# Solving the Dirac equation in an arbitrary metric [closed]

I want to solve Dirac equation in a metric like $$ds^2=g(u,v)\,du\,dv$$. The relations of $$u$$ and $$v$$ with Minkowski coordinates $$t$$ and $$x$$ are given by functions $$A$$ and $$B$$, $$t=A(u,v)$$ and $$x=B(u,v)$$.

First I want to find the Dirac equation in these coordinates and then I need its solution to find the Bogoliubov coefficients.

How can I use Mathematica for that?

I also installed the FeynCalc package, but I don't know how to use it. Is there any code for solving Dirac equation?

• i also want many things? Oct 9, 2018 at 21:57
• It's always best to start with a modest example. Click on the feyncalc tag beneath your question and see if you can find help there. Then formulate a concrete example as much as you can in copy-pastable code. That will increase your chances of getting a meaningful answer.
– Jens
Oct 10, 2018 at 2:36
• Also, what are $t$, $x$, $A$ and $B$? They do not appear in the equation. Oct 10, 2018 at 3:12
• I do not think the question is too broad, but it really requires more details and a demonstration of an attempt.
– Johu
Oct 10, 2018 at 6:35
• @aber, in order to dsolve or ndsolve with Mma there are some steps needed. Christoffel symbols, veilbeins, spin-connection, then the Dirac operator, then project out the chiral equations and then obtain a second order de that you can solve. An as the others mentioned, the more info you share will give you more chances for an answer. Also, I am sure that Feyncalc in not a necessity. Oct 10, 2018 at 10:31

Disclaimer: The code presented below is not entirely written by me. I found some pieces online, wrote some others, tweaked them a bit, and I am pasting the final thing that I am using in my notebooks. I am just answering as I have been dealing a lot with the Dirac lately, and hopefully, I can help a bit.

Firstly, let me give some piece of code that defines a metric, and calculates all the ingredients in order to compute the Dirac operator on a curved manifold.

I will use a $$5$$ dimensional AdS spacetime, which I write in the following way

$$ds^2 = \frac{du^2 + dx_{\mu} dx^{\mu}}{u^2}$$

where $$\mu$$ is a Minkowskian index; runs from $$0$$ to $$3$$.

Clear[coord, metric, inversemetric, affine, t, x, y, z, u]
(*The dimension n of the spacetime*)
n = 5;
coord = {t, x, y, z, u};
(*The metric with indices down*)

metric = {{-(1/u^2), 0, 0, 0, 0}, {0, 1/u^2, 0, 0, 0}, {0, 0, 1/u^2,
0, 0}, {0, 0, 0, 1/u^2, 0}, {0, 0, 0, 0, 1/u^2}};
metric // MatrixForm;
inversemetric = Simplify[Inverse[metric]];
inversemetric // MatrixForm;
(*Test N^o 1.*)
metric.inversemetric;
% // MatrixForm


The following bit is calculating and displaying the Christoffel symbols. Note that $$\Gamma[1,2,3]$$ is the symbol $$\Gamma^1_{23}$$ and also note that only the independent components are displayed.

(*The Christoffel symbols*)
(*Subscript[\[CapitalGamma]^\[Lambda], \
\[Mu]\[Nu]]=1/2g^\[Lambda]\[Sigma](\!$$\*SubscriptBox[\(\[PartialD]$$, $$\[Mu]$$]\
\*SubscriptBox[$$g$$, $$\[Sigma]\[Nu]$$]\)+\!$$\*SubscriptBox[\(\[PartialD]$$, $$\[Nu]$$]\
\*SubscriptBox[$$g$$, $$\[Sigma]\[Mu]$$]\)-\!$$\*SubscriptBox[\(\[PartialD]$$, $$\[Sigma]$$]\
\*SubscriptBox[$$g$$, $$\[Mu]\[Nu]$$]\))*)
affine :=
affine = Simplify[Table[(1/2)*Sum[(inversemetric[[i, s]])*
(D[metric[[s, j]], coord[[k]] ] +
D[metric[[s, k]], coord[[j]] ] -
D[metric[[j, k]], coord[[s]] ]), {s, 1, n}],
{i, 1, n}, {j, 1, n}, {k, 1, n}] ]
listaffine :=
Table[If[UnsameQ[affine[[i, j, k]],
0], {ToString[\[CapitalGamma][i, j, k]], affine[[i, j, k]]}] , {i,
1, n}, {j, 1, n}, {k, 1, j}]
TableForm[Partition[DeleteCases[Flatten[listaffine], Null], 2],
TableSpacing -> {2, 2}]


Here is the computation of the funf-beins and some consistency checks

Eup = { {1/u, 0, 0, 0, 0},
{0, 1/u, 0, 0, 0},
{0, 0, 1/u, 0, 0}, {0, 0, 0, 1/u, 0}, {0, 0, 0, 0, 1/u} };
Edown = Inverse[Eup];
(*Test N^o2.*)
(*Part I:\!$$\*SubsuperscriptBox[\(e$$, $$\[Mu]$$, $$a$$]\
\*SuperscriptBox[$$g$$, $$\[Mu]\[Nu]$$]
\*SubsuperscriptBox[$$e$$, $$\[Nu]$$, $$b$$]\) = \[Eta]^ab*)
(\[Eta] =
Table[
Eup[[a]].inversemetric.Eup[[b]], {a, 1, n}, {b, 1,
n} ] ) // MatrixForm
(*Part II: \!$$\*SubsuperscriptBox[\(e$$, $$\[Mu]$$, $$a$$]\
\*SubsuperscriptBox[$$e$$, $$a$$, $$\[Nu]$$]\) = Subsuperscript[\
\[Delta], \[Nu], \[Mu]]*)

Table[ Sum[Eup[[a, \[Mu]]] Edown[[a, \[Nu]]], {a, 1, n}], {\[Mu], 1,
n}, {\[Nu], 1, n} ] // MatrixForm
(*Part III:\!$$\*SubsuperscriptBox[\(e$$, $$\[Mu]$$, $$a$$]\
\*SubsuperscriptBox[$$e$$, $$b$$, $$\[Mu]$$]\) = Subsuperscript[\
\[Delta], b, a]*)

Table[ Sum[Eup[[a, \[Mu]]] Edown[[b, \[Mu]]], {\[Mu], 1, n}], {a, 1,
n}, {b, 1, n} ] // MatrixForm


This is the calculation of the spin-connection

(*The spin connection*)
(*\!$$\*SubsuperscriptBox[ SubscriptBox[\(\[CapitalOmega]$$, $$\[Mu]$$], $$b$$, $$a$$] = $$\*SubsuperscriptBox[\(e$$, $$a$$, $$\[Rho]$$]\
\*SubsuperscriptBox[$$e$$, $$\[Nu]$$, $$b$$]\
\*SubsuperscriptBox[$$\[CapitalGamma]$$, $$\[Mu]\[Rho]$$, $$\[Nu]$$] -
\*SubsuperscriptBox[$$e$$, $$a$$, $$\[Nu]$$]\
\*SubscriptBox[$$\[PartialD]$$, $$\[Mu]$$]
\*SubsuperscriptBox[$$e$$, $$\[Nu]$$, $$b$$]\)\)
*)
spinconnection := spinconnection = Table[
Sum[Edown[[a, q]] Eup[[b, \[Nu]]] affine[[\[Nu], \[Mu], q]], {q,
1, n}, {\[Nu], 1, n}]
- Sum[Edown[[a, \[Nu]]] \!$$\*SubscriptBox[\(\[PartialD]$$, $$coord[[\[Mu]]]$$]$$Eup[[ b, \ \[Nu]]]$$\), {\[Nu], 1, n}],
{\[Mu], 1, n}, {b, 1, n}, {a, 1, n} ]
listspinconnection :=
Table[If[UnsameQ[spinconnection[[i, j, k]],
0], {ToString[\[CapitalOmega][i, j, k]],
spinconnection[[i, j, k]]}] , {i, 1, n}, {j, 1, n}, {k, 1, j}]
TableForm[
Partition[DeleteCases[Flatten[listspinconnection], Null], 2],
TableSpacing -> {2, 2}]


And now, a check. If all quantities are well defined and calculated, the tetrad postulate should be satisfied.

(*Final Test*)
(*\!$$\*SubscriptBox[\(\[Del]$$, $$\[Mu]$$]
\*SubsuperscriptBox[$$e$$, $$\[Nu]$$, $$a$$]\)= \!$$\*SubscriptBox[\(\[PartialD]$$, $$\[Mu]$$]
\*SubsuperscriptBox[$$e$$, $$\[Nu]$$, $$a$$]\) + \!$$\*SubsuperscriptBox[ SubscriptBox[\(\[CapitalOmega]$$, $$\[Mu]$$], $$b$$, $$a$$]\
\*SubsuperscriptBox[$$e$$, $$\[Nu]$$, $$b$$]\)- \!$$\*SubsuperscriptBox[\(\[CapitalGamma]$$, $$\[Mu]\[Nu]$$, $$\[Rho]$$]
\*SubsuperscriptBox[$$e$$, $$\[Rho]$$, $$a$$]\) = 0*)

\!$$\*SubscriptBox[\(\[PartialD]$$, $$coord[[\[Mu]]]$$]$$Eup[[ a, \ \[Nu]]]$$\) +
Sum[spinconnection[[\[Mu], a, b]] Eup[[b, \[Nu]]], {b, 1, n}]
- Sum[ affine[[q, \[Mu], \[Nu]]] Eup[[a, q]], {q, 1, n} ] ,
{\[Mu], 1, n}, {\[Nu], 1, n}, {a, 1, n} ] // Flatten ;


So, having all these, you should be able to obtain the Dirac operator in AdS$$_5$$. It is given by

$$\gamma^A \nabla_A = u \gamma^{A} \partial_{A} - \frac{4}{2} \gamma^{u}$$

where in the above $$\gamma^{u}$$ stands for the chiral gamma matrix; the higher-dimensional analogue of $$\gamma^5$$ in $$4$$-dimensions and if you wanted to do AdS$$_{1+d}$$ in the last fraction you should have $$d$$. This is why I left it as $$4/2$$. Also, $$A$$ in the above is a world-volume index, takes all values.

I will not be showing the physics part of the problem, just taking the result.

You should be able to show that the Dirac equation, can be brought in a Klein-Gordon form. For the example at hand it reads -I am writing the result in (1+d)-dimensions and then specify it in the case $$d=4$$.

$$\left(u \gamma^{A} \partial_{A} - d u \partial_{u} - m^2 + \frac{d^2}{4} + \frac{d}{2}+m \gamma^{u} \right) \Psi(u,x^{\mu}) = 0$$

This can be solved analytically under some assumptions.

Since the OP has not any specific conditions let me illustrate a particular case.

Assume that you Fourier decompose the spinor in the Minkowski space, you set the spinor $$\Psi(u,x^{\mu})=f(u) e^{ikx}$$ and you apply $$d=4$$ for the AdS$$_5$$ case of study. This will result in obtaining a differential, using that $$k^2=-M^2$$ which are the eigenvalues from $$\gamma^{\mu}\partial_{\mu}$$, for the scalar function $$f(u)$$.

I am giving the code and the final result.

d := 4
dirac1 = z^2 D[f[z], {z, 2}] - d z D[f[z], z] + z^2 M^2 f[z] -
m^2 f[z] + (d^2/4 + d/2) f[z] + m f[z];
dirac2 = z^2 D[f[z], {z, 2}] - d z D[f[z], z] + z^2 M^2 f[z] -
m^2 f[z] + (d^2/4 + d/2) f[z] - m f[z];
sltn1 = DSolve[dirac1 == 0, f[z], z]
sltn2 = DSolve[dirac2 == 0, f[z], z]


where the two equations come from the two different eigenvalues of the $$\gamma^u$$.

Hope it helps a bit.

edit1: If you want the calculation for the second order differential equation from the first-order coupled ones, let me know, but please make a post in the Physics.S.E. It's not very difficult and similar techniques ought to be working in any spacetime.

edit2: For more complicated spacetimes, in example asymptotically AdS with a non-trivial dilaton flow, if you run the code as it is you might think that it does not work. Remember to perform Simplify or FullSimplify in the spin-connection and the tetrad postulate.