Using the Feshbach-Villars transformation, its possible to write the KG equation as two coupled equations in terms of two fields as below:(adsbygoogle = window.adsbygoogle || []).push({});

## i\partial_t \phi_1=-\frac{1}{2m} \nabla^2(\phi_1+\phi_2)+m\phi_1##

## i\partial_t \phi_2=\frac{1}{2m} \nabla^2(\phi_1+\phi_2)-m\phi_2##

Then we can combine the two fields into a column matrix ##\mathbf \Phi=(\phi_1 \ \ \ \phi_2)^T ## and write the above two equations as below:

## i\partial_t \mathbf \Phi=-\frac{1}{2m} \left( \begin{array}{cc} 1 \ \ \ \ \ \ 1 \\ -1 \ \ \ -1 \end{array} \right)\nabla^2 \mathbf \Phi+m\left( \begin{array}{cc}1 \ \ \ \ \ \ 0 \\ 0 \ \ \ -1 \end{array} \right) \mathbf \Phi=H \mathbf \Phi##

But the problem is, the matrix ## \left( \begin{array}{cc} 1 \ \ \ \ \ \ 1 \\ -1 \ \ \ -1 \end{array} \right) ## is not Hermitian. Bjorken and Drell(section 9.7) transform the Hamiltonian to a Hermitian one but I don't understand their method. Can anyone explain?

Thanks

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# Hermitian Hamiltonian for KG equation

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