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| -rw-r--r-- | bezout.tex | 27 |
1 files changed, 13 insertions, 14 deletions
@@ -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} |