# Dirac Operator in de Sitter background

I am attempting to write the Dirac operator in a curved background, and eventually solve the equation as a second order PDE, since I am attempting to bring it into Klein-Gordon form. Essentially what I am trying to do is for the Dirac operator $$$$D=\gamma^{a} e_{a}^{\mu}\left(\partial_{\mu}+\frac{1}{2} \sigma^{b c} \omega_{b c \mu}\right)=\gamma^{a} D_{a} \equiv \gamma^{\mu} D_{\mu}$$$$ have the Lichnerowicz formula $$$$D^{2}=\nabla^{2}-\frac{1}{4} R$$$$ where $$$$\nabla^{2} \equiv \frac{1}{\sqrt{g}} D_{\mu} \sqrt{g} g^{\mu \nu} D_{\nu}$$$$ essentially, $$D_{\mu}=\partial_{\mu}+\frac{1}{2} \sigma^{b c} \omega_{b c \mu}$$, $$\partial_{\mu}$$ is the partial/directional derivative and $$g=\det{g_{\mu \nu}}$$. The operator would be applied to the spinor $$\Psi$$, such that I could essentially write the Dirac equation in Klein-Gordon form as $$$$\left(D^{2} + m^{2}\right)\Psi=0$$$$ So far, I have the following code

xu = {t, x, y, z};
n = Length[xu];
gd = {{-1, 0, 0,
0}, {0, (1 - K*(y^2 + z^2))/(1 - K*(x^2 + y^2 + z^2)),
K*x*y/(1 - K*(x^2 + y^2 + z^2)),
K*x*z/(1 - K*(x^2 + y^2 + z^2))}, {0,
K*x*y/(1 - K*(x^2 + y^2 + z^2)), (1 - K*(x^2 + z^2))/(1 -
K*(x^2 + y^2 + z^2)), K*y*z/(1 - K*(x^2 + y^2 + z^2))}, {0,
K*x*z/(1 - K*(x^2 + y^2 + z^2)),
K*y*z/(1 - K*(x^2 + y^2 + z^2)), (1 - K*(x^2 + y^2))/(1 -
K*(x^2 + y^2 + z^2))}}; (*Let this be the metric*)
gu = Simplify[Inverse[gd]];(*Let this be its inverse*)
\[CapitalGamma] = 1/2 Table[
Sum[ gu[[\[Alpha], \[Beta]]] ( \!$$\*SubscriptBox[\(\[PartialD]$$, $$xu[[\[Mu]]]$$]$$gd[[\[Beta], \ \ \[Nu]]]$$\) +  \!$$\*SubscriptBox[\(\[PartialD]$$, $$xu[[\[Nu]]]$$]$$gd[[\[Beta], \ \ \[Mu]]]$$\) -  \!$$\*SubscriptBox[\(\[PartialD]$$, $$xu[[\[Beta]]]$$]$$gd[[\[Mu], \ \ \[Nu]]]$$\) ), {\[Beta], 1, n} ],
{\[Alpha], 1, n}, {\[Mu], 1, n}, {\[Nu], 1,
n} ];(*Christoffel symbols/affine connection*)
Simplify[Flatten[\[CapitalGamma]]];
Eud = {{1, 0, 0,
0}, {0, (y^2 + z^2 + x^2 Sqrt[1/(1 - K (x^2 + y^2 + z^2))])/(x^2 +
y^2 + z^2), (x y (-1 +
Sqrt[1/(1 - K (x^2 + y^2 + z^2))]))/(x^2 + y^2 +
z^2), (x z (-1 + Sqrt[1/(1 - K (x^2 + y^2 + z^2))]))/(x^2 +
y^2 + z^2)}, {0, (x y (-1 +
Sqrt[1/(1 - K (x^2 + y^2 + z^2))]))/(x^2 + y^2 + z^2), (x^2 +
z^2 + y^2 Sqrt[1/(1 - K (x^2 + y^2 + z^2))])/(x^2 + y^2 +
z^2), (y z (-1 + Sqrt[1/(1 - K (x^2 + y^2 + z^2))]))/(x^2 +
y^2 + z^2)}, {0, (x z (-1 +
Sqrt[1/(1 - K (x^2 + y^2 + z^2))]))/(x^2 + y^2 +
z^2), (y z (-1 + Sqrt[1/(1 - K (x^2 + y^2 + z^2))]))/(x^2 +
y^2 + z^2), (x^2 + y^2 +
z^2 Sqrt[1/(1 - K (x^2 + y^2 + z^2))])/(x^2 + y^2 +
z^2)}} ;(*Let this be the tetrad/vierbein*)
Edu = Simplify[
Inverse[Eud]]; (*The dual of the tetrad, i.e. the inverse vierbein*)
(\[Eta] =
Simplify[
Table[Eud[[a]].gu.Eud[[b]], {a, 1, n}, {b, 1, n}]]) // MatrixForm
Simplify[Table[ Sum[Eud[[a, \[Mu]]] Edu[[b, \[Mu]]], {\[Mu], 1, n}], {a, 1,
n}, {b, 1,
n} ]] // MatrixForm (*This checks the duality condition of the \
Omega := omega = Simplify[ Table[
Sum[
Edu[[a, \[Rho]]] Eud[[
b, \[Nu]]] \[CapitalGamma][[\[Nu], \[Mu], \[Rho]]], {\[Rho],
1, n}, {\[Nu], 1, n}]
- Sum[Edu[[a, \[Nu]]] \!$$\*SubscriptBox[\(\[PartialD]$$, $$xu[[\[Mu]]]$$]$$Eud[[ b, \ \[Nu]]]$$\), {\[Nu], 1, n}],
{\[Mu], 1, n}, {b, 1, n}, {a, 1,
n} ]] (*spin connection of the above metric*)
listOmega :=
Table[If[UnsameQ[Omega[[\[Mu], b, a]],
0], {ToString[\[CapitalOmega][\[Mu], b, a]],
Omega[[\[Mu], b, a]]}] , {\[Mu], 1, n}, {b, 1, n}, {a, 1, b}]
TableForm[Partition[DeleteCases[Flatten[listOmega], Null], 2],
TableSpacing -> {2, 2}]


I am a bit confused in writing the part involving the term $$\sigma^{bc}=\frac{i}{2}[\gamma^{b},\gamma^{c}]$$ in the spin connection. I am a bit at a loss in how to write the commutator, in order to define the operator which will inevitable act on the spinor. Must I use some sort of package such as FeynCalc or FeynRules? My ultimate goal is to define the Laplacian squared above, i.e. $$\nabla^{2}$$, which I will then use to derive the squared Dirac operator and esentially bring my Dirac equation to Klein-Gordon form. Any help would be greatly appreciated.

• I would suggest that you do a bit more algebra by hand for the term you are interested in and then plug everything in Mma. Maybe you would be interested in this:mathematica.stackexchange.com/questions/183495/… . I understand that this does not answer your question, but maybe it is an alternative that you have not considered
– user49048
Commented Apr 13, 2020 at 12:02
• Thank you so very much for your response; nonetheless, the post that you have referred to helped me in formulating the question that I just posted (I also got a few ideas from it). Thank you for you suggestion regarding the algebra, I went ahead and did that and found a very nice symmetry within my $\frac{1}{2} \sigma^{b c} \omega_{b c \mu}$. Yet, I would still like to type some code in order to make sure that my calculations are correct. Commented Apr 13, 2020 at 12:14
• glad you liked that post. Regarding the symmetry of the $\sigma^{bc} \omega_{bc \mu}$ and writing some code, I cannot help because I never did it using code and I am a bit swamped to sit and think. Sorry. Hopefully, someone else can help.
– user49048
Commented Apr 13, 2020 at 13:08
• Don't use the capital letters for any variables or constants since they usually do have special meanings in the system and it its the case with K being a default name for a summation index in symbolic sums. Commented Apr 13, 2020 at 16:56