From 34061638fbf86d3664faed0610ad23faa52dad3d Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 29 Jun 2022 11:20:34 +0200
Subject: killed old section.

---
 frsb_kac-rice.tex | 66 +++++--------------------------------------------------
 1 file changed, 6 insertions(+), 60 deletions(-)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 2015feb..b38feb3 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -1,13 +1,14 @@
 \documentclass[fleqn]{article}
 
 \usepackage{fullpage,amsmath,amssymb,latexsym,graphicx}
-\usepackage{appendix,xcolor}
+\usepackage{appendix}
+\usepackage[dvipsnames]{xcolor}
 \usepackage[
   colorlinks=true,
-  urlcolor=purple,
-  citecolor=purple,
-  filecolor=purple,
-  linkcolor=purple
+  urlcolor=MidnightBlue,
+  citecolor=MidnightBlue,
+  filecolor=MidnightBlue,
+  linkcolor=MidnightBlue
 ]{hyperref} % ref and cite links with pretty colors
 
 \begin{document}
@@ -794,61 +795,6 @@ Integrating by parts,
   &=\log[\hat\beta R_d^{-1}\lambda(q_{k-1})+1]+\frac{\hat\beta}{R_d}\int_{q_0^+}^{q_{k-1}}dq\,\frac1{\hat\beta R_d^{-1}\lambda(q)+1}
 \end{align*}
 
-\section{ A motivation for the ansatz}
-
- We may encode the original variables in a superspace variable:
-\begin{equation}
-  \phi_a(1)= s_a + \bar\eta_a\theta_1+\bar\theta_1\eta_a +  \hat s_a \bar \theta_1 \theta_1
-\end{equation}Here $\theta_a$, $\bar \theta_a$ are Grassmann variables, and we denote the full set of coordinates 
-in a  compact form as $1=  \theta_1 \overline\theta_1$, $d1=   d\theta_1 d\overline\theta_1$, etc.
-The correlations are encoded in
-\begin{equation}
-\begin{aligned}
-  \mathbb Q_{a,b}(1,2)&=\frac 1 N \phi_a(1)\cdot\phi_b (2) =
-Q_{ab}  -i\left[\bar\theta_1\theta_1+\bar\theta_2\theta_2\right]  R_{ab}
- +(\bar\theta_1\theta_2+\theta_1\bar\theta_2)F_{ab}
- + \bar\theta_1\theta_1 \bar \theta_2 \theta_2 D_{ab} \\ 
-&+  \text{odd terms in the $\bar \theta,\theta$}~.
-\end{aligned}
-\label{Q12}
-\end{equation}
-\begin{equation}
-  \overline{\Sigma(\epsilon,\mu)}
-  =\hat\beta\epsilon\lim_{n\to0}\frac1n\left[
-    \mu\int d1\sum_a^n\mathbb Q_{aa}(1,1)
-    +\int d2\,d1\,\frac12\sum_{ab}^n(1+\hat\beta\bar\theta_1\theta_1)f(\mathbb Q_{ab}(1,2))(1+\hat\beta\bar\theta_2\theta_2)
-    +\frac12\operatorname{sdet}\mathbb Q
-  \right]
-\end{equation}
-The odd and even fermion numbers decouple, so we can neglect all odd terms in $\theta,\bar{\theta}$.
-
-\cite{Annibale_2004_Coexistence}
-
-This encoding also works for dynamics, where the coordinates then read 
-$1= (\bar \theta,  \theta, t)$, etc.   The variables $\bar \theta \theta$ and $\bar \theta ' \theta'$ play
-the role of `times' in a superspace treatment. We have a long experience of
-making an ansatz for replicated quantum problems, which naturally involve a (Matsubara) time. The dependence on this time only holds for diagonal replica elements, a consequence of ultrametricity.  The analogy strongly
-suggests that only the diagonal ${\bf Q}_{aa}$ depend on the $\theta$'s. This boils down the ansatz \ref{ansatz}.
-Not surprisingly, and for the same reason as in the quantum case, this ansatz closes, as we shall see.For example, consider the convolution:
-
-\begin{equation}
-  \begin{aligned}
-  \int d3\,\mathbb Q_1(1,3)\mathbb Q_2(3,2)
-  =\int d3\,(
-  Q_1  -i(\bar\theta_1\theta_1+\bar\theta_3\theta_3)  R_1
-   +(\bar\theta_1\theta_3+\theta_1\bar\theta_3)F_1
- + \bar\theta_1\theta_1 \bar \theta_3 \theta_3 D_1
- ) \\ (
-  Q_2  -i(\bar\theta_3\theta_3+\bar\theta_2\theta_2)  R_2
-   +(\bar\theta_3\theta_2+\theta_3\bar\theta_2)F_2
- + \bar\theta_3\theta_3 \bar \theta_2 \theta_2 D_2
- ) \\
- =-i(Q_1R_2+R_1Q_2)
- +Q_1D_2\bar\theta_2\theta_2+D_1Q_2\bar\theta_1\theta_1
- -i\bar\theta_1\theta_1\bar\theta_2\theta_2R_1D_2
- -i\bar\theta_1\theta_1\bar\theta_2\theta_2D_1R_2
- \end{aligned}
-\end{equation}
 \end{appendix}
 
 
-- 
cgit v1.2.3-70-g09d2

