From 9471da284a8c6147c608ed1bc23675a2ec71c8ac Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 9 Mar 2021 15:52:39 +0100
Subject: New discussion of the constraint.

---
 bezout.tex | 18 +++++++++++++-----
 1 file changed, 13 insertions(+), 5 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index 9222449..1b9f04e 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -98,11 +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}
-We choose to constrain our model by $z^2=N$ rather than $|z|^2=N$ in order to
-preserve the analyticity of $H$. The nonholomorphic constraint also has a
-disturbing lack of critical points nearly everywhere: if $H$ were so
-constrained, then $0=\partial^* H=-p\epsilon z$ would only be satisfied for
-$\epsilon=0$.  
+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 itself, 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 real configuration space 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
+admits 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}
-- 
cgit v1.2.3-54-g00ecf

