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?
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*)
tetradpostulate = Table[
\!\(
\*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 ;
AllTrue[tetradpostulate, # == 0 &]
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.
feyncalctag 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 '18 at 02:36