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Sigma matrix

  1. Oct 10, 2009 #1
    Do you know where can I find more about [tex]\hat{\sigma}[/tex] matrix define like

    [tex]e^{-\beta \hat{H}}=e^{-\beta\hat{H}_0}\hat{\sigma}(\beta)\qquad \hat{H}=\hat{H}_0+\hat{V}[/tex]
     
  2. jcsd
  3. Oct 12, 2009 #2
    From analogy [tex]\hat{\sigma}[/tex] and [tex]\hat{S}[/tex] matrix you define Matzubara Green function with imaginary time. But I can't find [tex]\sigma[/tex] matrix in any book?
     
  4. Oct 25, 2009 #3
    Let say more about this. Maybe will start a discussion.

    [tex]
    e^{-\beta \hat{H}}=e^{-\beta\hat{H}_0}\hat{\sigma}(\beta)\qquad \hat{H}=\hat{H}_0+\hat{V}
    [/tex]


    [tex]\beta=\frac{1}{k_BT}[/tex]


    [tex]0\leq \tau\leq \beta=\frac{1}{k_BT}[/tex]

    [tex]
    e^{-\tau \hat{H}}=e^{-\tau\hat{H}_0}\hat{\sigma}(\tau)[/tex]


    [tex]\hat{\sigma}(\tau)=e^{\tau \hat{H}_0}e^{-\tau \hat{H}}[/tex]


    [tex]\frac{d\hat{\sigma}(\tau)}{d\tau}=-e^{\tau \hat{H}_0}\hat{V}e^{-\tau \hat{H}}
    =-e^{\tau \hat{H}_0}\hat{V}e^{-\tau \hat{H}_0}e^{\tau \hat{H}_0}e^{-\tau \hat{H}}=-\hat{V}_I(\tau)\hat{\sigma}(\tau)[/tex]
     
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