomena?

  38. Peter Woit says:

    anon,

    While it’s possible to argue that one can just ignore problems with handling gauge symmetry in vector-like theories like QCD (not that I’m convinced, one may very well learn something important by taking these problems seriously), this is not at all the case for chiral gauge theories (like the Standard Model…).

    Nakanishi,

    Yes, I’m afraid I agree with Peter Orland. I don’t believe the operator formalism for QCD is now well-defined in the sense that the path integral formalism is through lattice gauge theory. The lattice formalism gives a completely well-defined calculational set-up that one can even put on a computer (although the calculation may not be practical). A possible addition to my list of mysteries would be that of the relationship between the operator formalism and the path integral formalism. This is much trickier than often assumed and working out details of this might be very interesting.

    Z.

    gauge fields include both physical and gauge degrees of freedom. In some sense the problem is that there is no clean way to separate them.

  39. anon. says:

    Peter Orland, I don’t understand your comment “If you study gauge-invariant functionals (like Wilson loops) and somehow have a calculational scheme avoiding gauge fixing, you can bypass the issue altogether. Unfortunately, no one has such a scheme for QCD.” What about the lattice? Or do you just mean a continuum scheme?

  40. anon. says:

    I think the strongest hint that the Gribov problem might be relevant comes from evidence that at least in some gauges, configurations like center vortices that should be relevant for confinement (based on arguments that don’t depend on a choice of gauge) are localized on the Gribov horizon. There seems to be something interesting and poorly understood lurking there.

  41. Peter Orland says:

    anon.

    I meant a non-numerical (i.e. analytic) scheme, lattice or not.

  42. Z says:

    Thanks for the answer Peter.

  43. Rondeau says:

    “The same is true for the local gauge symmetry: finite-size
    local gauge transformations are irrelevant to the quantum
    gauge theory.”

    In perturbation theory, in principle, only infinitesimal
    gauge transformations are relevant because finite gauge
    transformations are just compositions of infinitesimal ones.
    But globally, not just locally, there are obstructions to
    that naive approach to gauge transformations.

    But even in the perturbative quantum theory there is at
    least one finite gauge transformation that is crucial: the
    one connecting unitary gauge (in which the theory is
    manifestly unitary), to a suitable covariant gauge (in which
    the theory is manifestly renormalizable). Without that
    transformation there is no consistent quantum theory.

  44. N. Nakanishi says:

    Peter Woit,

    You confuse the existence of the formalism and that of the method of numerical calculation. The former is a fundamental theoretical problem, while the latter is merely a practical one. The existence of the satisfactory formalism is very important as physics even if one cannot solve it. Indeed, the covariant operator formalism of the non-Abelian gauge theory clarified various theoretical problems (e.g., formulation of the condition of color confinement, resolution of the dilemma between the Higgs mechanism and the Goldstone theorem
    in the manifestly covariant theory, proof of the charge universality in the electroweak theory, etc.). On the other hand, I think that the lattice theory cannot show even the angular-momentum conservation law. The lattice theory discusses quark confinement but not color confinement.
    Anyway, you should read a review of the covariant operator formalism of BRS-quantized gauge theory. If you give me your mailing address, I am willing to send you a copy of my book.

    Rondeau,

    You confuse the problem at the operator level and that at the representation level. Since the path integral does not discriminate them, the people working only in the path-integral approach often misunderstand this point. The finite-size c-number transformations are not well-defined at the operator level. They are often recovered at the representation level; the physical equivalence (but not unitary equivalence!) of unitary gauge and covariant gauge can be shown after the representation space (i.e.,state-vector space) is introduced.

  45. Peter Orland says:

    OK, against my better judgement, I will jump back in. I am already regreting it.

    Mr. Nakanishi, stop telling other people they are confused. It does not make you look knowledegable. It just shows you have an agenda.

    So here are the issues:

    1) The lattice formalism does not require numerical simulation. It is just a gauge-invariant regularization of gauge theories. I repeat – numerical issues are irrelevant. Go and read Wilson’s paper from 1974. He does no simulations in that paper.

    2. It is not true that the lattice implies the use of the path integral. There is a Hamiltonian (Kogut-Susskind) lattice formalism where the same issues arise (in solving Gauss’ law).

    3. Let me say it again – the Gribov problem occurs in the Hamiltonian formalism too.

    4. The issue of large gauge transformations has nothing to do with one formalism (path integrals) or another (Hamiltonians). In the Hamiltonian approach, as I said before, it arises in the measure on the space of states. States are singlets under gauge transformations. ALL gauge transformations, not just infinitesimal ones. There is still the issue of large gauge transformations.

    This is fundamental to gauge theories and you can’t invent any formalism to avoid it. If you say some method removes it, the method is plain wrong (or at least, your interpretation of the method is wrong). For example, there will be no theta vacua in any formalism of gauge theories ignoring large gauge transformations. Different n-vacua are large-gauge transformation-equivalent to a pure gauge. That is how theta vacua arise. Once again, this is just as true in a Hamiltonian as a path-integral formalism

    5. Though one does not need to path integral in these arguments, why you dismiss path integrals mystifies me. I suggest you look at the books by Simon, Glimm and Jaffee on the subject. These people actually proved theorems about quantum mechanics and field theory, using path integrals.

    6. Now in this forum, I can’t prove any of this to you. All I can say that it has been generally known since the seventies. The blogosphere is not a good teaching environment. I am not able to prove it to you without our actually working through the math together (and I am pretty sure you would not be willing to do that anyway). So instead of repeating your pronouncements, you should read and understand some of the literature (not just the literature you are promoting).

    6. Last and most important. I don’t know if you are still doing research in this field. If you are, listen! I am saying something you should hear, even if you don’t want to.

  46. N. Nakanishi says:

    Mr. Orland,

    In this forum, discussions should be scientific ones. OK?

    1. It is Peter Woit who emphasized the importance of numerical aspect of the lattice theory.

    2. & 3. May I understand that what you call Hamiltonian formalism is the formulation in the Schroedinger picture? The Schroedinger picture is good in quantum mechanics (because of von Neumann’s theorem), but very bad in relativistic quantum field theory. The Schroedinger picture not only violates manifest covariance but also suffers from the difficulty of vacuum polarization. Even worse is the fact that it is unclear what representation is adopted.

    4. The covariant operator formalism of gauge theory is formulated in the Heisenberg picture. It is BRS invariant. It is not invariant under local gauge transformations. Hence the Gribov problem is totally irrelevant to it.
    The theta vacuum is a pathological consequence of the path-integral approach. What is physically sensible is the variation of the action but not the action integral itself.

    5. I know that path integral was used in the constructive field theory, but the models treated there are such simple ones as scalar field theory. The trouble with path integral arises in more realistic theories such as quantum gauge theory and quantum gravity. The path-integral approach cannot be a complete theory because within its own formalism one cannot prove the unitarity of the physical S-matrix. The hermiticity of the action cannot be defined in it.

    6. Non-scientific.

  47. Thomas says:

    > The theta vacuum is a pathological consequence of the path-integral approach.

    So there is no strong CP problem, theta vacua are just an illusion?

  48. Anon. says:

    “The covariant operator formalism of gauge theory is formulated in the Heisenberg picture. It is BRS invariant. It is not invariant under local gauge transformations.”

    Typically, one starts with gauge invariance, and then gets to BRST as a way to handle gauge invariance when quantizing. If you take the gauge-fixed BRST theory as the definition of your theory, then yes, there is no ambiguity. You are essentially truncating your definition of the theory at that level.

    As long as non-perturbative effects are hard to test experimentally, your theory might survive, I don’t know. Conceptually though, it sounds like you have no notion of a non-perturbative definition. You are just ignoring the unknown. Taking gauge invariance seriously all the way has lead to correct ideas in the past, eg.: Aharonov-Bohm effect, which is intimately tied to monopoles, theta angles and such.

    “The theta vacuum is a pathological consequence of the path-integral approach. What is physically sensible is the variation of the action but not the action integral itself.”

    That cannot be entirely true because we know (experimentally) that tunneling exists in quantum mechanics.

    More generally, path integrals are extremely natural from a lot many points of view. Anomalies, renormalization group, lattice gauge theory etc. are all most natural from a path integral viewpoint.

  49. Peter Woit says:

    Nakanishi,

    Thanks for the offer of a copy of your book, but I already have bought one, several years ago. It and Henneaux-Teitelboim are the only two books I know that address the issue of treating gauge symmetry via BRST, beyond Feynman diagrams. Henneaux-Teitelboim I think I understand quite well, your book much less so. Henneaux-Teitelboim try and connect the Hamiltonian and path integral formalism, but do so using (like others who try and do this) phase space path integrals that are ill-defined.

    Despite some effort, I still haven’t completely understood exactly what the assumptions are that go into the covariant operator formalism. I started my career working on lattice gauge theory, so one virtue it has for me is familiarity, but the other is that it is precisely well-defined (independently of its use in numerical calculations). One thing I learned by doing numerical calculations is that setting them up on a computer forces you to make sure you understand very precisely what you are doing, there is no room at all for ambiguity.

    The lattice approach to QCD is precisely defined, even if you can’t prove things about it. If the covariant operator formalism for QCD is as well-defined, and gives different answers, this would be very interesting. One problem you would have then is that as far as I know, no lattice computation that can be compared to experiment has given anything incompatible with what is observed. If you claim there are no theta-vacua, one thing you have to explain is the success of Witten’s argument for the eta-prime mass.

    There is conventional wisdom about how the path integral and operator formalisms are supposed to be two ways of treating the same theory, in principle compatible. If someone could sort out carefully what is going on here, that would be very interesting.

  50. Peter Orland says:

    In any case with no theta vacua, the U(1) problem in QCD has no solution.

    By the way, there is a formulation of unitarity within path-integral lattice gauge theories. It is called reflection positivity and was first considered in gauge theory by Osterwalder and Seiler.

    Whether or not one takes the Kugo-Ojima formulation seriously, I don’t believe that it forbids theta-vacua, unitary gauges or Gribov problems. If it does, it is purely perturbative. A formulation which does forbid such things is not going to be unitary, except in perturbation theory. An analogy is to treat the double-well Hamiltonian perturbatively, ignoring tunneling. The result for the spectrum is going to be wrong, unless tunneling (a nonperturbative process) is taken into account.

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