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| -rw-r--r-- | bezout.tex | 19 |
1 files changed, 10 insertions, 9 deletions
@@ -102,15 +102,16 @@ 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. +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. The critical points are of $H$ given by the solutions to the set of equations \begin{equation} \label{eq:polynomial} |