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Hermitian Hamiltonian for KG equation

  1. Oct 9, 2015 #1

    ShayanJ

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    Using the Feshbach-Villars transformation, its possible to write the KG equation as two coupled equations in terms of two fields as below:

    ## 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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  3. Oct 9, 2015 #2

    ChrisVer

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  4. Oct 9, 2015 #3
    In the beginning, I have the same problem at here. However, I think the Hermitian of this matrix should consider the infinity of momentum.
     
  5. Oct 10, 2015 #4

    ShayanJ

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    Although the method used in the reference ChrisVer pointed to is really nice, I still need to know how to transform the Hamiltonian to a Hermitian form because when we want to use this form of KG equation in perturbation theory, the form of the resulting perturbation series depends on the method used. Surely using the method in the above reference will introduce an extra factor in the series but what I'm trying the get out of KG equation, is the perturbation series without that factor.
    The reason I'm pursuing this is that in the textbook Quarks and Leptons by Halzen and Martin, section 3.6, they derive a perturbation series using an equation of the form ##(H_0 +V(\mathbf x,t))\psi=i \frac{\partial \psi}{\partial t}##, where the Hamiltonian is assumed to be Hermitian in the usual sense and then its eigenfunctions are used. I want to show that the KG equation can indeed be written in this form but now I'm stuck in transforming the Hamiltonian in the OP to a Hermitian form.
     
  6. Oct 10, 2015 #5

    ChrisVer

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    Would that be helpful then?
    http://arxiv.org/abs/0810.5643v4

    I am not sure that you can transform a hamiltonian to be hermiatian - most of the times you drop out the non-hermitian parts.
    What changes is the way you define hermitianity via : [itex]H^\dagger = a H a^{-1}[/itex] where [itex]a[/itex] a linear hermitian automorphism.
    http://arxiv.org/abs/1401.5255
    you can have a look here for some things concerning these.
     
  7. Oct 10, 2015 #6

    ShayanJ

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    This is from Bjorken and Drell:
     
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