# Problem replacing an ansatz in a field equation with spin-isospin matrices

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 (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 = r;x = \[Theta];x = z;x = t;


Then the Pauli matrices

\[Sigma]  = PauliMatrix; \[Sigma] =PauliMatrix; \[Sigma] = PauliMatrix;


Now we have the gamma matrices

\[Gamma] = -I \[Sigma]; \[Gamma] = -I \[Sigma]; \[Gamma] = \[Sigma];


The Spin matrices

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


The Isospin matrices

Do[\[Tau][i] = KroneckerProduct[IdentityMatrix, \[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]];


asd

\[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] = Sin[f[r]]*Cos[G1[r, \[Theta], z, t]];\[Phi]=Sin[f[r]]*Sin[G1[r, \[Theta], z, t]];\[Phi] = 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 =0;B = 0;B = 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].\[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.