I've been lately trying to replicate a calculation but it is not working. The main problem is simple. I just have to replace an ansatz in a equation, that is

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(sorry for putting an image of that ) actually we should get 3 equations, 2 of them are the same (the expretion for f) and the third one is the skyrme equation.

well i've been trying to solve this an here it is

First i difined the coordinates

x[1] = r;x[2] = \[Theta];x[3] = z;x[4] = t;

Then the Pauli matrices

\[Sigma] [1] = PauliMatrix[1]; \[Sigma][2] =PauliMatrix[2]; \[Sigma][3] = PauliMatrix[3];

Now we have the gamma matrices

\[Gamma][1] = -I \[Sigma][1]; \[Gamma][2] = -I \[Sigma][2]; \[Gamma][3] = \[Sigma][3];

The Spin matrices

Do[\[CapitalGamma][i] =KroneckerProduct[\[Gamma][i], IdentityMatrix[2]], {i, 1, 3}];

The Isospin matrices

Do[\[Tau][i] = KroneckerProduct[IdentityMatrix[2], \[Sigma][i]], {i,1, 3}];

Then, the spin-isospin eigenfunctions

\[Psi]\[Dagger] = ConstantArray[0, {1, 4}]; \[Psi]\[Dagger][[1, 1]] =Subscript[u, 1][r]* Exp[-I*l*G2[r, \[Theta], z, t]]; \[Psi]\[Dagger][[1, 2]]=-I*Subscript[u, 2][r]* Exp[-I*(l + n)*G2[r, \[Theta], z, t]]; \[Psi]\[Dagger][[1, 3]] =Subscript[v, 1][r]*  Exp[-I*(l + 1)*G2[r, \[Theta], z, t]];\[Psi]\[Dagger][[1, 4]] = -I*Subscript[v, 2][r]*Exp[-I*(l + n + 1)*G2[r, \[Theta], z, t]];


\[Psi] = ConstantArray[0, {4, 1}];\[Psi][[1, 1]] = Subscript[u, 1][r]*Exp[I*l*G2[r, \[Theta], z,t]];\[Psi][[2, 1]] = I*Subscript[u, 2][r]*Exp[I*(l + n)*G2[r, \[Theta], z, t]]; \[Psi][[3,1]] =Subscript[v, 1][r]*Exp[I*(l + 1)*G2[r, \[Theta], z, t]]; \[Psi][[4, 1]] = I*Subscript[v, 2][r]*Exp[I*(l + n + 1)*G2[r, \[Theta], z, t]];

Now the ansatz for the field $\phi$

\[Phi][1] = Sin[f[r]]*Cos[G1[r, \[Theta], z, t]];\[Phi][2]=Sin[f[r]]*Sin[G1[r, \[Theta], z, t]];\[Phi][3] = Cos[f[r]];

The Metric

h = ConstantArray[0, {4, 4}]; h[[3, 3]] = 1; h[[1, 1]] = 1; h[[2, 2]] = r*r; h[[4, 4]] = -1;dh =Det[h] // Simplify;invh = Simplify[Inverse[h]];raizmdh = Sqrt[-Det[h]];

And finally some parameters

G1[r, \[Theta], z, t] = (n \[Theta] + Pi/2); G2[r, \[Theta], z, t] = \[Theta]; n = 1; l = -1;B[1] =0;B[2] = 0;B[3] = B;

Here i defined the box (Laplacian)

Do[Boxx[\[Phi][i]] = (1/raizmdh) Sum[D[raizmdh invh[[mu, nu]] D[\[Phi][i], x[nu]], x[mu]], {mu, 1,3}, {nu, 1, 3}], {i, 1, 3}]

And finally we have the equation

Table[Boxx[\[Phi][i]] - 2 Sum[Signature[{i, j, k}] D[\[Phi][k], x[j]], {j, 1, 3}, {k, 1, 3}] +B[i] - g*\[Psi]\[Dagger].\[CapitalGamma][3].\[Tau][i].\[Psi], {i, 1, 3}]

Running the program i get something similar but not the right equation

It looks like the equation that i must to get but in general is wrong. I guess is a problem with the definition of the product with the spin-isospin matrix $\psi$ and the isospin matrix $\tau$. The isospin part of $\psi$ should talk only with isospin matrices and the spin matrices should talk with the spin part.

I think that is the main problem because i don't get the right equations. if someone that understand better this better may help me to find the problem i would be gratefull.


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