Lagrangian Subspace Theorem

TheoremProperties of Lagrangian Subspaces

Let $(V, \omega)$ be a $2n$ -dimensional symplectic vector space and $L \subseteq V$ a Lagrangian subspace (i.e., $L = L^\omega$ ). Then:

Dimension: $\dim L = n = \frac{1}{2}\dim V$
Complementary Lagrangians: There exists another Lagrangian $L'$ such that $V = L \oplus L'$ and $\omega|_{L \times L'}: L \times L' \to \mathbb{R}$ is a perfect pairing
Existence: Lagrangian subspaces always exist in any symplectic vector space

Furthermore, the space of all Lagrangian subspaces forms a smooth manifold called the Lagrangian Grassmannian $\Lambda(n)$ , diffeomorphic to $\text{U}(n)/\text{O}(n)$ .

Lagrangian subspaces are maximal isotropic subspaces, meaning they are the largest subspaces on which the symplectic form vanishes identically. They play a fundamental role in classical mechanics (as configuration spaces) and quantum mechanics (as quantum states in geometric quantization).

DefinitionTransverse Lagrangians

Two Lagrangian subspaces $L_1, L_2$ are transverse if $L_1 \cap L_2 = \{0\}$ , equivalently if $L_1 + L_2 = V$ .

In this case, every vector $v \in V$ has a unique decomposition $v = \ell_1 + \ell_2$ with $\ell_i \in L_i$ , and the map:

$\omega(-, -): L_1 \times L_2 \to \mathbb{R}$

defines a non-degenerate pairing, establishing isomorphisms $L_1 \cong L_2^*$ and $L_2 \cong L_1^*$ .

ExamplePosition and Momentum Lagrangians

In $(\mathbb{R}^{2n}, \omega_0)$ with coordinates $(q, p)$ , consider:

$L_q = \{(q, 0) : q \in \mathbb{R}^n\}, \quad L_p = \{(0, p) : p \in \mathbb{R}^n\}$

Both are Lagrangian subspaces with $L_q \cap L_p = \{0\}$ and $L_q + L_p = \mathbb{R}^{2n}$ . The pairing is:

$\omega_0((q, 0), (0, p)) = \sum_{i=1}^n q_i p_i = \langle q, p \rangle$

which identifies $L_p$ with the dual space $L_q^*$ .

Remark

The condition $L = L^\omega$ is equivalent to:

$L$ is isotropic: $\omega|_L = 0$
$L$ is maximal isotropic: if $W \supseteq L$ is isotropic, then $W = L$
$\dim L = n$ and $\omega|_L = 0$

The third characterization is most commonly used in practice.

The Lagrangian Grassmannian $\Lambda(n)$ parametrizes all Lagrangian subspaces. It has dimension $\frac{n(n+1)}{2}$ and carries a natural stratification based on the intersection dimensions with a reference Lagrangian.

ExampleLagrangians from Generating Functions

Given a smooth function $S: \mathbb{R}^n \to \mathbb{R}$ , its graph in phase space:

$L_S = \{(q, p) : p = \nabla S(q)\} \subseteq T^*\mathbb{R}^n \cong \mathbb{R}^{2n}$

is a Lagrangian subspace if $S$ is quadratic. For general $S$ , $L_S$ is a Lagrangian submanifold (not necessarily a linear subspace). This construction relates to classical mechanics where $S$ is the action.

DefinitionMaslov Index

The Maslov index measures the topology of paths in the Lagrangian Grassmannian. For a path of Lagrangians $L_t$ relative to a reference Lagrangian $L_0$ , the Maslov index counts (with signs and multiplicity) how many times $L_t$ intersects the singular set $\{L : L \cap L_0 \neq \{0\}\}$ .

The fundamental group $\pi_1(\Lambda(n)) \cong \mathbb{Z}$ is generated by the Maslov cycle.

Remark

Any two transverse Lagrangians determine a symplectic basis: if $L_1 = \text{span}\{e_1, \ldots, e_n\}$ and $L_2 = \text{span}\{f_1, \ldots, f_n\}$ with $L_1 \cap L_2 = \{0\}$ , we can choose bases such that $\omega(e_i, f_j) = \delta_{ij}$ .