Update the typeset of Wick contractions.

Author: myhsia

[email protected]

@@ -5,6 +5,7 @@
 \setlength \feynhandblobsize {10mm}
 \usepackage[natbib, style = phys, biblabel = brackets, sorting = none]{biblatex}
 \addbibresource{reference.bib}
+\usepackage{zref-clever, simpler-wick}
 
 \coverset{
   title   = Quantum Many-Body Theory,
@@ -55,6 +56,8 @@
 \newweek
 \include{./context/weeka_[2025-11-11]}
 \include{./context/10_.tex}
+\newweek
+\include{./context/weekb_[2025-11-18]}
 
 \printbibliography[title = {\refname\label{chap:bibliography}}]
 

[email protected]

@@ -107,7 +107,7 @@ \section{Contraction}
 
 Define the contraction of two operators
 \begin{equation}
-  \underbrace{A(t) B(t')}
+  \wick{\c A(t) \c B(t')}
 = T_\epsilon(A(t) B(t')) - \mathcal N(A(t)B(t'))
 \end{equation}
 The operator identity
@@ -119,7 +119,7 @@ \section{Operator level of ``contraction''}
 \begin{example}[Two operators]
   We just look at the RHS of the contraction
   \[
-    \underbrace{\gamma(t) \gamma^\dagger(')}
+    \wick{\c \gamma(t) \c \gamma^\dagger(')}
   = T_\epsilon(\gamma(t) \gamma^\dagger(t')) - N(\gamma(t) \gamma^\dagger(t'))
   \]
   On the operator level. Where
@@ -131,7 +131,7 @@ \section{Operator level of ``contraction''}
   \]
   Then, we have to define the actual contraction
   \[
-    \braket<\eta_0|\underbrace{\gamma \gamma^\dagger}|\eta_0>
+    \braket<\eta_0|\wick{\c \gamma \c \gamma^\dagger}|\eta_0>
   = \iu G^{0,c}[\gamma, \gamma^\dagger]
   \]
 \end{example}
@@ -147,7 +147,7 @@ \section{Operator level of ``contraction''}
   1234 + 2134 + \cdots
 = (12 + 21)34 + T_\epsilon (12)(34)
 \xlongequal{\text{Contraction tricks}}
-  (\overbrace{12} + N(12)) (34)
+  (\wick{\c11\c12} + N(12)) (34)
 \]
 So, the observable
 \[
@@ -160,9 +160,9 @@ \section{Operator level of ``contraction''}
   1243 + 2143 \to \iu g_{12}(43), \quad
 \]
 The combines to $\iu g_{12}$ and $\iu g_{34}$,
-i.e., $T_\epsilon(\overbrace{12}\overbrace{34})$.
+i.e., $T_\epsilon(\wick{\c1 1\c1 2\c1 3\c1 4})$.
 Also for $1324 + 3124$ and $1342 + 3142$, which can be combined into
-$T_\epsilon(1234)$. (overbrace 13 and 24).
+$T_\epsilon(\wick{\c1 1\c2 2\c1 3\c2 4})$.
 The LHS have $4!$ terms in total, and we will have $4\times3/2 = 6$ contractions
 \[
   g_{12} g_{34} \quad g_{13} g_{24} \quad g_{14} g_{23}
@@ -171,10 +171,10 @@ \section{Operator level of ``contraction''}
 \begin{multline}
   T_\epsilon(123 \cdots 2N)
 = N(12\cdots 2N) + \binom{N\text{-product}}{\text{with ONE contraction}}
-  N(\underbrace{12}34 \cdots)
-+ N(\underbrace{13}24 \cdots 2N) \\
+  N(\wick{\c11\c12}34 \cdots)
++ N(\wick{\c11\c13}24 \cdots 2N) \\
 + \binom{N\text{-product}}{\text{with TWO contraction}}
-+ N(\underbrace{12}\underbrace{34}56 \cdots 2N) + \cdots
++ N(\wick{\c11\c12\c13\c14}56 \cdots 2N) + \cdots
 + \{\text{total contractions}\}
 \end{multline}
 \begin{example}[$T_\epsilon(123456)$]
@@ -188,9 +188,9 @@ \section{Operator level of ``contraction''}
 \end{example}
 Do the induction $2n$/$2n + 2$-Wick's
 \[
-  (\underbrace{12}(34) + N(12)(34)) \cdot
-  (\underbrace{56} + N(56)) \to
-  \prod_{1, \cdots, 1}^n (\underbrace{i_1 i_2} + N(\\quad))
+  (\wick{\c11\c12}(34) + N(12)(34)) \cdot
+  (\wick{\c15\c16} + N(56)) \to
+  \prod_{1, \cdots, 1}^n (\wick{\c i_1 \c i_2} + N(\quad))
 \]
 Together, we have $n$-terms
 \[
@@ -210,7 +210,7 @@ \section{Operator level of ``contraction''}
 The canonical algebra
 \[
   \{c_i, c_j^\dagger\} = \delta_{ij}, \quad
-  \underbrace{\gamma \gamma^\dagger} = T_\epsilon - N(\cdots)
+  \wick{\c\gamma \c\gamma^\dagger} = T_\epsilon - N(\cdots)
   \to s^2 \Tr(s^+, s^-)
 \]
 One is canonical algebra, and one is the Gaussion states.
@@ -406,7 +406,7 @@ \section{Unlabeled Feyman diagram}
   COmbines them Topo ineq.
   \[
     G_\Gamma \to S, \quad
-    \cancel{\frac1{2^nn!m!}} \sum\{n||\text{contractionss}\} = \frac1s (\cdots)
+    \cancel{\frac1{2^nn!m!}} \sum\{n\|\text{contraction}\} = \frac1s (\cdots)
   \]
   which cancels by ``All poss value-inv-transf'', for $v = \frac12$.
 \end{example}

context/10.tex renamed to context/10_.tex

@@ -0,0 +1,38 @@
+% !TeX root = ../main.tex
+
+\section*{Homework \#11 [2025-11-18]}
+
+\begin{problem}[label = prob.11.1]\leavevmode
+  \begin{enumext}
+    \item Write the time-ordered product
+    \[
+      T_\epsilon \{a_{k\sigma}(t_1)a_{l\sigma'}^\dagger(t_2)
+                   a_{m\sigma}(t_3) a_{n\sigma'}^\dagger(t)3\}
+    \]
+    in terms of normal products and suitable contractions.
+    \item Express the expectation value of the time-ordered product in part
+    (a) in the ground state $\ket|\eta_0>$ of the unperturbed system in terms
+    of products of the free causal Green's functions.
+  \end{enumext}
+\end{problem}
+\begin{solution}
+  
+\end{solution}
+
+\begin{problem}
+  Evaluate explicitly the expectation value of the time-ordered product in
+  part 1 of \Autoref{prob.11.1} in the ground state $\eta|\eta_0>$ of the
+  unperturbed system for the special case $k = l = m = n$, $\sigma = \sigma'$
+  for
+  \begin{enumext}[columns = 2]
+    \item $t_1 > t_2 > t_3$,
+    \item $t_1 > t_3 > t_2$.
+  \end{enumext}
+  Check the results by direct calculation of the expectation values, i.e.,
+  without using Wick's theorem.
+\end{problem}
+\begin{solution}\leavevmode
+  \begin{enumext}
+    \item 
+  \end{enumext}
+\end{solution}