diff options
Diffstat (limited to 'content/know/concept/imaginary-time')
| -rw-r--r-- | content/know/concept/imaginary-time/index.pdc | 20 |
1 files changed, 15 insertions, 5 deletions
diff --git a/content/know/concept/imaginary-time/index.pdc b/content/know/concept/imaginary-time/index.pdc index 55f163a..b68afce 100644 --- a/content/know/concept/imaginary-time/index.pdc +++ b/content/know/concept/imaginary-time/index.pdc @@ -145,20 +145,30 @@ $$\begin{aligned} \hat{A}_I(\tau) \hat{K}_I(\tau, \tau') \hat{B}_I(\tau') \hat{K}_I(\tau', 0) \!\Big) \end{aligned}$$ -Assuming $\tau > \tau'$, -we introduce a time-ordering $\mathcal{T}$, -allowing us to reorder the operators inside, +We now introduce a time-ordering $\mathcal{T}$, +letting us reorder the (bosonic) $\hat{K}_I$-operators inside, and thereby reduce the expression considerably: $$\begin{aligned} - \expval*{\hat{A}_H \hat{B}_H} + \expval{\mathcal{T}\Big\{\hat{A}_H \hat{B}_H\Big\}} &= \frac{1}{Z} \Tr\!\Big( \mathcal{T} \Big\{ \hat{K}_I(\hbar \beta, \tau) \hat{K}_I(\tau, \tau') \hat{K}_I(\tau', 0) \hat{A}_I(\tau) \hat{B}_I(\tau') \Big\} \exp\!(-\beta \hat{H}_{0,S}) \Big) \\ - &= \frac{1}{Z} \Tr\!\Big( \mathcal{T} \Big\{ \hat{K}_I(\hbar \beta, 0) \hat{A}_I(\tau) \hat{B}_I(\tau') \Big\} \exp\!(-\beta \hat{H}_{0,S}) \Big) + &= \frac{1}{Z} \Tr\!\Big( \mathcal{T}\Big\{ \hat{K}_I(\hbar \beta, 0) \hat{A}_I(\tau) \hat{B}_I(\tau') \Big\} \exp\!(-\beta \hat{H}_{0,S}) \Big) \end{aligned}$$ Where $Z = \Tr\!\big(\exp\!(-\beta \hat{H}_S)\big) = \Tr\!\big(\hat{K}_I(\hbar \beta, 0) \exp\!(-\beta \hat{H}_{0,S})\big)$. +If we now define $\expval{}_0$ as the expectation value with respect +to the unperturbed equilibrium involving only $\hat{H}_{0,S}$, +we arrive at the following way of writing this time-ordered expectation: + +$$\begin{aligned} + \boxed{ + \expval{\mathcal{T}\Big\{\hat{A}_H \hat{B}_H\Big\}} + = \frac{\expval{\mathcal{T}\Big\{ \hat{K}_I(\hbar \beta, 0) \hat{A}_I(\tau) \hat{B}_I(\tau') \Big\}}_0}{\expval{\hat{K}_I(\hbar \beta, 0)}_0} + } +\end{aligned}$$ + For another application of imaginary time, see e.g. the [Matsubara Green's function](/know/concept/matsubara-greens-function/). |