From 55409dc28d44271a915e9197a423531a06a17d4a Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 9 Mar 2021 16:09:33 +0100
Subject: More working changes to constraint discussion.

---
 bezout.tex | 27 +++++++++++++--------------
 1 file changed, 13 insertions(+), 14 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index ff5a840..3a02f51 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -98,20 +98,19 @@ of Lagrange multipliers: introducing $\epsilon\in\mathbb C$, our energy is
 \begin{equation} \label{eq:constrained.hamiltonian}
   H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right).
 \end{equation}
-One might balk at taking the constraint as $z^2=N$---which might be more
-appropriately called a hyperbolic constraint---rather than $|z|^2=N$.  The
-reasoning is twofold. First, at every point $z$ the energy
-\eqref{eq:bare.hamiltonian} has a `radial' gradient of magnitude proportional
-to the energy, as $z\cdot\partial H_0=pH_0$. This trivial direction must be
-removed if critical points are to exist a any nonzero energy, and the
-constraint surface $z^2=N$ is the unique surface whose normal is parallel to
-$z$ and which contains the configuration space of the real $p$-spin model as a
-subspace. Second, taking the constraint to be the level set of a holomorphic
-function means the resulting configuration space is a \emph{bone fide} complex
-manifold, and therefore permits easy generalization of the integration
-techniques referenced above. The same cannot be said for the space defined by
-$|z|^2=N$, which is topologically the $(2N-1)$-sphere and cannot admit a
-complex structure.
+One might balk at the constraint $z^2=N$---which could appropriately be called
+a \emph{hyperbolic} constraint---by comparison with $|z|^2=N$.  The reasoning
+is twofold. First, at every point $z$ the energy \eqref{eq:bare.hamiltonian}
+has a `radial' gradient of magnitude proportional to the energy, since
+$z\cdot\partial H_0=pH_0$. This anomalous direction must be neglected if critical
+points are to exist a any nonzero energy, and the constraint surface $z^2=N$ is
+the unique surface whose normal is parallel to $z$ and which contains the
+configuration space of the real $p$-spin model as a subspace. Second, taking
+the constraint to be the level set of a holomorphic function means the
+resulting configuration space is a \emph{bone fide} complex manifold, and
+therefore permits easy generalization of the integration techniques referenced
+above. The same cannot be said for the space defined by $|z|^2=N$, which is
+topologically the $(2N-1)$-sphere and cannot admit a complex structure.
 
 The critical points are of $H$ given by the solutions to the set of equations
 \begin{equation} \label{eq:polynomial}
-- 
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