nLab polarization

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.

Idea

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

$$\phi^* \omega = \sum_{i = 1}^n \mathbf{d} q^i \wedge \mathbf{d}p_i$$

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.

Definition

The traditional notion of polarization applies to a symplectic manifold.

• Of 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:

• Of a Poisson Lie algebroid

• Of a Courant Lie 2-algebroid

• Of a symplectic Lie n-algebroid

Of a symplectic manifold

Let $(X, \omega)$ be a symplectic manifold.

Definition

A real polarization of $(X, \omega)$ is a foliation by Lagrangian submanifolds.

For instance (Weinstein, p. 9).

More generally

Definition

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)

Of a Poisson Lie algebroid

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

Definition

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.

Proposition

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 $(X, \pi)$.

(Ševera, section 4)

Remark

The dg-Lagrangian submanifolds also correspond to branes in the Poisson sigma-model (see there) on $(\mathfrak{P}, \omega)$.

Of a Courant Lie 2-algebroid

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

Definition

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.

Proposition

The dg-Lagrangian submanifolds of a Courant Lie 2-algebroid $(\mathfrak{C}, \omega)$ correspond to Dirac structures on $(\mathfrak{C}, \omega)$.

(Ševera, section 4)

Of a symplectic Lie $n$-algebroid

??

Of an $n$-plectic smooth $\infty$-groupoid

A simple notion of a real polarization for 2-plectic manifolds is considered within the context of higher geometric quantization in Rogers, Chap. 7.

Examples

Real polarizations

(…)

Kähler polarizations

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.

Convex polarizations

• p-convex polarization

Related concepts

Quantum states and wave functions

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”).

Bohr-Sommerfeld leaves

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.

Liouville integrability

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.

Refinement to higher geometric quantization

∞-Chern-Simons theory from binary and non-degenerate invariant polynomial

$n \in \mathbb{N}$	symplectic Lie n-algebroid	Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $(n+1)$-d sigma-model	higher symplectic geometry	$(n+1)$d sigma-model	dg-Lagrangian submanifold/ real polarization leaf	= brane	(n+1)-module of quantum states in codimension $(n+1)$	discussed in:
0	symplectic manifold	symplectic manifold	symplectic geometry		Lagrangian submanifold	–	ordinary space of states (in geometric quantization)	geometric quantization
1	Poisson Lie algebroid	symplectic groupoid	2-plectic geometry	Poisson sigma-model	coisotropic submanifold (of underlying Poisson manifold)	brane of Poisson sigma-model	2-module = category of modules over strict deformation quantiized algebra of observables	extended geometric quantization of 2d Chern-Simons theory
2	Courant Lie 2-algebroid	symplectic 2-groupoid	3-plectic geometry	Courant sigma-model	Dirac structure	D-brane in type II geometry
$n$	symplectic Lie n-algebroid	symplectic n-groupoid	(n+1)-plectic geometry	$d = n+1$ AKSZ sigma-model

(adapted from Ševera 00)

Refinement to higher quantization by push-forward

chromatic level	generalized cohomology theory / E-∞ ring	obstruction to orientation in generalized cohomology	generalized orientation/polarization	quantization	incarnation as quantum anomaly in higher gauge theory
1	complex K-theory $KU$	third integral SW class $W_3$	spinᶜ structure	K-theoretic geometric quantization	Freed-Witten anomaly
2	EO(n)	Stiefel-Whitney class $w_4$
2	integral Morava K-theory $\tilde K(2)$	seventh integral SW class $W_7$			Diaconescu-Moore-Witten anomaly in Kriz-Sati interpretation

Other

• canonical transformation

• polarized algebraic variety

References

Lecture notes include

• Alan Weinstein, Lectures on Symplectic manifolds Lecture 2 Lagrangian splittings, real and complex polarizations, Kähler manifolds, CBMS Regional Conference Series in Mathematics, AMS (1977)

• Sean Bates, Alan Weinstein, Lectures on the geometry of quantization, pdf

• Kristin Shaw, An introduction to polarizations (pdf)

and section 4 and 5 of

• Matthias Blau, Symplectic Geometry and Geometric Quantization (pdf)

or section 5 of

• A. Echeverria-Enriquez, M.C. Munoz-Lecanda, N. Roman-Roy, C. Victoria-Monge, Mathematical Foundations of Geometric Quantization Extracta Math. 13 (1998) 135-238 (arXiv:math-ph/9904008)

or

• Yuichi Nohara, Independence of Polarization in Geometric Quantization (pdf)

Lagrangian submanifolds of L-infinity algebroids are considered in

• Pavol Ševera, Some title containing the words “homotopy” and “symplectic”, e.g. this one (arXiv:0105080)

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

• Gabriel Catren, On the Relation Between Gauge and Phase Symmetries, Foundations of Physics 44 (12):1317-1335 (2014) (web)

A candidate for polarizations for higher geometric quantization in $n$-plectic geometry is discussed in Chapter 7 of

• Chris Rogers, Higher symplectic geometry PhD thesis (arXiv:1106.4068).