# Sigma matrix

1. Oct 10, 2009

### Petar Mali

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

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

2. Oct 12, 2009

### 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?

3. Oct 25, 2009

### Petar Mali

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

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

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

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

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

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