Understanding the \hat{\sigma} Matrix

[Petar Mali]

Do you know where can I find more about \hat{\sigma} matrix define like

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

 

[Petar Mali]

From analogy \hat{\sigma} and \hat{S} matrix you define Matzubara Green function with imaginary time. But I can't find \sigma matrix in any book?

 

[Petar Mali]

Let say more about this. Maybe will start a discussion.

<br /> e^{-\beta \hat{H}}=e^{-\beta\hat{H}_0}\hat{\sigma}(\beta)\qquad \hat{H}=\hat{H}_0+\hat{V}<br />


\beta=\frac{1}{k_BT}


0\leq \tau\leq \beta=\frac{1}{k_BT}

<br /> e^{-\tau \hat{H}}=e^{-\tau\hat{H}_0}\hat{\sigma}(\tau)


\hat{\sigma}(\tau)=e^{\tau \hat{H}_0}e^{-\tau \hat{H}}


\frac{d\hat{\sigma}(\tau)}{d\tau}=-e^{\tau \hat{H}_0}\hat{V}e^{-\tau \hat{H}}<br /> =-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)