Symplectic Vector Space - Standard Symplectic Space

Standard Symplectic Space

The standard symplectic space is R2n with the symplectic form given by a nonsingular, skew-symmetric matrix. Typically ω is chosen to be the block matrix

where I_n is the n × n identity matrix. In terms of basis vectors (x₁, ..., x_n, y₁, ..., y_n):

A modified version of the Gram-Schmidt process shows that any finite-dimensional symplectic vector space has a basis such that ω takes this form, often called a Darboux basis.

There is another way to interpret this standard symplectic form. Since the model space Rn used above carries much canonical structure which might easily lead to misinterpretation, we will use "anonymous" vector spaces instead. Let V be a real vector space of dimension n and V∗ its dual space. Now consider the direct sum W := V ⊕ V∗ of these spaces equipped with the following form:

Now choose any basis (v₁, ..., v_n) of V and consider its dual basis

We can interpret the basis vectors as lying in W if we write x_i = (v_i, 0) and y_i = (0, v_i∗). Taken together, these form a complete basis of W,

The form ω defined here can be shown to have the same properties as in the beginning of this section; in other words, every symplectic structure is isomorphic to one of the form V ⊕ V∗. The subspace V is not unique, and a choice of subspace V is called a polarization. The subspaces that give such an isomorphism are called Lagrangian subspaces or simply Lagrangians.

Explicitly, given a Lagrangian subspace (as defined below), then a choice of basis (x₁, ..., x_n) defines a dual basis for a complement, by