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

1 files changed, 53 insertions, 50 deletions

diff --git a/source/know/concept/matsubara-greens-function/index.md b/source/know/concept/matsubara-greens-function/index.md
index 81dd360..fdcadb3 100644
--- a/source/know/concept/matsubara-greens-function/index.md
+++ b/source/know/concept/matsubara-greens-function/index.md
@@ -21,10 +21,10 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where the expectation value $\Expval{}$ is with respect to thermodynamic equilibrium,
-and $\mathcal{T}$ is the [time-ordered product](/know/concept/time-ordered-product/) pseudo-operator.
-Because the Hamiltonian $\hat{H}$ cannot depend on the imaginary time,
-$C_{AB}$ is a function of the difference $\tau \!-\! \tau'$ only:
+Where the expectation value $$\Expval{}$$ is with respect to thermodynamic equilibrium,
+and $$\mathcal{T}$$ is the [time-ordered product](/know/concept/time-ordered-product/) pseudo-operator.
+Because the Hamiltonian $$\hat{H}$$ cannot depend on the imaginary time,
+$$C_{AB}$$ is a function of the difference $$\tau \!-\! \tau'$$ only:
 
 $$\begin{aligned}
     C_{AB}(\tau, \tau')
@@ -36,9 +36,9 @@ $$\begin{aligned}
     &= - \frac{1}{\hbar Z} \Tr\!\Big( e^{-\beta \hat{H}} e^{(\tau - \tau') \hat{H} / \hbar} \hat{A} e^{-(\tau - \tau') \hat{H} / \hbar} \hat{B} \Big)
 \end{aligned}$$
 
-For $\tau > \tau'$, we see by expanding in the many-particle eigenstates $\Ket{n}$
-that we need to demand $\hbar \beta > \tau \!-\! \tau'$ to prevent
-$C_{AB}$ from diverging for increasing temperatures:
+For $$\tau > \tau'$$, we see by expanding in the many-particle eigenstates $$\Ket{n}$$
+that we need to demand $$\hbar \beta > \tau \!-\! \tau'$$ to prevent
+$$C_{AB}$$ from diverging for increasing temperatures:
 
 $$\begin{aligned}
     C_{AB}(\tau \!-\! \tau')
@@ -48,8 +48,8 @@ $$\begin{aligned}
     &= - \frac{1}{\hbar Z} \sum_{n} \Matrixel{n}{\hat{A} e^{-(\tau - \tau') \hat{H} / \hbar} \hat{B}}{n} e^{-\beta E_n} e^{(\tau - \tau') E_n / \hbar}
 \end{aligned}$$
 
-And likewise, for $\tau < \tau'$,
-we must demand that $\tau \!-\! \tau' > -\hbar \beta$
+And likewise, for $$\tau < \tau'$$,
+we must demand that $$\tau \!-\! \tau' > -\hbar \beta$$
 for the same reason:
 
 $$\begin{aligned}
@@ -61,14 +61,14 @@ $$\begin{aligned}
     &= \mp \frac{1}{\hbar Z} \sum_{n} \Matrixel{n}{\hat{B} e^{(\tau - \tau') \hat{H} / \hbar} \hat{A}}{n} e^{-\beta E_n} e^{- (\tau - \tau') E_n / \hbar}
 \end{aligned}$$
 
-With $-$ for bosons, and $+$ for fermions,
-due to the time-ordered product for $\tau > \tau'$.
+With $$-$$ for bosons, and $$+$$ for fermions,
+due to the time-ordered product for $$\tau > \tau'$$.
 
-On this domain $[-\hbar \beta, \hbar \beta]$,
-the Matsubara Green's function $C_{AB}$
+On this domain $$[-\hbar \beta, \hbar \beta]$$,
+the Matsubara Green's function $$C_{AB}$$
 obeys a useful shift relation:
-it is $\hbar \beta$-periodic for bosons,
-and $\hbar \beta$-antiperiodic for fermions:
+it is $$\hbar \beta$$-periodic for bosons,
+and $$\hbar \beta$$-antiperiodic for fermions:
 
 $$\begin{aligned}
     \boxed{
@@ -88,8 +88,8 @@ $$\begin{aligned}
 <label for="proof-period">Proof</label>
 <div class="hidden" markdown="1">
 <label for="proof-period">Proof.</label>
-First $\tau \!-\! \tau' < 0$.
-We insert the argument $\tau \!-\! \tau' \!+\! \hbar \beta$,
+First $$\tau \!-\! \tau' < 0$$.
+We insert the argument $$\tau \!-\! \tau' \!+\! \hbar \beta$$,
 and use the cyclic property:
 
 $$\begin{aligned}
@@ -105,8 +105,8 @@ $$\begin{aligned}
     &= - \frac{1}{\hbar Z} \Tr\!\Big( e^{-\beta \hat{H}} \hat{B}(\tau') \hat{A}(\tau) \Big)
 \end{aligned}$$
 
-Since $\tau < \tau'$ by assumption,
-we can bring back the time-ordered product $\mathcal{T}$:
+Since $$\tau < \tau'$$ by assumption,
+we can bring back the time-ordered product $$\mathcal{T}$$:
 
 $$\begin{aligned}
     C_{AB}(\tau \!-\! \tau' \!+\! \hbar \beta)
@@ -115,7 +115,7 @@ $$\begin{aligned}
     &= \pm C_{AB}(\tau \!-\! \tau')
 \end{aligned}$$
 
-Moving on to $\tau \!-\! \tau' > 0$, the proof is perfectly analogous:
+Moving on to $$\tau \!-\! \tau' > 0$$, the proof is perfectly analogous:
 
 $$\begin{aligned}
     C_{AB}(\tau \!-\! \tau' \!-\! \hbar \beta)
@@ -133,16 +133,17 @@ $$\begin{aligned}
     \\
     &= \pm C_{AB}(\tau \!-\! \tau')
 \end{aligned}$$
+
 </div>
 </div>
 
-Due to this limited domain $\tau \in [-\hbar \beta, \hbar \beta]$,
+Due to this limited domain $$\tau \in [-\hbar \beta, \hbar \beta]$$,
 the [Fourier transform](/know/concept/fourier-transform/)
-of $C_{AB}(\tau)$ consists of discrete frequencies
-$k_n \equiv n \pi / (\hbar \beta)$.
+of $$C_{AB}(\tau)$$ consists of discrete frequencies
+$$k_n \equiv n \pi / (\hbar \beta)$$.
 The forward and inverse Fourier transforms
-are therefore defined as given below (with $\tau' = 0$).
-It is convention to write $C_{AB}(i k_n)$ instead of $C_{AB}(k_n)$:
+are therefore defined as given below (with $$\tau' = 0$$).
+It is convention to write $$C_{AB}(i k_n)$$ instead of $$C_{AB}(k_n)$$:
 
 $$\begin{aligned}
     \boxed{
@@ -162,8 +163,8 @@ $$\begin{aligned}
 <div class="hidden" markdown="1">
 <label for="proof-FT-def">Proof.</label>
 We will prove that one is indeed the inverse of the other.
-We demand that the inverse FT of the forward FT of $C_{AB}(\tau)$
-is simply $C_{AB}(\tau)$ again:
+We demand that the inverse FT of the forward FT of $$C_{AB}(\tau)$$
+is simply $$C_{AB}(\tau)$$ again:
 
 $$\begin{aligned}
     C_{AB}(\tau)
@@ -197,11 +198,12 @@ $$\begin{aligned}
     \\
     &= C_{AB}(\tau)
 \end{aligned}$$
+
 </div>
 </div>
 
-Let us now define the **Matsubara frequencies** $\omega_n$
-as a species-dependent subset of $k_n$:
+Let us now define the **Matsubara frequencies** $$\omega_n$$
+as a species-dependent subset of $$k_n$$:
 
 $$\begin{aligned}
     \boxed{
@@ -232,7 +234,7 @@ $$\begin{aligned}
 <div class="hidden" markdown="1">
 <label for="proof-FT-alt">Proof.</label>
 We split the integral, shift its limits,
-and use the (anti)periodicity of $C_{AB}$:
+and use the (anti)periodicity of $$C_{AB}$$:
 
 $$\begin{aligned}
     C_{AB}(i k_n)
@@ -247,9 +249,9 @@ $$\begin{aligned}
     &= \frac{1}{2} \big( 1 \pm e^{-i k_n \hbar \beta} \big) \int_0^{\hbar \beta} C_{AB}(\tau) \: e^{i k_n \tau} \dd{\tau}
 \end{aligned}$$
 
-With $+$ for bosons, and $-$ for fermions. Since $k_n \equiv n \pi / (\hbar \beta)$,
-we know $e^{-i k_n \hbar \beta} \in \{-1, 1\}$,
-so for bosons all odd $n$ vanish, and for fermions all even $n$,
+With $$+$$ for bosons, and $$-$$ for fermions. Since $$k_n \equiv n \pi / (\hbar \beta)$$,
+we know $$e^{-i k_n \hbar \beta} \in \{-1, 1\}$$,
+so for bosons all odd $$n$$ vanish, and for fermions all even $$n$$,
 yielding the desired result.
 
 For the other case, we simply shift the first integral's limits instead of the seconds':
@@ -263,11 +265,12 @@ $$\begin{aligned}
     \\
     &= \frac{1}{2} \big( 1 \pm e^{-i k_n \hbar \beta} \big) \int_{-\hbar \beta}^0 C_{AB}(\tau) \: e^{i k_n \tau} \dd{\tau}
 \end{aligned}$$
+
 </div>
 </div>
 
 If we actually evaluate this,
-we obtain the following form of $C_{AB}$,
+we obtain the following form of $$C_{AB}$$,
 which is almost identical to the
 [Lehmann representation](/know/concept/lehmann-representation/)
 of the "ordinary" retarded and advanced Green's functions:
@@ -285,8 +288,8 @@ $$\begin{aligned}
 <label for="proof-Lehmann">Proof</label>
 <div class="hidden" markdown="1">
 <label for="proof-Lehmann">Proof.</label>
-For $\tau \!-\! \tau' > 0$, we start by expanding
-in the many-particle eigenstates $\Ket{n}$:
+For $$\tau \!-\! \tau' > 0$$, we start by expanding
+in the many-particle eigenstates $$\Ket{n}$$:
 
 $$\begin{aligned}
     C_{AB}(\tau \!-\! \tau')
@@ -300,7 +303,7 @@ $$\begin{aligned}
     \matrixel{n'}{\hat{B}}{n} e^{(E_n - E_{n'})(\tau - \tau') / \hbar}
 \end{aligned}$$
 
-We take the Fourier transform by integrating over $[0, \hbar \beta]$:
+We take the Fourier transform by integrating over $$[0, \hbar \beta]$$:
 
 $$\begin{aligned}
     C_{AB}(i \omega_m)
@@ -320,8 +323,8 @@ $$\begin{aligned}
     \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big)
 \end{aligned}$$
 
-Moving on to $\tau \!-\! \tau' < 0$,
-we again expand in the many-particle eigenstates $\Ket{n}$:
+Moving on to $$\tau \!-\! \tau' < 0$$,
+we again expand in the many-particle eigenstates $$\Ket{n}$$:
 
 $$\begin{aligned}
     C_{AB}(\tau \!-\! \tau')
@@ -335,8 +338,8 @@ $$\begin{aligned}
     \matrixel{n'}{\hat{A}}{n} e^{-(E_n - E_{n'})(\tau - \tau') / \hbar}
 \end{aligned}$$
 
-Since $\tau \!-\! \tau' < 0$ this time,
-we take the Fourier transform over $[-\hbar \beta, 0]$:
+Since $$\tau \!-\! \tau' < 0$$ this time,
+we take the Fourier transform over $$[-\hbar \beta, 0]$$:
 
 $$\begin{aligned}
     C_{AB}(i \omega_m)
@@ -359,17 +362,17 @@ $$\begin{aligned}
     \Big( e^{- \beta E_{n'}} \mp e^{-\beta E_n} \Big)
 \end{aligned}$$
 
-Where swapping $n$ and $n'$ gives the desired result.
+Where swapping $$n$$ and $$n'$$ gives the desired result.
 </div>
 </div>
 
-This gives us the primary use of the Matsubara Green's function $C_{AB}$:
-calculating the retarded $C_{AB}^R$ and advanced $C_{AB}^A$.
-Once we have an expression for Matsubara's $C_{AB}$,
-we can recover $C_{AB}^R$ and $C_{AB}^A$ by substituting
-$i \omega_m \to \omega \!+\! i \eta$ and $i \omega_m \to \omega \!-\! i \eta$ respectively.
+This gives us the primary use of the Matsubara Green's function $$C_{AB}$$:
+calculating the retarded $$C_{AB}^R$$ and advanced $$C_{AB}^A$$.
+Once we have an expression for Matsubara's $$C_{AB}$$,
+we can recover $$C_{AB}^R$$ and $$C_{AB}^A$$ by substituting
+$$i \omega_m \to \omega \!+\! i \eta$$ and $$i \omega_m \to \omega \!-\! i \eta$$ respectively.
 
-In general, we can define the **canonical Green's function** $C_{AB}(z)$
+In general, we can define the **canonical Green's function** $$C_{AB}(z)$$
 on the complex plane:
 
 $$\begin{aligned}
@@ -380,8 +383,8 @@ $$\begin{aligned}
 
 This is a [holomorphic function](/know/concept/holomorphic-function/),
 except for poles on the real axis.
-It turns out that $C_{AB}(z)$ must have these properties
-for the substitution $i \omega_n \to \omega \!\pm\! i \eta$ to be valid.
+It turns out that $$C_{AB}(z)$$ must have these properties
+for the substitution $$i \omega_n \to \omega \!\pm\! i \eta$$ to be valid.