- #1
blue2script
- 47
- 0
Hello everybody,
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
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