# Pathological expression for field strength contractions in a curved background

I am trying to define in Mathematica the quantity $$\star F^{\mu}=\frac{1}{2}\epsilon^{\mu\alpha\beta}F_{\alpha\beta}$$, where $$F_{\mu\nu}=\nabla_\mu A_\nu-\nabla_\nu A_\mu+\left[A_\mu,A_\nu\right]$$. Here $$\nabla$$ is the spacetime covariant derivative of the metric $$ds^2=-dt^2+R^2dr^2+L^2d\phi^2$$. In the field strength $$F_{\mu\nu}$$ the gauge field $$A_\mu$$ is defined as $$A=A^j_\mu t_j dx^\mu$$ where $$t_j=i\sigma_j$$ with $$\sigma_j$$ the $$j=1,2,3$$ Pauli matrices and $$A_\mu=\lambda(x) U^{-1}\partial_\mu U\,\quad U(x)=\cos[\alpha(x)]I_{2\times 2}+\sin[\alpha(x)]n^it_i,$$ with $$n^1=\sin\left[\Theta(x)\right]\cos\left[\Phi(x)\right],\quad n^2=\sin\left[\Theta(x)\right]\sin\left[\Phi(x)\right],\quad n^3=\cos\left[\Theta(x)\right].$$ When I compute $$\star F^\mu$$ with the previous functions, I obtain an expression of the form Sin[1[t, r, \[Phi]]] or Sin[2 1[t, r, \[Phi]]], which seems pathological. This is my question.

Let me show my code below and explain it, step by step.

First, the spacetime is defined as

Coords = {t, r, \[Phi]};
gdd = {{-1, 0, 0}, {0, \[Rho]^2, 0}, {0, 0, L^2}}
gUU = Inverse[gdd]
dimM = Dimensions[gdd][[1]]


Then I define the General Relativity tensors, i.e., Christoffel Symbols, Riemann tensors and so on (this is not important, however, I put them for completeness and perhaps this code will be helpful for somebody)

\[CapitalGamma]Udd =
1/2 Table[
Sum[gUU[[\[Mu] + 1, \[Tau] +
1]] (D[gdd[[\[Lambda] + 1, \[Tau] + 1]],
Coords[[\[Nu] + 1]]] +
D[gdd[[\[Nu] + 1, \[Tau] + 1]], Coords[[\[Lambda] + 1]]] -
D[gdd[[\[Nu] + 1, \[Lambda] + 1]],
Coords[[\[Tau] + 1]]]), {\[Tau], 0, dimM - 1}], {\[Mu], 0,
dimM - 1}, {\[Nu], 0, dimM - 1}, {\[Lambda], 0, dimM - 1}] //
Simplify;
RUddd = Table[
D[\[CapitalGamma]Udd[[\[Alpha] + 1, \[Beta] + 1, \[Delta] + 1]],
Coords[[\[Gamma] + 1]]], {\[Alpha], 0, dimM - 1}, {\[Beta], 0,
dimM - 1}, {\[Gamma], 0, dimM - 1}, {\[Delta], 0, dimM - 1}] -
Table[D[\[CapitalGamma]Udd[[\[Alpha] + 1, \[Beta] + 1, \[Gamma] +
1]], Coords[[\[Delta] + 1]]], {\[Alpha], 0,
dimM - 1}, {\[Beta], 0, dimM - 1}, {\[Gamma], 0,
dimM - 1}, {\[Delta], 0, dimM - 1}] +
Table[Sum[\[CapitalGamma]Udd[[\[Mu] + 1, \[Beta] + 1, \[Delta] +
1]] \[CapitalGamma]Udd[[\[Alpha] + 1, \[Mu] + 1, \[Gamma] +
1]], {\[Mu], 0, dimM - 1}], {\[Alpha], 0,
dimM - 1}, {\[Beta], 0, dimM - 1}, {\[Gamma], 0,
dimM - 1}, {\[Delta], 0, dimM - 1}] -
Table[Sum[\[CapitalGamma]Udd[[\[Mu] + 1, \[Beta] + 1, \[Gamma] +
1]] \[CapitalGamma]Udd[[\[Alpha] + 1, \[Mu] + 1, \[Delta] +
1]], {\[Mu], 0, dimM - 1}], {\[Alpha], 0,
dimM - 1}, {\[Beta], 0, dimM - 1}, {\[Gamma], 0,
dimM - 1}, {\[Delta], 0, dimM - 1}] // Simplify;
Rdd = Table[
Sum[RUddd[[\[Alpha] + 1, \[Beta] + 1, \[Alpha] + 1, \[Delta] +
1]], {\[Alpha], 0, dimM - 1}], {\[Beta], 0,
dimM - 1}, {\[Delta], 0, dimM - 1}] // Simplify;
R = Sum[gUU[[\[Beta] + 1, \[Delta] + 1]] Rdd[[\[Beta] + 1, \[Delta] +
1]], {\[Beta], 0, dimM - 1}, {\[Delta], 0, dimM - 1}] //
Simplify;
Einsteindd = Rdd - 1/2 gdd R;


Here begins the important part. I define the Pauli matrices, the normal vector $$n^i$$, the matrix $$U$$, its inverse and the gauge field $$A_\mu$$

ColorCoords = {t, r, \[Phi]};
dimC = 3;
t1 = I PauliMatrix[1];
t2 = I PauliMatrix[2];
t3 = I PauliMatrix[3];
td = {{t1}, {t2}, {t3}};
tdd = -(1/2) Table[
Tr[td[[a + 1, 1]].td[[b + 1, 1]]], {a, 0, dimC - 1}, {b, 0,
dimC - 1}];
tUU = Inverse[tdd];
tU = Sum[Table[tUU[[a + 1, b + 1]] td[[b + 1]], {b, 0, dimC - 1}], {a,
0, dimC - 1}];
nU = {{Sin[\[CapitalTheta][t, r, \[Phi]]] Cos[\[CapitalPhi][t,
r, \[Phi]]]}, {Sin[\[CapitalTheta][t,
r, \[Phi]]] Sin[\[CapitalPhi][t,
r, \[Phi]]]}, {Cos[\[CapitalTheta][t, r, \[Phi]]]}};
nd = Table[
Sum[nU[[a + 1]] tdd[[a + 1, b + 1]], {a, 0, dimC - 1}], {b, 0,
dimC - 1}];
U = Cos[\[Alpha][t, r, \[Phi]]] IdentityMatrix[2] +
Sin[\[Alpha][t, r, \[Phi]]] Sum[
nU[[a + 1, 1]] td[[a + 1, 1]], {a, 0, dimC - 1}] // Simplify;
Uinv = Inverse[U] // Simplify;
Ad = \[Lambda][t, r, \[Phi]] Table[
Uinv.D[U, ColorCoords[[\[Mu] + 1]]], {\[Mu], 0, dimC - 1}] //
FullSimplify;


Defining now the covariant derivative of the gauge field $$\nabla_\mu A_\nu$$

covDAd = Table[D[Ad, Coords[[\[Mu] + 1]]], {\[Mu], 0, dimM - 1}] -
Sum[Table[\[CapitalGamma]Udd[[\[Lambda]1 + 1, \[Mu] + 1, \[Nu] +
1]] Ad[[\[Lambda]1 + 1]], {\[Mu], 0, dimM - 1}, {\[Nu], 0,
dimM - 1}], {\[Lambda]1, 0, dimM - 1}] // Simplify;


we are now in a position to define the field strength

Fdd = covDAd - Transpose[covDAd] +
Ad[[\[Nu] + 1]].Ad[[\[Mu] + 1]], {\[Mu], 0, dimM - 1}, {\[Nu], 0,
dimM - 1}];


Now we are going to define the components of the dual field strength $$*F^\mu$$. In three-dimensions, the Levi-Civita symbol is defined as

LCd = Normal[LeviCivitaTensor[3]];
LCU = Normal[-LeviCivitaTensor[3]];
\[Epsilon]UUU = LCU;
\[Epsilon]ddd = LCd;


then, we define $$\star F^\mu=\frac{1}{2}\epsilon^{\mu\alpha\beta}F_{\alpha\beta}$$ as

dualFU = 1/
2 Table[Sum[\[Epsilon]UUU[[\[Mu] + 1, \[Alpha] + 1, \[Beta] +
1]] Fdd[[\[Alpha] + 1, \[Beta] + 1]], {\[Alpha], 0,
dimM - 1}, {\[Beta], 0, dimM - 1}], {\[Mu], 0, dimM - 1}];


Observe that if we compute dualFU[[1, 1]] we quickly obtain terms of the form Cos[2 1[t, r, \[Phi]]], Sin[2 1[t, r, \[Phi]]] and so on. What am I doing wrong?

• I tried to run your code, but ran into errors due to nU and tdd not being defined, causing Part to complain . These affect the evaluation of nd and U . If I push on anyway and evaluate the remaining terms, I get dualFU = {{{0,0},{0,0}},{{0,0},{0,0}},{{0,0},{0,0}}} and dualFU[[1,1]] = {0,0}. It's likely the missing terms will alter this result however, since you appear to compute derivatives of U, etc.
– jdp
Dec 18, 2022 at 2:09
• I edited the question and added the definition of tdd, so tell me now if you are able to compile the code. Dec 18, 2022 at 9:54
• There are typos due to mixture of definition Ad = \[Lambda][t, r, \[Phi]] ... and index $\lambda$ in covDAd. Try covDAd = Table[D[Ad, Coords[[\[Mu] + 1]]], {\[Mu], 0, dimM - 1}] - Sum[Table[\[CapitalGamma]Udd[[\[Lambda]1 + 1, \[Mu] + 1, \[Nu] + 1]] Ad[[\[Lambda]1 + 1]], {\[Mu], 0, dimM - 1}, {\[Nu], 0, dimM - 1}], {\[Lambda]1, 0, dimM - 1}] // Simplify; Dec 18, 2022 at 12:15
• Ok, I corrected the typos, but the error persists in dualFU. Dec 18, 2022 at 12:22
• Maybe you need to refresh kernel? Dec 18, 2022 at 13:57

    dualFU =