Today I have posted a paper on arxiv. It will appear on monday. This paper was required to my by Terry Tao to supplement the proof of the mapping theorem showing that indeed it holds.

If you cannot hold the paper is 09032357v1: preprint. Don’t trust that number as may change.

The argument may be put up with very simple words: If you trust Smilga’s solutions that depend only on time, a Lorentz boost will fit the bill.


16 Responses to Posted!

  1. Rafael says:

    Dear Marco,

    I followed this discussion with great interest. Hence you got a gluon propagator that nicely fits Lattice’s calculations and “decoupling” SD solutions (in deep infrared) as well as reproduces YM glueball spectrum in concordance with other approaches, I think it is not by mere mathematical chance.
    However, as you well know, IR QCD is THE open problem of theoretical physics nowadays and anyone claiming to have found a “shortcut” to its understanding should be properly scrutinized by scientific community… I think its a natural and health process. On the other hand, unpoliteness and destructive comments (as you have received here in your blog) are simply unacceptable, these people should formally appologize!

    As a follower of your blog, I read your paper and would like to say that a doubt remain, maybe you could clarify this. You wrote:

    “From these equations we immediately see that our chosen solution is again a solution moving the agreement to order O(1/g 2 ). We can repeat this operation to any order confirming that our solution for SU(2), obtained by a Smilga’s choice, is indeed an exact solution of Yang-Mills equations…”

    Would it be possible to explicitly write down the generic term of this order-n expansion and show that your solution also solves it (i.e. for all orders in (1/g))?

    Thanks by keeping this nice place on the web.
    Best regards,


  2. mfrasca says:

    Dear Rafael,

    Thank you very much for your comments that I have always appreciated. Some improved understandings come out from your writings in this blog and this is one of the main reasons why I opened and maintain it.

    I am aware of dynamics of scientific community. I have always tried to follow the rules as strictly as possible as this is the only way to get my ideas accepted, if right. Indeed, during these twenty years of activity, I have got a very satisfactorily response as you can judge by my file of publications.

    About that sentence you are right, but for my needs I can just stop at 1/g. I did this for a couple of reasons. Firstly, I wanted to get Yang-Mills equations at the leading order in a gradient expansion. Secondly, I wanted to show in this paper the meaning of this expansion and its power through this case. But the general case is quite simple to obtain: remove any spatial dependence in the components of the field and you are left just with the leading order. The solution of these equations is surely an exact solution. Then, do a Lorentz boost and you are turned to an exact solution of the full equations.

    I will keep on doing my best. Thank you again.

    Best regards,


  3. unit says:

    Dear Marco,
    I’ll take some time to read your paper. In the meanwhile there’s something I noticed: eqn 14 gives eqn 10 with a rotation from the t axis to p, which in general is clearly NOT a Lorentz boost. The two are Lorentz-related only if p is timelike. Do you agree?

  4. mfrasca says:

    Dear unit,

    I appreciate a lot your interest for my work. I hope it will be of some help to you.

    Please, explain better what you mean using latex. This is done using two dollars and writing latex attached to the first one. Inside you can put your latex formula.

    Do you mean that I cannot pass from the rest frame to a moving one changing e^{imt} to e^{ip\cdot x}? I think this is standard field theory but maybe I have not understood properly what you mean.


  5. unit says:

    Hi Marco,
    hope not wasting your time. Probably I’m just wrong.
    Eqn 10 follows from Eqn 14 if $x_0 = p \cdot x’ $ for some Lorentz transfomation $x -> x’$. Do you agree? This transformation exists only if $p$ is timelike, of course there’s no Lorentz transformation that switch $x_0$ with $x_1 ‘$. i.e. you cannot obtain Eqn 10 with $x_1’$ as argument coming from eqn 14.

    Bye and thanks,


  6. unit says:

    Sorry, my fault. I tought that p in your paper was a general vector, not just a timelike one.


  7. Terence Tao says:

    Dear Marco,

    I think I have isolated the root problem here. You seem to be operating under the assumption that the ansatz A^1_1=A^2_2=A^3_3=\phi (with all other components zero) is a kind of “free lunch” that one can impose on top of the Yang-Mills equations without losing the other properties (e.g. invariances) that these equations enjoy. Unfortunately, there is no free lunch in this business, and the ansatz in fact breaks many of the symmetries you are relying on. For instance:

    * The ansatz breaks the variational characterisation of Yang-Mills: extremisers of Yang-Mills using the ansatz as constraint are not necessarily extremisers globally. This is the problem with your first proof, as we have agreed upon.

    * The ansatz breaks the gauge invariance of Yang-Mills: applying a gauge transform A_\mu \mapsto G A_\mu G^{-1} - \frac{1}{g} G \partial_\mu G^{-1} to a solution obeying the ansatz gives rise to a new solution that does not obey the ansatz any more. This is the problem with the second proof you have proposed.

    * The ansatz breaks the Lorentz invariance of Yang-Mills: applying a Lorentz transformation x^\mu \mapsto \Lambda^\mu_\nu x^\nu to a solution obeying the ansatz leads to a solution that no longer obeys the ansatz. (For instance, solutions obeying the ansatz have a vanishing time axis component, but of course the time axis is not Lorentz-invariant.) This is the problem with the proof you have posted above (specifically, the use of Lorentz invariance on page 6).

    In order to forestall any further effort expended on proving a result which is, in fact, false, let me give a proof that the mapping theorem does not work, expanding upon my Wikipedia comment. As with your paper, I will use expansions. Observe that the scalar field \phi(t,x_1,x_2,x_3) = x_1 x_2 obeys the scalar equation \partial^\mu \partial_\mu \phi = \phi^3 to order O(|x|^6) near the spacetime origin. Applying standard power series expansion methods (or the Cauchy-Kowalewski theorem) we can thus find a (local) exact solution to the scalar equation \partial^\mu \partial_\mu \phi = \phi^3 which has asymptotics \phi = x_1 x_2 + O(|x|^6) near that origin.

    Now we substitute in your ansatz. This gives a putative Yang-Mills field with A^1_1=A^2_2=A^3_3=x_1 x_2 + O(|x|^6) with all other components zero.

    Now let us compute the a=1, \nu=2 component of the Yang-Mills equations

    \partial^\mu\partial_\mu A^a_\nu-\left(1-\frac{1}{\alpha}\right)\partial_\nu(\partial^\mu A^a_\mu)+gf^{abc}A^{b\mu}(\partial_\mu A^c_\nu-\partial_\nu A^c_\mu)+gf^{abc}\partial^\mu(A^b_\mu A^c_\nu)+g^2f^{abc}f^{cde}A^{b\mu}A^d_\mu A^e_\nu = 0

    at the origin. All terms except for the -(1-\frac{1}{\alpha}) \partial_2 \partial^1 A^1_1 term vanish at the origin, thus leaving a non-zero term of -(1-\frac{1}{\alpha}). Thus the ansatz does not map solutions of the scalar equation to the Yang-Mills equation.

  8. mfrasca says:

    Dear Terry,

    Thank you very much for having some time to discuss this matter. I should say, looking at your comment in Wikipedia that I have never expected that was yours, mostly because there was a lot of noise at that time.

    Let me say that you are right but I keep on thinking that you are meaning something else. The reason is the following. When you have a set of PDEs, can we agree that there are exact solutions having no spatial dependence? I think you agreed on this. These kind of solutions always exist and this is true also in this case. This is proved in my computations.

    Indeed, let me write the full set of equations

    \partial^\mu\partial_\mu A^a_\nu-\left(1-\frac{1}{\alpha}\right)\partial_\nu(\partial^\mu A^a_\mu)+gf^{abc}A^{b\mu}(\partial_\mu A^c_\nu-\partial_\nu A^c_\mu)+gf^{abc}\partial^\mu(A^b_\mu A^c_\nu)+g^2f^{abc}f^{cde}A^{b\mu}A^d_\mu A^e_\nu = 0

    Now, we look for solutions depending only on time. This is a well-known technique in physics ( in Dispersive Wiki there is an entry I have written about on BKL conjecture

    and this is a gradient expansion physicists do with general relativity). You can also do the opposite as for soliton solutions when exist. You will be left with

    \frac{1}{\alpha}\partial_\theta^2 A^{a(0)}_0-f^{abc}A^{b(0)}_i\partial_\theta A^{c(0)}_i+f^{abc}\partial_\theta(A^{b(0)}_0A^{c(0)}_0)+f^{abc}f^{cde}A^{b\mu(0)}A^{d(0)}_\mu A^{e(0)}_0 = 0


    \partial_\theta^2 A^{a(0)}_k+f^{abc}A^{b(0)}_0\partial_\theta A^{c(0)}_k+f^{abc}\partial_\theta(A^{b(0)}_0A^{c(0)}_k)+f^{abc}f^{cde}A^{b\mu(0)}A^{d(0)}_\mu A^{e(0)}_k = 0.

    The point is the following. If I choose A_1^1=A_2^2=A_3^3=\phi(t) the equation are identically true. I have solved the full set of Yang-Mills equations. Do you agree that this is an exact solution?

    Then a set of solutions of Yang-Mills equations exist that prove me right. But this cannot be true for all the solutions of the scalar field (you proved that right now), and I think that this is the main point on which we should agree. Mostly important, this is the set of solutions I need to get the spectrum of the theory!

    I cannot agree with your point 3 above. I have never pretended that a Lorentz transformation preserves the ansatz. What I need is that gives a solution of Yang-Mills equations proportional to the one of the scalar field as I need. The mapping is preserved not the ansatz.

    Let me summarize all the matter here:

    1) Given a set of PDEs I can always find a set of exact solutions depending only on time reducing this set to a set of ODEs. There is a lot of research about Yang-Mills equations done in this way since eighties.

    2) A set of integrable solutions of the above ODEs set is obtained solving the equation of a quartic scalar field mapping in this way each other with Yang-Mills equations.

    3) From these solutions, I can get a full set of exact solutions by Lorentz transformations.

    The existence of this set of solutions proves me right.

    Let me repeat here the algorithm.

    1) Take Y-M equations changes to ODEs.

    2) Do a Smilga’s choice obtain a quartic scalar field equation and solve it.

    3) Do a Lorentz transformation and obtain an exact solution to full Y-M equations.

    4) Change Y-M with any set of PDEs you like and solve the corresponding equations taking some set of equal components. You have solved them.

    Sorry, your solution is not in this algorithm.

    I would like to know if you agree on the very existence of these solutions. Because, if this is true I am right and we are talking about fixing the boundaries (how far can we trust such a mapping?). Otherwise, please, let me know.


  9. Luboš Motl says:

    Dear Terry,

    all your observations about the no-free-lunch are “spiritually” correct but if you look at the exact terms that spoil the validity of the Yang-Mills equations for Marco’s configuration, the technical reason why the Yang-Mills structure contradicts Marco’s configuration’s being a solution is slightly different than any of the particular “broken beauty” entries you wrote.

    His configuration would be a solution if the terms coming from the gauge-fixing term, proportional to (partial_m A^m)^2, could be dropped. You can check it – check directly the validity of the YM equations of motion with the scalar Ansatz. For Feynman gauge, xi=1 (or alpha=1 in Marco’s notation), you would actually solve the YM equations: the Feynman gauge makes the gauge field resemble 4 scalars, indeed.

    It follows that his configurations would solve the YM equations of motion (only) if it satisfied the Lorentz gauge, partial_m A^m = 0, too. But Marco’s configuration doesn’t satisfy the Lorentz gauge and that’s the real problem.

    If you impose both Marco’s Ansatz and the Lorentz gauge on a configuration, you can easily see that this configuration must be x,y,z-independent, too.

    (More explicitly, the “wrong” term violating the equations, coming from the Lorentz gauge-fixing term, goes like partial_nu partial_a phi – some components of it – while all other terms’ tensor structure is proportional to delta(a,mu) so they can’t cancel it.)

    The x,y,z-independence condition actually puts you into the Smilga class of solutions because Smilga considered configurations that only depended on time. For this very limited and arguably not too interesting subclass of configurations, Marco’s trick can be applied. That also answers the question that you may be interested in, whether Smilga shares Marco’s errors. He actually doesn’t because he only looks at the limited x,y,z-independent subclass of configurations.

    For Marco: if a solution is just a boost of another solution, it surely doesn’t deserve a special paper for its own sake. It is still “static”, albeit in a different reference frame.

    Best wishes

  10. mfrasca says:

    Dear Lubos,

    The point was if there is any subset of Yang-Mills solutions that can be shared with a quartic scalar field. In the original Wiki’s comment by Terry this point was questioned. But this exists as is easy to prove and is obtained through Smilga’s choice and this contradict original skepticism. Smilga’s solutions are all I need to work out my argument.

    Of course, I need the mapping to hold in a 1/g expansion and this is again true. If you check my paper I have not given explicitly the mapping set whose existence was questioned by Terry. But this set exists and all the arguments of my paper are correct. I do not think Terry moved one step further with respect to the mapping theorem, otherwise we would agree about the correctness of the argument and we would just question what are the limitations of the mapping theorem.

    The point here is if my Dispersive Wiki entry was right or not and if Terry’s claim that the mapping theorem is false holds. But as it exists a subset of solutions shared by both theories, this claim is too strong.


  11. unit says:

    Dear Marco,

    May I ask you a question? In my opinion, as I think Tao said, your mapping doesn’t commute with Lorentz boosts, so boosted Smilga’s solutions differ from your eqn 10. In the case of Smilga’s solutions you have to transform the vector A, not the scalar /phi, so the results are different. Or not?

    Thanks for the patience,


  12. carlbrannen says:

    Luboš’s comment is particularly insightful, to the effect that Smilga’s choice requires no dependence on position.

    There is a whole field of quantum mechanics that is devoted to the approximation where one ignores spatial dependence. As soon as you do this, by Fourier transform, you also ignore momentum. This is quantum information theory. Even though position and momentum are gone, time and energy are still important, and so one can do physics with models with this restriction.

    Normally quantum information theory deals with quantum mechanics rather than quantum field theory, but it is possible to define Feynman diagrams on qubits. See the arXiv version of Physical Review A 76, 06106 (2007), “Quantum Electrodynamics of Qubits” for a QED version that seems related to Smilga’s choice for QFT.

    A way of physically justifying Smilga’s choice is that it would apply to a situation where the wave function of the quark has spread out so far that it no longer has any position dependence. Or one could think about QCD in a small box with boundary conditions as is done in lattice QCD. In any case, if QCD is universally valid, then it must also be valid in this quantum information approximation.

    Along this line, I’m currently arguing with the referees at Phys Math Central over my paper on hadrons from January. This paper uses quantum information theory explicitly. The reviewer is arguing that it has no contact with QCD. I pointed out that in QED, the Lamb shift is calculated as a correction to a QM model of the hydrogen atom where the 1/r potential has no direct contact with QED; to get direct contact from a QFT to a QM potential one must solve a problem that is very difficult to solve. I’m guessing that the absence of any dependence on position may also be a source of contention, if they think of it, but this is the nature of quantum information calculations.

  13. mfrasca says:

    Dear unit,

    I do not know why I have to recover your comments in my spam basket. Maybe is there a problem with your ISP and WordPress?

    Of course, you are right. The point is quite different. It does not matter if the ansatz is not preserved. A class of solutions exists that grant the mapping. This contradicts the idea that the mapping never exists as suggested by Terry.

    This way of solving PDEs is quite old and largely used in physics.


  14. unit says:

    Hi Marco,

    probably your spam filter is smarter than you think and recognize stupid questions :).
    I turned off Tor, does it work now?

    Anyway, it’s not clear to me which class of solutions are you referring to. If you were referring to those described by eqn. 10, the argument used at page 6 is not valid, as previously said by Tao, because your mapping doesn’t preserve Lorentz invariance. I described this point as I understand it in my last comment, and I think you agree.

    If you are referring to the class of boosted Smilga’s solution instead, I don’t see the relationship with scalar field solutions.
    Maybe you just need Smilga’s solutions with only time dependance? If this were the case you could do your calculations in a reference frame in which all the spatial derivatives disappear, and everything becomes trivial, isn’t it? There’s no added value in boosting these solutions.

    Can you explain me this point in detail?

    Thank you,


  15. mfrasca says:

    Hi unit,

    Now it works.

    Smilga’s solutions are


    being \phi(t) a solution of equation


    This is a just a possible selection out of many. This is an exact solution of Yang-Mills equations and maps on the solutions of scalar field equation. For this case the mapping theorem is true against Terry’s skepticism. Now, let us turn on the spatial part. In a series whose ordering parameter is 1/g one has that the Smilga’s solutions are the leading order term and this is what I need for my proof of a quantum Yang-Mills theory.

    Terry’s criticism about breaking of Lorentz invariance has been criticized by me and Lubos. I cannot pretend the ansatz is preserved by a Lorentz transformation. This should be a standard way to solve PDEs. A Lorentz transformation change A_\mu^a(t)\rightarrow \hat A_\mu^a(x) and now you will have \hat A_\mu^a(x)=\eta_\mu^a\phi(x). So, if you do a boost along x direction one has


    and m=\mu g^{1 \over 2} the field mass. I cannot see here any violation whatsoever and this is a solution yet.

    All this works as it happens when you do a gradient expansion and this kind of expansion is a strong coupling expansion. Terry did not negate this, he kept this in Dispersive Wiki, and being this a scientific result of mine of great moment let me be happy.

    Of course, I have no hope to turn Terry’s ideas and I do not have his authority. I can only insist that he said something wrong that everybody can read in the discussion of Wikipedia about Yang-Mills theory.



  16. Rafael says:

    Just to show to some of your readers how the possibility of a “temporal mapping” would be specially interesting for QFT at finite temperature:
    Regards, Rafael.

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