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| -rw-r--r-- | bezout.tex | 18 |
1 files changed, 13 insertions, 5 deletions
@@ -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} |