This entry is about polarization of phase spaces (or of any symplectic manifold) into canonical “position” coordinates and canonical momenta. Different concepts of a similar name include the polarization identity (such as in an inner product space or a Jordan algebra) or wave polarization (such as polarized light). On the other hand, the concept of polarized algebraic variety is closely related.
geometric quantization higher geometric quantization
geometry of physics: Lagrangians and Action functionals + Geometric Quantization
prequantum circle n-bundle = extended Lagrangian
prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology
For a symplectic manifold $(X, \omega)$ regarded as the phase space of a physical system, a choice of polarization is, locally, a choice of decomposition of the coordinates on $X$ into “canonical coordinates” and “canonical momenta”.
The archtypical example is that where $X = T^* \Sigma$ is the cotangent bundle of a manifold $\Sigma$. In this case the canonical canonical coordinates are those parameterizing $\Sigma$ itself, while the canonical canonical momenta are coordinates on each fiber of the cotangent bundle.
But for general symplectic manifolds there is no such canonical choice of coordinates and momenta. Moreover, in general there is not even a global notion of canonical momenta. Instead, a choice of (real) polarization is a foliation of phase space by Lagrangian submanifolds and then
the “canonical coordinates” are coordinates on the corresponding leaf space (parameterizing the leaves);
the “canonical momenta” are coordinates along each leaf. If there is no typical leaf then this is not a globally defined notion, only the polarization itself is.
Locally this is a choice of coordinate patch $\phi : \mathbb{R}^{2n} \to X$ such that the symplectic form takes the form
where the $\{q^i : \mathbb{R}^{2n} \simeq \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}\}$ are the canonical coordinates on the first $\mathbb{R}^n$-factor of the Cartesian space $\mathbb{R}^{2n}$, and where $\{p_o : \mathbb{R}^{2n} \simeq \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}\}$ are canonical coordinates on the second $\mathbb{R}^n$-factor.
The traditional notion of polarization applies to a symplectic manifold.
Symplectic manifold are the lowest step in a tower of notions in higher symplectic geometry which proceeds with n-plectic geometry for all $n$ and manifolds refined to smooth infinity-groupoids. The next simplest cases in this tower are symplectic Lie n-algebroids, which for $n=1$ are Poisson Lie algebroids and for $n = 2$ are Courant Lie 2-algebroids:
Let $(X, \omega)$ be a symplectic manifold.
A real polarization of $(X, \omega)$ is a foliation by Lagrangian submanifolds.
For instance (Weinstein, p. 9).
More generally
A polarization of $(X,\omega)$ is a choice of involutive Lagrangian subbundle $\mathcal{P} \hookrightarrow T_{\mathbb{C}} X$ of of the complexified tangent bundle of $X$.
For instance (Bates-Weinstein, def. 7.4)
A Poisson Lie algebroid $\mathfrak{P}$ is a symplectic Lie n-algebroid for $n = 1$. Regarding its Chevalley-Eilenberg algebra as the algebra of functions on a dg-manifold, that dg-manifold carries a graded symplectic form $\omega$. One can then say
A dg-Lagrangian submanifold of $(\mathfrak{P}, \omega)$ is also called a $\Lambda$-structure. (Ševera, section 4).
Hence we might say real polarization of $(\mathfrak{P}, \omega)$ is a foliation by dg-Lagrangian submanifolds.
For $(X, \pi)$ the Poisson manifold underlying a Poisson Lie algebroid $(\mathfrak{P}, \omega)$, a dg-Lagrangian submanifold of $(\mathfrak{P}, \omega)$ corresponds to a coisotropic submanifold of $(X, \pi)$.
The dg-Lagrangian submanifolds also correspond to branes in the Poisson sigma-model (see there) on $(\mathfrak{P}, \omega)$.
A Courant Lie algebroid $\mathfrak{C}$ is a symplectic Lie n-algebroid for $n = 2$. Regarding its Chevalley-Eilenberg algebra as the algebra of functions on a dg-manifold, that dg-manifold carries a graded symplectic form $\omega$. One can then say
A dg-Lagrangian submanifold of $(\mathfrak{C}, \omega)$ is also called a $\Lambda$-structure. (Ševera, section 4).
Hence we might say real polarization of $(\mathfrak{C}, \omega)$ is a foliation by dg-Lagrangian submanifolds.
The dg-Lagrangian submanifolds of a Courant Lie 2-algebroid $(\mathfrak{C}, \omega)$ correspond to Dirac structures on $(\mathfrak{C}, \omega)$.
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A simple notion of a real polarization for 2-plectic manifolds is considered within the context of higher geometric quantization in Rogers, Chap. 7.
(…)
If the symplectic manifold $(X,\omega)$ lifts to the structure of a Kähler manifold $(X, J, g)$, hence with Riemannian metric $g(-,-) = \omega(-,I(-))$, then the holomorphic/antiholomorphic decomposition induced by the complex manifold structure is a polarization of $(X,\omega)$. Polarizations of this form are therefore called Kähler polarizations.
Upon (geometric) quantization of the physical system described by the symplectic manifold $(X, \omega)$ a quantum state is supposed to be a function on $X$ – or rather a section of a prequantum line bundle which is a “wave-function that only depends on the canonical coordinates”, not on the canonical momenta. In terms of polarizations this is formalized by saying that a quantum state is a section which is covariant constant along the leaves of the polarization (along the “momentum direction”).
After a choice of prequantum line bundle $\nabla$ lifting $\omega$, a Bohr-Sommerfeld leaf of a (real) polarization is a leaf on which the prequantum line bundle is not just flat, but also trivializable as a circle bundle with connection.
If a polarization on an $2n$-dimensional symplectic manifold is generated from $n$ Hamiltonian vector fields whose Hamiltonians commute with each other under the Poisson bracket (and one of them is regarded as that generating time evolution of a mechanical system) then one speaks of a Liouville integrable system.
∞-Chern-Simons theory from binary and non-degenerate invariant polynomial
(adapted from Ševera 00)
Lecture notes include
and section 4 and 5 of
or section 5 of
or
Lagrangian submanifolds of L-infinity algebroids are considered in
In the case that the polarization integrates to the action of a Lie group $G$ one may think of passing to polarized sections as equivlent to passing to $G$-gauge equivalence classes. This point of view is highlighted in
A candidate for polarizations for higher geometric quantization in $n$-plectic geometry is discussed in Chapter 7 of
Last revised on June 25, 2022 at 08:10:15. See the history of this page for a list of all contributions to it.