Expand knowledge base, reorganize Green's functions

1 files changed, 5 insertions, 6 deletions

diff --git a/content/know/concept/lehmann-representation/index.pdc b/content/know/concept/lehmann-representation/index.pdc
index f38f803..5808934 100644
--- a/content/know/concept/lehmann-representation/index.pdc
+++ b/content/know/concept/lehmann-representation/index.pdc
@@ -18,9 +18,8 @@ is an alternative way to write the [Green's functions](/know/concept/greens-func
 obtained by expanding in the many-particle eigenstates
 under the assumption of a time-independent Hamiltonian $\hat{H}$.
 
-We start by writing out the
-greater Green's function $G_{\nu \nu'}(t, t')$,
-and then expanding its thermal expectation value $\expval{}$
+First, we write out the greater Green's function $G_{\nu \nu'}(t, t')$,
+and then expand its expected value $\expval{}$ (at thermodynamic equilibrium)
 into a sum of many-particle basis states $\ket{n}$:
 
 $$\begin{aligned}
@@ -29,9 +28,9 @@ $$\begin{aligned}
     &= - \frac{i}{\hbar Z} \sum_{n} \matrixel**{n}{\hat{c}_\nu(t) \hat{c}_{\nu'}^\dagger(t') e^{-\beta \hat{H}}}{n}
 \end{aligned}$$
 
-Where $\beta = 1 / (k_B T)$, and $Z$ is the partition function
-(see [canonical ensemble](/know/concept/canonical-ensemble/));
-the operator $e^{\beta \hat{H}}$ gives the weight of each term at thermal equilibrium.
+Where $\beta = 1 / (k_B T)$, and $Z$ is the grand partition function
+(see [grand canonical ensemble](/know/concept/grand-canonical-ensemble/));
+the operator $e^{\beta \hat{H}}$ gives the weight of each term at equilibrium.
 Since $\ket{n}$ is an eigenstate of $\hat{H}$ with energy $E_n$,
 this gives us a factor of $e^{\beta E_n}$.
 Furthermore, we are in the [Heisenberg picture](/know/concept/heisenberg-picture/),