From 8a9fb5fef2a97af3274290e512816e1a4cac0c02 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Mon, 24 Jan 2022 19:29:00 +0100
Subject: Rewrite "Lindhard function", split off "dielectric function"

---
 content/know/concept/greens-functions/index.pdc | 49 ++++++++++++++-----------
 1 file changed, 27 insertions(+), 22 deletions(-)

(limited to 'content/know/concept/greens-functions')

diff --git a/content/know/concept/greens-functions/index.pdc b/content/know/concept/greens-functions/index.pdc
index b3c9ede..92f0fcf 100644
--- a/content/know/concept/greens-functions/index.pdc
+++ b/content/know/concept/greens-functions/index.pdc
@@ -32,12 +32,9 @@ If the two operators are single-particle creation/annihilation operators,
 then we get the **single-particle Green's functions**,
 for which the symbol $G$ is used.
 
-The **time-ordered** or **causal Green's function** $G_{\nu \nu'}$
-is defined as follows,
+The **time-ordered** or **causal Green's function** $G_{\nu \nu'}$ is as follows,
 where $\mathcal{T}$ is the [time-ordered product](/know/concept/time-ordered-product/),
-the expectation value $\expval{}$ is
-with respect to thermodynamic equilibrium,
-$\nu$ and $\nu'$ are labels of single-particle states,
+$\nu$ and $\nu'$ are single-particle states,
 and $\hat{c}_\nu$ annihilates a particle from $\nu$, etc.:
 
 $$\begin{aligned}
@@ -47,6 +44,24 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
+The expectation value $\expval{}$ is
+with respect to thermodynamic equilibrium.
+This is sometimes in the [canonical ensemble](/know/concept/canonical-ensemble/)
+(for some two-particle Green's functions, see below),
+but usually in the [grand canonical ensemble](/know/concept/grand-canonical-ensemble/),
+since we are adding/removing particles.
+In the latter case, we assume that the chemical potential $\mu$
+is already included in the Hamiltonian $\hat{H}$.
+Explicitly, for a complete set of many-particle states $\ket{\Psi_n}$, we have:
+
+$$\begin{aligned}
+    G_{\nu \nu'}(t, t')
+    &= -\frac{i}{\hbar Z} \Tr\!\Big( \mathcal{T} \Big\{ \hat{c}_{\nu}(t) \: \hat{c}_{\nu'}^\dagger(t')\Big\} \: e^{- \beta \hat{H}} \Big)
+    \\
+    &= -\frac{i}{\hbar Z} \sum_{n}
+    \matrixel**{\Psi_n}{\mathcal{T} \Big\{ \hat{c}_{\nu}(t) \: \hat{c}_{\nu'}^\dagger(t')\Big\} \: e^{- \beta \hat{H}}}{\Psi_n}
+\end{aligned}$$
+
 Arguably more prevalent are
 the **retarded Green's function** $G_{\nu \nu'}^R$
 and the **advanced Green's function** $G_{\nu \nu'}^A$
@@ -67,10 +82,10 @@ $$\begin{aligned}
 Where $\Theta$ is a [Heaviside function](/know/concept/heaviside-step-function/),
 and $[,]_{\mp}$ is a commutator for bosons,
 and an anticommutator for fermions.
-We are in the [Heisenberg picture](/know/concept/heisenberg-picture/),
-hence $\hat{c}_\nu$ and $\hat{c}_{\nu'}^\dagger$ are time-dependent,
-but keep in mind that time-dependent Hamiltonians are allowed,
-so it might not be trivial.
+Depending on the context,
+we could either be in the [Heisenberg picture](/know/concept/heisenberg-picture/)
+or in the [interaction picture](/know/concept/interaction-picture/),
+hence $\hat{c}_\nu$ and $\hat{c}_{\nu'}^\dagger$ are time-dependent.
 
 Furthermore, the **greater Green's function** $G_{\nu \nu'}^>$
 and **lesser Green's function** $G_{\nu \nu'}^<$ are:
@@ -146,16 +161,7 @@ $\expval*{\hat{A}(t) \hat{B}(t')}$ only depends on $t - t'$
 for arbitrary $\hat{A}$ and $\hat{B}$,
 and it trivially follows that the Green's functions do too.
 
-Suppose that the system started in thermodynamic equilibrium.
-This could sometimes be in the [canonical ensemble](/know/concept/canonical-ensemble/)
-(for two-particle Green's functions, see below),
-but usually it will be in the
-[grand canonical ensemble](/know/concept/grand-canonical-ensemble/),
-since we are adding/removing particles.
-In the latter case, we assume that the chemical potential $\mu$
-is already included in the Hamiltonian.
-
-In any case, at equilibrium, we know that the
+In (grand) canonical equilibrium, we know that the
 [density operator](/know/concept/density-operator/)
 $\hat{\rho}$ is as follows:
 
@@ -163,9 +169,8 @@ $$\begin{aligned}
     \hat{\rho} = \frac{1}{Z} \exp\!(- \beta \hat{H})
 \end{aligned}$$
 
-Where $Z \equiv \Tr\!(\exp\!(- \beta \hat{H}))$ is the partition function.
-In that case, the expected value of the product
-of the time-independent operators $\hat{A}$ and $\hat{B}$ is calculated like so:
+The expected value of the product
+of the time-independent operators $\hat{A}$ and $\hat{B}$ is then:
 
 $$\begin{aligned}
     \expval*{\hat{A}(t) \hat{B}(t')}
-- 
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