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.

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


$$\beta=\frac{1}{k_BT}$$


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

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


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


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