- #1

blue2script

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I have a problem here related to QFT in a research project. I end up with some Dirac equation with space-time dependent mass in 2 spatial dimensions.

More mathematically, the PDE to solve is

[tex]

\left( {i\left( {\sigma ^i \otimes I_2 } \right)\partial _i + g_y \varphi ^a \left( {I_2 \otimes \sigma ^a } \right)} \right)\psi = 0

[/tex]

where [tex]\varphi = \varphi\left(x,y\right)[/tex] is a (given) function of x,y.

More explicit, the system looks like:

[tex]

\left[ {i\left( {\begin{array}{*{20}c}

0 & 1 & 0 & 0 \\

1 & 0 & 0 & 0 \\

0 & 0 & 0 & 1 \\

0 & 0 & 1 & 0 \\

\end{array} } \right)\partial _1 + i\left( {\begin{array}{*{20}c}

0 & { - i} & 0 & 0 \\

i & 0 & 0 & 0 \\

0 & 0 & 0 & { - i} \\

0 & 0 & i & 0 \\

\end{array} } \right)\partial _2 + g_y \left\{ {\varphi ^1 \left( {\begin{array}{*{20}c}

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1 \\

1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

\end{array} } \right) + \varphi ^2 \left( {\begin{array}{*{20}c}

0 & 0 & { - i} & 0 \\

0 & 0 & 0 & { - i} \\

i & 0 & 0 & 0 \\

0 & i & 0 & 0 \\

\end{array} } \right) + \varphi ^3 \left( {\begin{array}{*{20}c}

1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & { - 1} & 0 \\

0 & 0 & 0 & { - 1} \\

\end{array} } \right)} \right\}} \right]\left( {\begin{array}{*{20}c}

{\psi _1 } \\

{\psi _2 } \\

{\psi _3 } \\

{\psi _4 } \\

\end{array} } \right) = \left( {\begin{array}{*{20}c}

0 \\

0 \\

0 \\

0 \\

\end{array} } \right)

[/tex]

with [tex]\psi_i = \psi_i\left(x,y\right)[/tex]. I tried some Fourier-method, but that wouldn't work out. I am completely stuck here and have no idea how to proceed. Could anyone give me a hint?

A big thanks in advance!

Blue2script