- #1
blue2script
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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
<br /> \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<br />
where \varphi = \varphi\left(x,y\right) is a (given) function of x,y.
More explicit, the system looks like:
<br /> \left[ {i\left( {\begin{array}{*{20}c}<br /> 0 & 1 & 0 & 0 \\<br /> 1 & 0 & 0 & 0 \\<br /> 0 & 0 & 0 & 1 \\<br /> 0 & 0 & 1 & 0 \\<br /> <br /> \end{array} } \right)\partial _1 + i\left( {\begin{array}{*{20}c}<br /> 0 & { - i} & 0 & 0 \\<br /> i & 0 & 0 & 0 \\<br /> 0 & 0 & 0 & { - i} \\<br /> 0 & 0 & i & 0 \\<br /> <br /> \end{array} } \right)\partial _2 + g_y \left\{ {\varphi ^1 \left( {\begin{array}{*{20}c}<br /> 0 & 0 & 1 & 0 \\<br /> 0 & 0 & 0 & 1 \\<br /> 1 & 0 & 0 & 0 \\<br /> 0 & 1 & 0 & 0 \\<br /> <br /> \end{array} } \right) + \varphi ^2 \left( {\begin{array}{*{20}c}<br /> 0 & 0 & { - i} & 0 \\<br /> 0 & 0 & 0 & { - i} \\<br /> i & 0 & 0 & 0 \\<br /> 0 & i & 0 & 0 \\<br /> <br /> \end{array} } \right) + \varphi ^3 \left( {\begin{array}{*{20}c}<br /> 1 & 0 & 0 & 0 \\<br /> 0 & 1 & 0 & 0 \\<br /> 0 & 0 & { - 1} & 0 \\<br /> 0 & 0 & 0 & { - 1} \\<br /> <br /> \end{array} } \right)} \right\}} \right]\left( {\begin{array}{*{20}c}<br /> {\psi _1 } \\<br /> {\psi _2 } \\<br /> {\psi _3 } \\<br /> {\psi _4 } \\<br /> <br /> \end{array} } \right) = \left( {\begin{array}{*{20}c}<br /> 0 \\<br /> 0 \\<br /> 0 \\<br /> 0 \\<br /> <br /> \end{array} } \right)<br />
with \psi_i = \psi_i\left(x,y\right). 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
<br /> \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<br />
where \varphi = \varphi\left(x,y\right) is a (given) function of x,y.
More explicit, the system looks like:
<br /> \left[ {i\left( {\begin{array}{*{20}c}<br /> 0 & 1 & 0 & 0 \\<br /> 1 & 0 & 0 & 0 \\<br /> 0 & 0 & 0 & 1 \\<br /> 0 & 0 & 1 & 0 \\<br /> <br /> \end{array} } \right)\partial _1 + i\left( {\begin{array}{*{20}c}<br /> 0 & { - i} & 0 & 0 \\<br /> i & 0 & 0 & 0 \\<br /> 0 & 0 & 0 & { - i} \\<br /> 0 & 0 & i & 0 \\<br /> <br /> \end{array} } \right)\partial _2 + g_y \left\{ {\varphi ^1 \left( {\begin{array}{*{20}c}<br /> 0 & 0 & 1 & 0 \\<br /> 0 & 0 & 0 & 1 \\<br /> 1 & 0 & 0 & 0 \\<br /> 0 & 1 & 0 & 0 \\<br /> <br /> \end{array} } \right) + \varphi ^2 \left( {\begin{array}{*{20}c}<br /> 0 & 0 & { - i} & 0 \\<br /> 0 & 0 & 0 & { - i} \\<br /> i & 0 & 0 & 0 \\<br /> 0 & i & 0 & 0 \\<br /> <br /> \end{array} } \right) + \varphi ^3 \left( {\begin{array}{*{20}c}<br /> 1 & 0 & 0 & 0 \\<br /> 0 & 1 & 0 & 0 \\<br /> 0 & 0 & { - 1} & 0 \\<br /> 0 & 0 & 0 & { - 1} \\<br /> <br /> \end{array} } \right)} \right\}} \right]\left( {\begin{array}{*{20}c}<br /> {\psi _1 } \\<br /> {\psi _2 } \\<br /> {\psi _3 } \\<br /> {\psi _4 } \\<br /> <br /> \end{array} } \right) = \left( {\begin{array}{*{20}c}<br /> 0 \\<br /> 0 \\<br /> 0 \\<br /> 0 \\<br /> <br /> \end{array} } \right)<br />
with \psi_i = \psi_i\left(x,y\right). 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