From 2a7f289f573e50ac92e1f549fb450dcb0751b313 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 7 Dec 2020 12:45:28 +0100
Subject: Added constrained Hamiltonian and defined ε.
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---
 bezout.tex | 17 +++++++++++------
 1 file changed, 11 insertions(+), 6 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index 4abb8e2..53cc5e9 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -28,13 +28,18 @@
 
 \maketitle
 
-\begin{equation} \label{eq:hamiltonian}
-  H = \frac1{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p}
+\begin{equation} \label{eq:bare.hamiltonian}
+  H_0 = \frac1{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p},
 \end{equation}
-where the $z$ are constrained by $z\cdot z=N$ and $J$ is a symmetric
-tensor whose elements are complex normal with $\langle|J|^2\rangle=p!/2N^{p-1}$
-and $\langle J^2\rangle=\kappa\langle|J|^2\rangle$ for complex parameter
-$|\kappa|<1$.
+where the $z$ are constrained by $z\cdot z=N$ and $J$ is a symmetric tensor
+whose elements are complex normal with $\langle|J|^2\rangle=p!/2N^{p-1}$ and
+$\langle J^2\rangle=\kappa\langle|J|^2\rangle$ for complex parameter
+$|\kappa|<1$. The constraint is enforced using the method of Lagrange
+multipliers: introducing the $\epsilon\in\mathbb C$, this gives
+\begin{equation} \label{eq:constrained.hamiltonian}
+  H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right).
+\end{equation}
+At any critical point $\epsilon=H/N$, the average energy.
 
 \bibliographystyle{apsrev4-2}
 \bibliography{bezout}
-- 
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