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diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index c322e8c..0896da4 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} @@ -797,61 +798,6 @@ 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} |