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\begin{document}

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\section*{Bohr-Sommerfeld leaf}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{geometric_quantization}{}\paragraph*{{Geometric quantization}}\label{geometric_quantization}

[[!include geometric quantization - contents]]

\hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry}

[[!include symplectic geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{HarmonicOscillator}{Harmonic oscillator}\dotfill \pageref*{HarmonicOscillator} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In the context of [[geometric quantization]] of a [[symplectic manifold]] $(X, \omega)$, a \emph{Bohr-Sommerfeld leaf} is a [[Lagrangian submanifold]] of $X$ on which not only the [[symplectic form]] $\omega$ vanishes, but on which also a given [[prequantum bundle|prequantization]] $\nabla$ of $\omega$ is trivializable.

Therefore given a [[real polarization]] of $(X,\omega)$, hence a [[foliation]] by [[Lagrangian submanifolds]], the Bohr-Sommerfeld leaves form a discrete subset of the [[leaf space]]. The discreteness of this subset is essentially the formal incarnation of ``quantization'' and this is what [[Bohr]] and [[Sommerfeld]] originally considered (in less abstract terms, the archetypical example was the harmonic oscillator as discussed \hyperlink{HarmonicOscillator}{below}).

(There is a correction to this picture, given by the fact that a [[quantum states]]/[[semiclassical states]], involve not just Lagrangian submanifolds/Bohr-Sommerfeld leaves, but moreover [[half-densities]] over these. These are to satisfy an additional condition, encoded by the \emph{[[metaplectic correction]]}.)

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $(X,\omega)$ be a ([[presymplectic manifold|pre-]])[[symplectic manifold]] and $\nabla$ a [[prequantum bundle|prequatization]], hence a $U(1)$-[[principal connection]] on $X$ with [[curvature]] 2-form $F_\nabla = \omega$.

\begin{defn}
\label{}\hypertarget{}{}
A [[Lagrangian submanifold]] $L \hookrightarrow X$ is a \textbf{Bohr-Sommerfeld leaf} if the restriction $\nabla|_L$ of the [[prequantum connection]] to $L$ is trivializable there, hence if its [[cohomology]] class vanishes in [[ordinary differential cohomology]]

\begin{displaymath}
[\nabla_L] = 0 \in H^2_{conn}(X)
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
For every [[isotropic submanifold]], hence in particular every [[Lagrangian submanifold]], $L \hookrightarrow X$ the restriction $\nabla|_L$ is necessarily already a [[flat connection]]. As discussed there, flat connections are equivalently encoded in the [[holonomy]] of their [[parallel transport]]: a flat connection is trivializable as a connection precisely if its [[holonomy]] is trivial. Therefore a Bohr-Sommerfeld leaf is equivalently a Lagrangian submanifold $L$ such that $\nabla|_L$ has trivial holonomy. In this form the Bohr-Sommerfeld condition is usually stated in the literature.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The Bohr-Sommerfeld condition is the natural lift of the [[Lagrangian subspace]]-condition to prequantum geometry:

When expressed in terms of [[smooth infinity-groupoid|smooth]] [[moduli stacks]] (see at \emph{[[geometry of physics]]} for background), the ([[presymplectic manifold|pre-]])[[symplectic manifold|symplectic structure]] is equivalently a map

\begin{displaymath}
\omega \;\colon\; X \to \Omega^2_{cl}
\end{displaymath}
and a prequantization $\nabla$ is equivalently a lift $\nabla$ in the diagram

\begin{displaymath}
\itexarray{
    && \mathbf{B}U(1)_{conn}
    \\
    & {}^{\mathllap{\nabla}}\nearrow & \downarrow^{\mathrlap{F_{(-)}}}
    \\
    X &\stackrel{\omega}{\to}& \Omega^2_{cl}(X)
  }
  \,.
\end{displaymath}
The condition on an [[isotropic submanifold]] $L \hookrightarrow X$ is that the composite map

\begin{displaymath}
\omega|_L
  \;\colon\;
  \itexarray{
    L &\hookrightarrow &  X &\stackrel{\omega}{\to}& \Omega^2_{cl}
  }
\end{displaymath}
is trivial in $H(L,\Omega^2_{cl}) = \Omega^2_{cl}(L)$ (and $L$ being Lagrangian means that it is maximal with this property). Then $L$ is Bohr-Sommerfeld if moreover the restriction of the prequantum lift

\begin{displaymath}
\nabla|_L
  \;\colon\;
  \itexarray{
    L &\hookrightarrow &  X &\stackrel{\nabla}{\to}& \mathbf{B}U(1)_{conn}
  }
\end{displaymath}
is trivial in $H(L, \mathbf{B}U(1)_{conn}) = H^2_{conn}(X)$.

\end{remark}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{HarmonicOscillator}{}\subsubsection*{{Harmonic oscillator}}\label{HarmonicOscillator}

For the single 1-dimensional [[Harmonic oscillator]], [[phase space]] is the [[symplectic manifold]] $\mathbb{R}^2$ equipped with the symplectic form

\begin{displaymath}
\omega = d_{dR} t \wedge d\theta
  \,,
\end{displaymath}
where on $\mathbb{R}^2- \mathbb{R}_+$ $(t, \theta)$ are the canonical [[polar coordinates]].

We may choose the trivial [[prequantum line bundle]] with [[connection on a bundle|connection]] given by the globally defined [[differential form|differential 1-form]]

\begin{displaymath}
\Theta := t \wedge d\theta
  \,.
\end{displaymath}
Then a [[polarization]] is given by the [[foliation]] whose [[leaves]] are the [[submanifolds]] of constant $t$.

The [[covariant derivative]] along any leaf acts as

\begin{displaymath}
(\nabla_\Theta \sigma)(t, \theta) 
  = (\frac{\partial}{\partial \theta} \sigma)(t, \theta)
  - i t \sigma(t, \theta)
 \,.
\end{displaymath}
The covariantly sections covariantly constat on a leaf hence must be of the form

\begin{displaymath}
\sigma(t, \theta) = a(t) \exp( i t \theta)
  \,.
\end{displaymath}
For this to be well-defined as a globally defined section on the whole leaf the condition

\begin{displaymath}
t = 2 \pi k \; \; k \in \mathbb{Z}
\end{displaymath}
has to hold. Hence the Bohr-Sommerfeld leaves here are the [[circles]] of radius $2 \pi k$ in $\mathbb{R}^2$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Guillemin-Sternberg)}

If a [[polarization]] of $X$ is a [[regular fibration]] with [[compact space|compact]] [[leaves]] over a [[simply connected]] base $B$, then the Bohr-Sommerfeld leaves form a discrete subset given by

\begin{displaymath}
\{F_BS\} = \{
    p \in X | (f_1(p), \cdots, f_n(p)) \in \mathbb{Z}^n
  \}
\end{displaymath}
where the $\{f_i\}$ are global [[action coordinates]] on the base space $B$.

\end{theorem}
\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Sniatycki)}

If the leaf space $B$ is [[Hausdorff space|Hausdorff]] and the projection $X \to B$ has [[compact space|compact]] fibers, then the [[dimension]] of the [[space of states (in geometric quantization)|space of quantum states]] is given by the number of Bohr-Sommerfeld leaves.

\end{theorem}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item J. niatycki, \emph{Wave functions relative to a real polarization}, Internat. J. Theoret. Phys., 14(4):277--288 (1975)


\item J. niatycki, \emph{Geometric Quantization and Quantum Mechanics}, volume 30 of Applied Mathematical Sciences. Springer-Verlag, New York (1980)


\item Mark Hamilton, \emph{Locally toric manifolds and singular Bohr-Sommerfeld leaves}


\item Eva Miranda, \emph{From action-angle coordinates to geometric quantization and back} (2011) (\href{http://fdis2011.uni-jena.de/Talks/Eva%20Miranda.pdf}{pdf})



\end{itemize}
[[!redirects Bohr-Sommerfeld leaves]]

[[!redirects BS-leaf]] [[!redirects BS-leaves]]

[[!redirects Bohr-Sommerfeld quantization]]



\end{document}