# Which exact solutions of the classical Yang-Mills equations are known?

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I'm interested in the pure gauge (no matter fields) case on Minkowski spacetime with simple gauge groups. It would be nice if someone can find a review article discussing all such solutions

EDIT: I think these are relevant to the physics of corresponding QFTs in the high energy / small scale regime. This is because the path integral for a pure gauge Yang-Mills theory is of the form

$$\int \exp\left(\frac{iS[A]}{g^2 \hbar}\right) DA$$

where $g$ is the coupling constant. In high energies we have the renormalization group behavior $g \to 0$ (asymptotic freedom) which can be equivalently described by fixing $g$ and letting $\hbar \to 0$.

EDIT: For the purpose of this question, an "exact" solution is a solution in closed form modulo single variable functions defined by certain ODEs and initial / boundary conditions.

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retagged Apr 19, 2014
In my humble opinion, if such a solution exists, it is good for nothing - it cannot be physical. It needs at least renormalizations and maybe solving the infrared problem. I can only offer you a toy model to explain the difficulties: https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B4Db4rFq72mLYzAwMzY0MGMtZmE0Yy00OGIwLTgzYzItOGJlZjllNGFmYWM5&hl=en_US

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@Vladimir: perhaps by "pure gauge" the OP means "without matter", not a solution which is pure gauge in the sense that it is a gauge transformation of the trivial solution $A=0$. At least that's the only reading that makes the question nontrivial.

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Maybe some description of the motivation will help. The long distance properties of pure gauge theories in Minkowski space are strongly quantum mechanical, so it's hard to imagine there are many physical motivations for looking at classical solutions. Perhaps there is a mathematical physics motivation, some special structure those solutions have, or maybe these are toy models for other non-linear differential equations, etc. etc.

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Moshe's comment is a bit abstract. More concretely, consider QCD. The low energy solution are bound states of quarks, arising from the anti-screening defect of the vacuum, i.e. its ability to pull virtual pairs out of the vacuum. This has no classical analogy, so no classical solution will display this effect.

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@SHuntsman: Doesn't Rubakov's book deal overwhelmingly with solutions in euclidean signature?

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@genneth, I was also trying to direct the search for an answer. So, based on the above-mentioned reasons, I doubt any theoretical HEP person would have a motivation to work on this. On the other hand, I know some numerical relativists who looked at the YM equations at great detail to check their methods of solving the Einstein equations. This would be a different branch of the literature. Besides, long-list questions without motivation are not as much fun to answer, at least for me...

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Squark: the argument in your edit works in Euclidean signature, where you can apply saddle point methods. You are looking then for Euclidean solutions of finite action, instantons. On another point: you’d have to specify which kind of solution you are interested in, otherwise set any initial data on a Cauchy surface and let it evolve - this is a classical solution, and there are many of those.

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Also, I am not sure what is the motivation for “exact” solution, and what you mean by that. Do you mean closed form solution? many of the known solutions for YM + matter for example are not “exact” in that sense since they require numerical solution of an ODE.

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@Moshe: It seems to me that saddle points methods can be applied regardless of asymptotic freedom but I may be missing something. I think my argument should work in Minkowski spacetime too. In fact, I suspect the following: to each classical solution of the YM equations there should correspond a state in the QFT s.t. the expectation values of (gauge invariant) observables are close to their classical values at a neighbourhood of a given spacetime point small w.r.t. the length scale of the QFT. Of course this should be made more precise

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@Moshe: Regarding "exact", I agree it's a somewhat vague notion. I think it's a good idea to define "exact" as "reducible to ODE"

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@Squark, saddle point methods don't apply to oscillatory integrals, even one dimensional integrals (instead you have steepest descent method, which is more subtle). Generally classical solutions correspond approximately to coherent states, or other configurations of many quanta. Hard to imagine, though perhaps not impossible, that asymptotic freedom helps you there. Perhaps if you cook up a special situation, but I cannot think of one right now.

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Also, classical YM has no scale, so what would set the scale (say the mass) of any solution? Either it is a modulus, or it is zero or infinity,

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@Moshe, I'm not using saddle points methods directly. Decreasing the bare coupling constant is equivalent to decreasing hbar. It is also equivalent to decreasing the renormalized coupling constant as a fixed scale lambda. Decreasing the renormalized coupling constant is equivalent to changing the scale (zooming in/out) in a non-trivial way defined by the renormalization group. However small coupling constant is guaranteed to correspond to small length scale due to asymptotic freedom. Hence zooming in to small length scales is the same as decreasing hbar.

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let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/1626/discussion-between-squark-and-moshe)

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@Moshe, regarding scale, classical solutions obviously come in families generate by dilation. The only exception is if the solution is scale-invariant but then it has to be singular at the origin. Why is that a problem?

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I am having difficulty with chat. Let me just say that it is not clear to me that smooth classical solutions, large collection of soft quanta, benefit from asymptotic freedom. I think cooking up a situation where classical solutions are relevant might give you a hint what type of solutions may exist. Beyond that, I really have nothing useful to say...

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Wu and Yang (1968) found a static solution to the sourceless SU(2) Yang-Mills equations, (please, see the following two relatively recent articles containing a rather detailed description of the solution: Marinho, Oliveira, Carlson, Frederico and Ngome The solution constitutes of a generalization of the Abelian Dirac monopole. The vector potential is given by:

$A_i^a=g \epsilon_{iaj}x^j\frac{f(r)}{r^2}$

where $f(r)$ satisfies a nonlinear radial equation (The Wu-Yang equation) obtained from the substitution this ansatz into the Yang-Mills field equations.The Wu-Yang monopole has a singularity at the origin, in which the magnetic energy density diverges. The first article contains references to phenomenological works involving the Wu-Yang monopole.

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answered Oct 18, 2011 by (4,355 points)
OK, but what about nonsingular solutions?

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If you allow the function $f$ to be time dependent $f(r,t)$, it is known that this Ansatz admits regular solutions. I couldn't find references for these solutions, but I think that it is no to hard to reproduce them.

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There is an old review, Alfred Actor, Classical solutions of SU(2) Yang—Mills theories, Rev. Mod. Phys. 51, 461–525 (1979), that provides some of the known solutions of SU(2) gauge theory in Minkowski (monopoles, plane waves, etc) and Euclidean space (instantons and their cousins). For general gauge groups you can get solutions by embedding SU(2)'s. For instantons the most general solution is known, first worked out by ADHM (Atiyah, Hitchin, Drinfeld, Manin) for the classical groups SU,SO,Sp, and then by Bernard, Christ, Guth, Weinberg for exceptional groups. The latest twist on the instanton story is the construction of solutions with non-trivial holomony: Periodic instantons with nontrivial holonomy''. Kraan, van Baal, Nucl.Phys. B533 (1998) 627-659. There is a nice set of lecture notes by David Tong on topological solutions with different co-dimension (instantons, monopoles, vortices, domain walls). Note, however, that except for instantons these solutions typically require extra scalars and broken U(1)'s, as you may find in susy gauge theories.

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answered Oct 19, 2011 by (720 points)
The OP wanted Loretzian solutions of pure gauge theory. I know, this does not leave much.

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A number of exact solutions (mostly those that are invariant under a certain subgroup of the full symmetry group of the system in question) for the SU(2) Yang Mills equations, including the case of Minkowski signature, can be also found in this survey paper:

R.Z. Zhdanov, V.I. Lahno, Symmetry and Exact Solutions of the Maxwell and SU(2) Yang-Mills Equations, arXiv:hep-th/0405286

Also, if one is willing to keep the Minkowski signature while allowing for a global topology other than R4, this paper could be of interest:

A.D. Popov, Explicit Non-Abelian Monopoles and Instantons in SU (N) Pure Yang-Mills Theory, Phys. Rev. D77 (2008), 125026, arxiv:0803.3320

answered Apr 22, 2014 by (95 points)
edited Apr 22, 2014
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Exact solutions could not be the right way to understand infrared behavior of Yang-Mills theory. As we know from quantum field theory, we can start with some approximation (weak coupling). With this in mind, it can be proved that the following holds (see http://arxiv.org/abs/0903.2357) for a gauge coupling going formally to infinity

$$A_\mu^a(x)=\eta_\mu^a\phi(x)+O(1/\sqrt{N}g)$$

being $\eta_\mu^a$ a set of constants and $\phi(x)$ a solution to the equation

$$\Box\phi(x)+\lambda\phi(x)^3=0.$$

provided $\lambda=Ng^2$. This is the content of the so called mapping theorem. The relevant aspect of this theorem is that one can provide a set of exact solutions for the scalar field in the form

$$\phi(x)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p\cdot x+\theta,i)$$

being $\mu$ and $\theta$ constants, sn a Jacobi elliptic function and provided the following dispersion relation holds

$$p^2=\mu^2\left(\frac{\lambda}{2}\right)^\frac{1}{2}.$$

That is, one has massive classical solutions even if we started from massless equations. So, we can start from these classical approximate solutions to build up an infrared quantum field theory for the Yang-Mills field and displaying in this way a mass gap (see http://arxiv.org/abs/1011.3643).

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answered Nov 8, 2011 by (345 points)
Terry Tao has pointed out -- see http://en.wikipedia.org/wiki/Talk:Yang%E2%80%93Mills_theory#Removed_.22Integrable_solutions_of_classical_Yang-Mills_equations_and_QFT.22" -- that the so-called 'mapping theorem' is false. Have you actually checked that this is a solution to the Yang-Mills equations?

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Terry agreed with the proof given in the paper cited above. Please, see http://wiki.math.toronto.edu/DispersiveWiki/index.php/Talk:Yang-Mills_equations : This is not an exact solution but, as I prove, an asymptotic one.

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1st, I'm alluding the the UV behavior not to the IR one. 2nd I'm not sure I fully understand what you're saying here. Are you saying that solutions of this non-linear PDE which is the equation of motion for massless phi^4 theory yield approximate solutions of the Yang-Mills equation? And that the approximation is good for large N?

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@Squark. Ok, sorry for being unhelpful here. Anyhow, this is a theorem. Let me state quite differently, with a reference to Smilga's book http://www.amazon.com/LECTURES-QUANTUM-CHROMODYNAMICS-V-Smilga/dp/book-citations/9810243316. One can seek solutions in the form $A_0=0$ and $A_i$ just depending on t and not $\bf x$. One can find exact solutions by taking $$A_1^1(t)=A_2^2(t)=A(t)$$ and one has for the equations of motion $$\ddot A(t)+2g^2 A(t)^3=0.$$ This the equation of the quartic scalar field and twos just remap each other. This is observation is the starting point for my theorem.

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