I did it for you

11/03/2009

It is very easy to show, from Yang-Mills equations, how to obtain a scalar field equation through the Smilga’s choice. Let us write down 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

using the choice A_1^1=A_2^2=A_3^3=\phi. This is really a great simplification. Smilga, in his book, already checked this for us but we give here the full computation. From above eqautions, the only critical term is the following

f^{abc}A^{b\mu}(\partial_\mu A^c_\nu-\partial_\nu A^c_\mu)

as this term would produce terms deviating from the known form of the scalar theory. For SU(2) we have f^{abc}=\epsilon^{abc} the fully-antisymmetric Levi-Civita tensor. This means that we will have

\epsilon^{a1c}A^{11}(\partial_1A_\nu^c-\partial_\nu A_1^c)+

\epsilon^{a2c}A^{22}(\partial_2A_\nu^c-\partial_\nu A_2^c)+

\epsilon^{a3c}A^{33}(\partial_3A_\nu^c-\partial_\nu A_3^c).

Where we have used largely Smilga’s choice. Now do the following. Take the following components to evolve \nu=1 a=1, \nu=2  a=2 and \nu=3, a=3. It easy to see that the possible harmful term is zero with the Smilgaì’s choice. Now, for the cubic term you should use the useful relation

\epsilon^{abc}\epsilon^{cde}=\delta_{ad}\delta_{be}-\delta_{ae}\delta_{bd}

and you will get back the quartic term.

The gauge fixing term can be easily disposed of through a rescaling of spatial variables while the kinematic term gives the right contribution. You will get three identical equations for the scalar field.

Of course, Smilga in his book already did this and I repeated his computations after the Editor of PLB asked for a revision having the referee already put out this problem. The Editorial work was done very well and two referees read the paper emphasizing errors where they were.

Finally, Tao’s critcism does not apply as I said. This does not mean that what he says is wrong. This means that does not apply to my case.

Update: As the question of the gauge fixing term appears so relevant, let me fix it once and for all. Firstly, I would like to point out that these solutions belong to a class of solutions in the Maximal Abelian Gauge (MAG). But let us forget about this and consider the question of gauge fixing. This term is arbitrarily introduced in the Lagrangian of the field in order to fix the gauge when a quantization procedure is applied. Due to gauge invariance and the fact that becomes an exact differential after partial integration, it useful to have it there for the above aims. The form that it  takes is

\frac{1}{\alpha}(\partial A)^2

and is put directly into the Lagrangian. How does this term become with the Smilga’s choice? One has

\frac{1}{\alpha}\sum_{i=1}^3(\partial_iA_i^i)^2

and the final effect is a pure rescaling into the space variables of the scalar field. In this way the argument is made consistent. One cannot take the other way around for the very nature of this term and claiming the result is wrong.

This particular class of solutions belongs to the subgroup of SU(N) given by the direct product of U(1). This is a property of MAG and all the matter is really consistent and works.

Finally, I invite people commenting this and other posts to limit herself to polite responses and in the realm of scientific discussion. Of course, doing something wrong happens and happened to anyone working in a scientifc endeavour for the simple reason that she is really doing things. People that only do useless criticisms boiling down to personal offenses are kindly invited to refrain from further interventions.

Update 2: I will get a paper published about this matter. Please, check here.

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