nLab Bohr-Sommerfeld leaf

Idea

In the context of geometric quantization of a symplectic manifold $(X, \omega)$, a Bohr-Sommerfeld leaf is a Lagrangian submanifold of $X$ on which not only the symplectic form $\omega$ vanishes, but on which also a given 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 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 metaplectic correction.)

Definition

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

Definition

A Lagrangian submanifold $L \hookrightarrow X$ is a 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

$$[\nabla_L] = 0 \in H^2_{conn}(X) \,.$$

Remark

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.

Remark

The Bohr-Sommerfeld condition is the natural lift of the Lagrangian subspace-condition to prequantum geometry:

When expressed in terms of smooth moduli stacks (see at geometry of physics for background), the (pre-)symplectic structure is equivalently a map

$$\omega \;\colon\; X \to \Omega^2_{cl}$$

and a prequantization $\nabla$ is equivalently a lift $\nabla$ in the diagram

$$\array{ && \mathbf{B}U(1)_{conn} \\ & {}^{\mathllap{\nabla}}\nearrow & \downarrow^{\mathrlap{F_{(-)}}} \\ X &\stackrel{\omega}{\to}& \Omega^2_{cl}(X) } \,.$$

The condition on an isotropic submanifold $L \hookrightarrow X$ is that the composite map

$$\omega|_L \;\colon\; \array{ L &\hookrightarrow & X &\stackrel{\omega}{\to}& \Omega^2_{cl} }$$

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

$$\nabla|_L \;\colon\; \array{ L &\hookrightarrow & X &\stackrel{\nabla}{\to}& \mathbf{B}U(1)_{conn} }$$

is trivial in $H(L, \mathbf{B}U(1)_{conn}) = H^2_{conn}(X)$.

Examples

Harmonic oscillator

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

$$\omega = d_{dR} t \wedge d\theta \,,$$

where on $\mathbb{R}^2- \mathbb{R}_+$ $(t, \theta)$ are the canonical polar coordinates.

We may choose the trivial prequantum line bundle with connection given by the globally defined differential 1-form

$$\Theta := t \wedge d\theta \,.$$

Then a polarization is given by the foliation whose leaves are the submanifolds of constant $t$.

The covariant derivative along any leaf acts as

$$(\nabla_\Theta \sigma)(t, \theta) = (\frac{\partial}{\partial \theta} \sigma)(t, \theta) - i t \sigma(t, \theta) \,.$$

The covariantly sections covariantly constat on a leaf hence must be of the form

$$\sigma(t, \theta) = a(t) \exp( i t \theta) \,.$$

For this to be well-defined as a globally defined section on the whole leaf the condition

$$t = 2 \pi k \; \; k \in \mathbb{Z}$$

has to hold. Hence the Bohr-Sommerfeld leaves here are the circles of radius $2 \pi k$ in $\mathbb{R}^2$.

Properties

Theorem

(Guillemin-Sternberg)

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

$$\{F_BS\} = \{ p \in X | (f_1(p), \cdots, f_n(p)) \in \mathbb{Z}^n \}$$

where the $\{f_i\}$ are global action coordinates? on the base space $B$.

Theorem

(Sniatycki)

If the leaf space $B$ is Hausdorff and the projection $X \to B$ has compact fibers, then the dimension of the space of quantum states is given by the number of Bohr-Sommerfeld leaves.

References

Named after Niels Bohr and Arnold Sommerfeld.

For historical background see also

• Wikipedia, Old quantum theory.

Discussion in the modern formalism of geometric quantization:

• Jędrzej Śniatycki, Wave functions relative to a real polarization, Internat. J. Theoret. Phys., 14 4 (1975) 277-288 [doi:10.1007/BF01807689)]

• Jędrzej Śniatycki, Geometric Quantization and Quantum Mechanics, Applied Mathematical Sciences 30, Springer-Verlag (1980) [doi:10.1007/978-1-4612-6066-0]

• Mark Hamilton, Locally toric manifolds and singular Bohr-Sommerfeld leaves

• Eva Miranda, From action-angle coordinates to geometric quantization and back (2011) (pdf)

and with application to conformal blocks:

• Andrei Tyurin, On Bohr-Sommerfeld bases, Izvestiya: Mathematics 64 5 (2000) 1033–1064 [arXiv:math/9909084, doi:10.1070/IM2000v064n05ABEH000308, pdf]