From 6505b1fb3399ec4bff97aabda2554764bf305d0f Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 21 Nov 2021 18:02:35 +0100
Subject: Expand knowledge base

---
 content/know/concept/imaginary-time/index.pdc | 20 +++++++++++++++-----
 1 file changed, 15 insertions(+), 5 deletions(-)

(limited to 'content/know/concept/imaginary-time')

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/).
 
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