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1 files changed, 46 insertions, 22 deletions

diff --git a/content/know/concept/greens-functions/index.pdc b/content/know/concept/greens-functions/index.pdc
index 2f86e63..b3c9ede 100644
--- a/content/know/concept/greens-functions/index.pdc
+++ b/content/know/concept/greens-functions/index.pdc
@@ -32,16 +32,28 @@ 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 **retarded Green's function** $G_{\nu \nu'}^R$
-and the **advanced Green's function** $G_{\nu \nu'}^A$
-are defined like so,
-where the expectation value $\expval{}$ is
+The **time-ordered** or **causal Green's function** $G_{\nu \nu'}$
+is defined 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,
 and $\hat{c}_\nu$ annihilates a particle from $\nu$, etc.:
 
 $$\begin{aligned}
     \boxed{
+        G_{\nu \nu'}(t, t')
+        \equiv -\frac{i}{\hbar} \expval{\mathcal{T} \Big\{ \hat{c}_{\nu}(t) \: \hat{c}_{\nu'}^\dagger(t') \Big\}}
+    }
+\end{aligned}$$
+
+Arguably more prevalent are
+the **retarded Green's function** $G_{\nu \nu'}^R$
+and the **advanced Green's function** $G_{\nu \nu'}^A$
+which are defined like so:
+
+$$\begin{aligned}
+    \boxed{
         \begin{aligned}
             G_{\nu \nu'}^R(t, t')
             &\equiv -\frac{i}{\hbar} \Theta(t - t') \expval{\comm*{\hat{c}_{\nu}(t)}{\hat{c}_{\nu'}^\dagger(t')}_{\mp}}
@@ -75,15 +87,19 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where $-$ is for bosons, and $+$ is for fermions.
-The retarded and advanced Green's functions can thus be expressed as follows:
+Where $-$ is for bosons, and $+$ for fermions.
+With this, the causal, retarded and advanced Green's functions
+can thus be expressed as follows:
 
 $$\begin{aligned}
+    G_{\nu \nu'}(t, t')
+    &= \Theta(t - t') \: G_{\nu \nu'}^>(t, t') + \Theta(t' - t) \: G_{\nu \nu'}^<(t, t')
+    \\
     G_{\nu \nu'}^R(t, t')
-    &= \Theta(t - t') \Big( G_{\nu \nu'}^>(t, t') - G_{\nu \nu'}^<(t, t') \Big)
+    &= \Theta(t - t') \big( G_{\nu \nu'}^>(t, t') - G_{\nu \nu'}^<(t, t') \big)
     \\
     G_{\nu \nu'}^A(t, t')
-    &= \Theta(t' - t) \Big( G_{\nu \nu'}^<(t, t') - G_{\nu \nu'}^>(t, t') \Big)
+    &= \Theta(t' - t) \big( G_{\nu \nu'}^<(t, t') - G_{\nu \nu'}^>(t, t') \big)
 \end{aligned}$$
 
 If the Hamiltonian involves interactions,
@@ -93,14 +109,14 @@ In that case, instead of a label $\nu$,
 we use the spin $s$ and position $\vb{r}$, leading to:
 
 $$\begin{aligned}
-    G_{ss'}^R(\vb{r}, t; \vb{r}', t')
-    &= -\frac{i}{\hbar} \Theta(t - t') \expval{\comm*{\hat{\Psi}_{s}(\vb{r}, t)}{\hat{\Psi}_{s'}^\dagger(\vb{r}', t')}_{\mp}}
+    G_{ss'}(\vb{r}, t; \vb{r}', t')
+    &= -\frac{i}{\hbar} \Theta(t - t') \expval{\mathcal{T}\Big\{ \hat{\Psi}_{s}(\vb{r}, t) \hat{\Psi}_{s'}^\dagger(\vb{r}', t') \Big\}}
     \\
-    &= \sum_{\nu \nu'} \psi_\nu(\vb{r}) \: \psi^*_{\nu'}(\vb{r}') \: G_{\nu \nu'}^R(t, t')
+    &= \sum_{\nu \nu'} \psi_\nu(\vb{r}) \: \psi^*_{\nu'}(\vb{r}') \: G_{\nu \nu'}(t, t')
 \end{aligned}$$
 
-And analogously for $G_{ss'}^A$, $G_{ss'}^>$ and $G_{ss'}^<$.
-Note that the time-dependence is given to the old $G_{\nu \nu'}^R$,
+And analogously for $G_{ss'}^R$, $G_{ss'}^A$, $G_{ss'}^>$ and $G_{ss'}^<$.
+Note that the time-dependence is given to the old $G_{\nu \nu'}$,
 i.e. to $\hat{c}_\nu$ and $\hat{c}_{\nu'}^\dagger$,
 because we are in the Heisenberg picture.
 
@@ -108,7 +124,9 @@ If the Hamiltonian is time-independent,
 then it can be shown that all the Green's functions
 only depend on the time-difference $t - t'$:
 
-$$\begin{aligned}
+$$\begin{gathered}
+    G_{\nu \nu'}(t, t') = G_{\nu \nu'}(t - t')
+    \\
     G_{\nu \nu'}^R(t, t') = G_{\nu \nu'}^R(t - t')
     \qquad \quad
     G_{\nu \nu'}^A(t, t') = G_{\nu \nu'}^A(t - t')
@@ -116,7 +134,7 @@ $$\begin{aligned}
     G_{\nu \nu'}^>(t, t') = G_{\nu \nu'}^>(t - t')
     \qquad \quad
     G_{\nu \nu'}^<(t, t') = G_{\nu \nu'}^<(t - t')
-\end{aligned}$$
+\end{gathered}$$
 
 <div class="accordion">
 <input type="checkbox" id="proof-time-diff"/>
@@ -324,16 +342,20 @@ i.e. the Hamiltonian only contains kinetic energy.
 
 ## Two-particle functions
 
-The above can be generalized to two arbitrary operators $\hat{A}$ and $\hat{B}$,
+We generalize the above to two arbitrary operators $\hat{A}$ and $\hat{B}$,
 giving us the **two-particle Green's functions**,
 or just **correlation functions**.
-The **retarded correlation function** $C_{AB}^R$
-and the **advanced correlation function** $C_{AB}^A$ are defined as
+The **causal correlation function** $C_{AB}$,
+the **retarded correlation function** $C_{AB}^R$
+and the **advanced correlation function** $C_{AB}^A$ are defined as follows
 (in the Heisenberg picture):
 
 $$\begin{aligned}
     \boxed{
         \begin{aligned}
+            C_{AB}(t, t')
+            &\equiv -\frac{i}{\hbar} \expval{\mathcal{T}\Big\{\hat{A}(t) \hat{B}(t')\Big\}}
+            \\
             C_{AB}^R(t, t')
             &\equiv -\frac{i}{\hbar} \Theta(t - t') \expval{\comm*{\hat{A}(t)}{\hat{B}(t')}_{\mp}}
             \\
@@ -350,13 +372,15 @@ of two single-particle creation/annihilation operators.
 
 Like for the single-particle Green's functions,
 if the Hamiltonian is time-independent,
-then it can be shown that $C_{AB}^R$ and $C_{AB}^A$
+then it can be shown that the two-particle functions
 only depend on the time-difference $t - t'$:
 
 $$\begin{aligned}
-    G_{\nu \nu'}^>(t, t') = G_{\nu \nu'}^>(t - t')
-    \qquad \quad
-    G_{\nu \nu'}^<(t, t') = G_{\nu \nu'}^<(t - t')
+    G_{\nu \nu'}(t, t') = G_{\nu \nu'}(t \!-\! t')
+    \qquad
+    G_{\nu \nu'}^R(t, t') = G_{\nu \nu'}^>(t \!-\! t')
+    \qquad
+    G_{\nu \nu'}^A(t, t') = G_{\nu \nu'}^<(t \!-\! t')
 \end{aligned}$$