Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

1 files changed, 34 insertions, 34 deletions

diff --git a/source/know/concept/lehmann-representation/index.md b/source/know/concept/lehmann-representation/index.md
index cfc6838..74bd457 100644
--- a/source/know/concept/lehmann-representation/index.md
+++ b/source/know/concept/lehmann-representation/index.md
@@ -11,11 +11,11 @@ layout: "concept"
 In many-body quantum theory, the **Lehmann representation**
 is an alternative way to write the [Green's functions](/know/concept/greens-functions/),
 obtained by expanding in the many-particle eigenstates
-under the assumption of a time-independent Hamiltonian $\hat{H}$.
+under the assumption of a time-independent Hamiltonian $$\hat{H}$$.
 
-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 $\Ket{n}$:
+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 $$\Ket{n}$$:
 
 $$\begin{aligned}
     G_{\nu \nu'}^>(t, t')
@@ -23,13 +23,13 @@ $$\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 grand partition function
+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 $e^{\beta E_n}$.
+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/),
-so we write out the time-dependence of $\hat{c}_\nu$ and $\hat{c}_{\nu'}^\dagger$:
+so we write out the time-dependence of $$\hat{c}_\nu$$ and $$\hat{c}_{\nu'}^\dagger$$:
 
 $$\begin{aligned}
     G_{\nu \nu'}^>(t, t')
@@ -40,10 +40,10 @@ $$\begin{aligned}
     \Matrixel{n}{e^{i \hat{H} (t - t') / \hbar} \hat{c}_\nu e^{- i \hat{H} (t - t') / \hbar} \hat{c}_{\nu'}^\dagger}{n}
 \end{aligned}$$
 
-Where we used that the trace $\Tr\!(x) = \sum_{n} \matrixel{n}{x}{n}$
-is invariant under cyclic permutations of $x$.
-The $\Ket{n}$ form a basis of eigenstates of $\hat{H}$,
-so we insert an identity operator $\sum_{n'} \Ket{n'} \Bra{n'}$:
+Where we used that the trace $$\Tr\!(x) = \sum_{n} \matrixel{n}{x}{n}$$
+is invariant under cyclic permutations of $$x$$.
+The $$\Ket{n}$$ form a basis of eigenstates of $$\hat{H}$$,
+so we insert an identity operator $$\sum_{n'} \Ket{n'} \Bra{n'}$$:
 
 $$\begin{aligned}
     G_{\nu \nu'}^>(t - t')
@@ -54,10 +54,10 @@ $$\begin{aligned}
     \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} e^{i (E_n - E_{n'}) (t - t') / \hbar}
 \end{aligned}$$
 
-Note that $G_{\nu \nu'}^>$ now only depends on the time difference $t - t'$,
-because $\hat{H}$ is time-independent.
+Note that $$G_{\nu \nu'}^>$$ now only depends on the time difference $$t - t'$$,
+because $$\hat{H}$$ is time-independent.
 Next, we take the [Fourier transform](/know/concept/fourier-transform/)
-$t \to \omega$ (with $t' = 0$):
+$$t \to \omega$$ (with $$t' = 0$$):
 
 $$\begin{aligned}
     G_{\nu \nu'}^>(\omega)
@@ -66,9 +66,9 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Here, we recognize the integral
-as a [Dirac delta function](/know/concept/dirac-delta-function/) $\delta$,
-thereby introducing a factor of $2 \pi$,
-and arriving at the Lehmann representation of $G_{\nu \nu'}^>$:
+as a [Dirac delta function](/know/concept/dirac-delta-function/) $$\delta$$,
+thereby introducing a factor of $$2 \pi$$,
+and arriving at the Lehmann representation of $$G_{\nu \nu'}^>$$:
 
 $$\begin{aligned}
     \boxed{
@@ -78,7 +78,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-We now go through the same process for the lesser Green's function $G_{\nu \nu'}^<(t, t')$:
+We now go through the same process for the lesser Green's function $$G_{\nu \nu'}^<(t, t')$$:
 
 $$\begin{aligned}
     G_{\nu \nu'}^<(t - t')
@@ -88,7 +88,7 @@ $$\begin{aligned}
    e^{i (E_{n'} - E_n) (t - t') / \hbar}
 \end{aligned}$$
 
-Where $-$ is for bosons, and $+$ for fermions.
+Where $$-$$ is for bosons, and $$+$$ for fermions.
 Fourier transforming yields the following:
 
 $$\begin{aligned}
@@ -97,8 +97,8 @@ $$\begin{aligned}
     \: \delta(E_{n'} - E_n + \hbar \omega)
 \end{aligned}$$
 
-We swap $n$ and $n'$, leading to the following
-Lehmann representation of $G_{\nu \nu'}^<$:
+We swap $$n$$ and $$n'$$, leading to the following
+Lehmann representation of $$G_{\nu \nu'}^<$$:
 
 $$\begin{aligned}
     \boxed{
@@ -108,8 +108,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Due to the delta function $\delta$,
-each term is only nonzero for $E_n' = E_n + \hbar \omega$,
+Due to the delta function $$\delta$$,
+each term is only nonzero for $$E_n' = E_n + \hbar \omega$$,
 so we write:
 
 $$\begin{aligned}
@@ -119,7 +119,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Therefore, we arrive at the following useful relation
-between $G_{\nu \nu'}^<$ and $G_{\nu \nu'}^>$:
+between $$G_{\nu \nu'}^<$$ and $$G_{\nu \nu'}^>$$:
 
 $$\begin{aligned}
     \boxed{
@@ -129,7 +129,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Moving on, let us do the same for
-the retarded Green's function $G_{\nu \nu'}^R(t, t')$, given by:
+the retarded Green's function $$G_{\nu \nu'}^R(t, t')$$, given by:
 
 $$\begin{aligned}
     G_{\nu \nu'}^R(t \!-\! t')
@@ -141,7 +141,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 We take the Fourier transform, but to ensure convergence,
-we must introduce an infinitesimal positive $\eta \to 0^+$ to the exponent
+we must introduce an infinitesimal positive $$\eta \to 0^+$$ to the exponent
 (and eventually take the limit):
 
 $$\begin{aligned}
@@ -155,7 +155,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Leading us to the following Lehmann representation
-of the retarded Green's function $G_{\nu \nu'}^R$:
+of the retarded Green's function $$G_{\nu \nu'}^R$$:
 
 $$\begin{aligned}
     \boxed{
@@ -166,7 +166,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Finally, we go through the same steps for the advanced Green's function $G_{\nu \nu'}^A(t, t')$:
+Finally, we go through the same steps for the advanced Green's function $$G_{\nu \nu'}^A(t, t')$$:
 
 $$\begin{aligned}
     G_{\nu \nu'}^A(t \!-\! t')
@@ -177,7 +177,7 @@ $$\begin{aligned}
      \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) e^{i (E_n - E_{n'}) (t - t') / \hbar}
 \end{aligned}$$
 
-For the Fourier transform, we must again introduce $\eta \to 0^+$
+For the Fourier transform, we must again introduce $$\eta \to 0^+$$
 (although note the sign):
 
 $$\begin{aligned}
@@ -191,7 +191,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Therefore, the Lehmann representation of
-the advanced Green's function $G_{\nu \nu'}^A$ is as follows:
+the advanced Green's function $$G_{\nu \nu'}^A$$ is as follows:
 
 $$\begin{aligned}
     \boxed{
@@ -211,8 +211,8 @@ $$\begin{aligned}
     \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big)
 \end{aligned}$$
 
-Note the subscripts $\nu$ and $\nu'$.
-Comparing this to $G_{\nu \nu'}^R$ gives us another useful relation:
+Note the subscripts $$\nu$$ and $$\nu'$$.
+Comparing this to $$G_{\nu \nu'}^R$$ gives us another useful relation:
 
 $$\begin{aligned}
     \boxed{