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

1 files changed, 55 insertions, 50 deletions

diff --git a/source/know/concept/feynman-diagram/index.md b/source/know/concept/feynman-diagram/index.md
index d117562..c36e7df 100644
--- a/source/know/concept/feynman-diagram/index.md
+++ b/source/know/concept/feynman-diagram/index.md
@@ -13,13 +13,13 @@ When doing calculations in the context of condensed matter physics and quantum f
 that would be tedious or error-prone to work with directly.
 This article is about condensed matter physics.
 
-Suppose we have a many-particle Hamiltonian $\hat{H} = \hat{H}_0 + \hat{H}_1$,
-consisting of an "easy" term $\hat{H}_0$,
-and then a "difficult" term $\hat{H}_1$
+Suppose we have a many-particle Hamiltonian $$\hat{H} = \hat{H}_0 + \hat{H}_1$$,
+consisting of an "easy" term $$\hat{H}_0$$,
+and then a "difficult" term $$\hat{H}_1$$
 with time-dependent and/or interacting parts.
-Let $\Ket{\Phi_0}$ be a known eigenstate (or superposition thereof)
-of the easily solvable part $\hat{H}_0$,
-with respect to which we will take expectation values $\Expval{}$.
+Let $$\Ket{\Phi_0}$$ be a known eigenstate (or superposition thereof)
+of the easily solvable part $$\hat{H}_0$$,
+with respect to which we will take expectation values $$\Expval{}$$.
 
 Below, we go through the most notable components of Feynman diagrams
 and how to translate them into a mathematical expression.
@@ -28,26 +28,27 @@ and how to translate them into a mathematical expression.
 ## Real space
 
 The most common component is a **fermion line**, which represents
-a [Green's function](/know/concept/greens-functions/) $G^0$
-for the simple Hamiltonian $\hat{H}_0$.
+a [Green's function](/know/concept/greens-functions/) $$G^0$$
+for the simple Hamiltonian $$\hat{H}_0$$.
 Any type of Green's function is possible in theory (e.g. a retarded),
 but usually the *causal* function is used.
-Let the subscript $I$ refer to the
+Let the subscript $$I$$ refer to the
 [interaction picture](/know/concept/interaction-picture/),
-and $\mathcal{T}\{\}$ denote the
+and $$\mathcal{T}\{\}$$ denote the
 [time-ordered product](/know/concept/time-ordered-product/):
 
 <a href="freegf.png">
 <img src="freegf.png" style="width:60%">
 </a>
+
 $$\begin{aligned}
     = i \hbar G_{s_2 s_1}^0(\vb{r}_2, t_2; \vb{r}_1, t_1)
     = \Expval{\mathcal{T} \Big\{ \hat{\Psi}_{s_2 I}(\vb{r}_2, t_2) \hat{\Psi}_{s_1 I}^\dagger(\vb{r}_1, t_1) \Big\}}
 \end{aligned}$$
 
 The arrow points in the direction of time, or more generally,
-from the point of creation $\hat{\Psi}{}^\dagger$
-to the point of annihilation $\hat{\Psi}$.
+from the point of creation $$\hat{\Psi}{}^\dagger$$
+to the point of annihilation $$\hat{\Psi}$$.
 The dots at the ends are called **vertices**,
 which represent points in space and time with a spin.
 Vertices can be
@@ -55,19 +56,20 @@ Vertices can be
 or **external** (either one Green's function entering OR one leaving).
 
 Less common is a **heavy fermion line**, representing
-a causal Green's function $G$ for the entire Hamiltonian $\hat{H}$,
-where the subscript $H$ refers to the [Heisenberg picture](/know/concept/heisenberg-picture/):
+a causal Green's function $$G$$ for the entire Hamiltonian $$\hat{H}$$,
+where the subscript $$H$$ refers to the [Heisenberg picture](/know/concept/heisenberg-picture/):
 
 <a href="fullgf.png">
 <img src="fullgf.png" style="width:60%">
 </a>
+
 $$\begin{aligned}
     = i \hbar G_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1)
     = \Expval{\mathcal{T} \Big\{ \hat{\Psi}_{s_2 H}(\vb{r}_2, t_2) \hat{\Psi}_{s_1 H}^\dagger(\vb{r}_1, t_1) \Big\}}
 \end{aligned}$$
 
 Next, an **interaction line** or **boson line** represents
-a two-body interaction operator $\hat{W}$ (in $\hat{H}_1$),
+a two-body interaction operator $$\hat{W}$$ (in $$\hat{H}_1$$),
 which we assume to be instantaneous, i.e. time-independent
 (in quantum field theory this is *not* assumed),
 hence it starts and ends at the same time,
@@ -76,16 +78,17 @@ and no arrow is drawn:
 <a href="interaction.png">
 <img src="interaction.png" style="width:60%">
 </a>
+
 $$\begin{aligned}
     = \frac{1}{i \hbar} W_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1)
     = \frac{1}{i \hbar} W(\vb{r}_2, \vb{r}_1; t_1) \: \delta(t_2 - t_1)
 \end{aligned}$$
 
 We have chosen to disallow spin flipping,
-so $W$ does not depend on $s_1$ or $s_2$.
-For reference, this function $W$
+so $$W$$ does not depend on $$s_1$$ or $$s_2$$.
+For reference, this function $$W$$
 has a time-dependence coming only from the interaction picture,
-and is to be used as follows to get the full two-body operator $\hat{W}$:
+and is to be used as follows to get the full two-body operator $$\hat{W}$$:
 
 $$\begin{aligned}
     \hat{W}
@@ -93,12 +96,13 @@ $$\begin{aligned}
     W(\vb{r}_1, \vb{r}_2) \hat{\Psi}_{s_2}(\vb{r}_2) \hat{\Psi}_{s_1}(\vb{r}_1) \dd{\vb{r}_1} \dd{\vb{r}_2}
 \end{aligned}$$
 
-One-body (time-dependent) operators $\hat{V}$ in $\hat{H}_1$
+One-body (time-dependent) operators $$\hat{V}$$ in $$\hat{H}_1$$
 are instead represented by a special vertex:
 
 <a href="perturbation.png">
 <img src="perturbation.png" style="width:35%">
 </a>
+
 $$\begin{aligned}
     = \frac{1}{i \hbar} V_s(\vb{r}, t)
 \end{aligned}$$
@@ -114,27 +118,27 @@ a.  Each vertex must be connected to one or two fermion lines,
     at most one of which leaves,
     and at most one of which enters.
 b.  Each internal vertex contains at most one "event";
-    which could be $V$ or $W$.
+    which could be $$V$$ or $$W$$.
 
 Finally, we need some additional rules to convert
 diagrams into mathematical expressions:
 
 1.  Disallow spin flipping by multiplying
-    each internal vertex by $\delta_{s_\mathrm{in} s_\mathrm{out}}$.
-2.  If both ends of a line are at the same time (always the case for $W$),
-    an infinitesimal $\eta \to 0^+$ must be added
+    each internal vertex by $$\delta_{s_\mathrm{in} s_\mathrm{out}}$$.
+2.  If both ends of a line are at the same time (always the case for $$W$$),
+    an infinitesimal $$\eta \to 0^+$$ must be added
     to the time of all creation operators,
-    so e.g. $G(t, t) \to G(t, t\!+\!\eta)$.
-3.  Integrate over spacetime coordinates $(\vb{r}, t)$
-    and sum over the spin $s$ of all internal vertices,
+    so e.g. $$G(t, t) \to G(t, t\!+\!\eta)$$.
+3.  Integrate over spacetime coordinates $$(\vb{r}, t)$$
+    and sum over the spin $$s$$ of all internal vertices,
     but not external ones.
-4.  Multiply the result by $(-1)^F$,
-    where $F$ is the number of closed fermion loops.
+4.  Multiply the result by $$(-1)^F$$,
+    where $$F$$ is the number of closed fermion loops.
 5.  Depending on the context, additional constant factors may be required;
     sometimes they are changed on-the-fly during a calculation.
 
 Note that rules 4 and 5 are convention,
-just like the factors $i \hbar$ in $G^0$, $G$, $V$ and $W$;
+just like the factors $$i \hbar$$ in $$G^0$$, $$G$$, $$V$$ and $$W$$;
 it simply turns out to be nicer to do it this way
 when using Feynman diagrams in the wild.
 
@@ -171,11 +175,12 @@ in the Fourier transform definitions for space and time.
 
 Working in Fourier space allows us to simplify calculations.
 Consider the following diagram and the resulting expression,
-where $\tilde{\vb{r}} = (\vb{r}, t)$, and $\tilde{\vb{k}} = (\vb{k}, \omega)$:
+where $$\tilde{\vb{r}} = (\vb{r}, t)$$, and $$\tilde{\vb{k}} = (\vb{k}, \omega)$$:
 
 <a href="conservation.png">
 <img src="conservation.png" style="width:40%">
 </a>
+
 $$\begin{aligned}
     &= (i \hbar)^3 \sum_{s s'} \!\!\iint \dd{\tilde{\vb{r}}} \dd{\tilde{\vb{r}}'}
     G_{s_1's}^0(\tilde{\vb{r}}_1', \tilde{\vb{r}}) G_{s s_1}^0(\tilde{\vb{r}}, \tilde{\vb{r}}_1) \delta_{s_1 s_1'}
@@ -230,8 +235,8 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Conveniently, the Dirac delta functions that appear from the integrals
-represent conservation of wavevector $\vb{k}$ (momentum $\hbar \vb{k}$)
-and angular frequency $\omega$ (energy $\hbar \omega$).
+represent conservation of wavevector $$\vb{k}$$ (momentum $$\hbar \vb{k}$$)
+and angular frequency $$\omega$$ (energy $$\hbar \omega$$).
 
 In Fourier space, it makes more sense
 to regard the incoming energies and momenta and spins as given,
@@ -246,21 +251,21 @@ $$\begin{aligned}
     \: G_{s_2}^0(\tilde{\vb{q}}_1 \!+\! \tilde{\vb{p}}) \: G_{s_2}^0(\tilde{\vb{q}}_1)
 \end{aligned}$$
 
-Therefore, we say that fermion lines represent $i \hbar G_{s}^0(\vb{k}, \omega)$,
-interaction lines $W(\vb{k}) / i \hbar$, etc.,
+Therefore, we say that fermion lines represent $$i \hbar G_{s}^0(\vb{k}, \omega)$$,
+interaction lines $$W(\vb{k}) / i \hbar$$, etc.,
 and the other interpretation rules are modified to the following:
 
-1.  Each line has a momentum $\vb{k}$ and energy $\omega$,
-    and each fermion line has a spin $s$;
+1.  Each line has a momentum $$\vb{k}$$ and energy $$\omega$$,
+    and each fermion line has a spin $$s$$;
     these must all be conserved at each vertex.
 2.  If both ends of a *fermion* line would be at the same time,
-    multiply it by $e^{i \omega \eta}$,
-    where $\eta \to 0^+$ is a positive infinitesimal,
-    so e.g. $G(\tau, \tau) \to e^{i \omega \eta} G(\tau, \tau)$.
-3.  Integrate over all internal $(\vb{k}, \omega)$,
-    and sum over all internal spins $s$.
-    Let each $(\vb{k}, \omega)$ integral contribute a factor $1 / (2 \pi)^4$.
-4.  Multiply the end result by $(-1)^F$, where $F$ is the number of closed fermion loops.
+    multiply it by $$e^{i \omega \eta}$$,
+    where $$\eta \to 0^+$$ is a positive infinitesimal,
+    so e.g. $$G(\tau, \tau) \to e^{i \omega \eta} G(\tau, \tau)$$.
+3.  Integrate over all internal $$(\vb{k}, \omega)$$,
+    and sum over all internal spins $$s$$.
+    Let each $$(\vb{k}, \omega)$$ integral contribute a factor $$1 / (2 \pi)^4$$.
+4.  Multiply the end result by $$(-1)^F$$, where $$F$$ is the number of closed fermion loops.
 5.  Depending on the context, additional constant factors may be required;
     sometimes they are changed on-the-fly during a calculation.
 
@@ -288,7 +293,7 @@ $$\begin{aligned}
     = \Expval{\mathcal{T} \Big\{ \hat{\Psi}_H(\vb{r}_2, \tau_2) \hat{\Psi}_H^\dagger(\vb{r}_1, \tau_1) \Big\}}
 \end{aligned}$$
 
-Where the time-ordering is with respect to $\tau$.
+Where the time-ordering is with respect to $$\tau$$.
 Interaction lines are modified like so:
 
 $$\begin{aligned}
@@ -298,14 +303,14 @@ $$\begin{aligned}
     = -\frac{1}{\hbar} W(\vb{r}_2, \vb{r}_1; \tau_1) \delta(\tau_2 \!-\! \tau_1)
 \end{aligned}$$
 
-One-body $V$-vertices are usually not used,
+One-body $$V$$-vertices are usually not used,
 because they are intended for real-time-dependent operators,
-but in theory they would get a factor $-1/\hbar$ too.
+but in theory they would get a factor $$-1/\hbar$$ too.
 
 For imaginary time, the Fourier transform is defined differently,
 and a distinction must be made between
-fermionic Matsubara frequencies $i \omega_n^f$ (for $G$ and $G^0$)
-and bosonic Matsubara ones $i \omega_n^b$ (for $W$).
+fermionic Matsubara frequencies $$i \omega_n^f$$ (for $$G$$ and $$G^0$$)
+and bosonic Matsubara ones $$i \omega_n^b$$ (for $$W$$).
 This distinction is compatible with frequency conservation,
 since a sum of two fermionic frequencies is always bosonic.
 We have:
@@ -322,8 +327,8 @@ $$\begin{aligned}
 
 The interpretation in Fourier space is the same,
 except that each internal integral/sum
-instead gives a constant $1 / \big(\hbar \beta (2 \pi)^3\big)$,
-and same-time fermion lines need a factor of $e^{i \omega_n^f \eta}$.
+instead gives a constant $$1 / \big(\hbar \beta (2 \pi)^3\big)$$,
+and same-time fermion lines need a factor of $$e^{i \omega_n^f \eta}$$.