math lessons - Symplectic vector space

In mathematics, a symplectic vector space is a vector space V equipped with a nondegenerate, skew-symmetric, bilinear form ω called the symplectic form.

Explicitly, a symplectic form is a bilinear form ω : V × V → R which is

Skew-symmetric: ω(u, v) = −ω(v, u) for all u, v ∈ V,
Non-degenerate: if ω(u, v) = 0 for all v ∈ V then u = 0.

Working in a fixed basis ω can be represented by a matrix. The two conditions above say that this matrix must be skew-symmetric and nonsingular. (Note that this is not the same thing as a symplectic matrix.)

A nondegenerate skew-symmetric bilinear form behaves quite differently from a nondegenerate symmetric bilinear form, such as the dot product on euclidean vector spaces. With a euclidean form g, we have g(v,v) > 0 for all nonzero vectors v, whereas a symplectic form ω satisfies ω(v,v) = 0.

If V is finite-dimensional then its dimension must necessarily be even since every skew-symmetric matrix of odd size has determinant zero.

Contents

1 Standard symplectic space

2 Symplectic transformations

3 Subspaces

4 Related topics

Standard symplectic space

The standard symplectic space is R²ⁿ with the symplectic form given by the block matrix

$\omega = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix}$

where I_n is the n × n identity matrix. In terms of basis vectors

$(x_1, \ldots, x_n, y_1, \ldots, y_n)$ :

$\omega(x_i, y_j) = -\omega(y_j, x_i) = \delta_{ij}\,$

$\omega(x_i, x_j) = \omega(y_i, y_j) = 0\,$ .

A modified version of the Gram-Schmidt process shows that any finite-dimensional symplectic vector space has such a basis, 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 $W := V \oplus V^*$ of these spaces equipped with the following form:

ω((x,η),(y,ξ)) = ξ(x) - η(y)

Now choose any basis $(x_1, \ldots, x_n)$ of V and consider its dual basis

$(x^*_1, \ldots, x^*_n)$ .

We can interpret the basis vectors as lying in W if we write $(x i,0)$ and $(0, x^*_i)$ instead of $x i$ and $x^*_i$ . Combining these bases of V and V* (in that order) in this way, we obtain a basis of W. If we relabel that basis again to

$(x_1, \ldots, x_n, y_1, \ldots, y_n)$ ,

the form $ω$ has the same properties as in the beginning of this section.

Symplectic transformations

A linear symplectic transformation of V is a linear transformation A : V → V such that

ω(Au, Av) = ω(u, v).

That is, it is a linear transformation which preserves the symplectic form. The group of all symplectic transformations of V is called the symplectic group, denoted Sp(V). In matrix form symplectic transformations are given by symplectic matrices.

Subspaces

Let W be a linear subspace of V. Define the symplectic complement of W to be the subspace

$W^{\perp} = \{v\in V \mid \omega(v,w) = 0 \mbox{ for all } w\in W\}$

The symplectic complement satisfies

$(W^{\perp})^{\perp} = W$

and

$\dim W + \dim W^\perp = \dim V$

However, unlike orthogonal complements, $W^\perp \cap W$ need not be 0. We distinguish four cases.

W is symplectic if $W^\perp \cap W = \{0\}$ . This is true iff ω restricts to a nondegenerate form on W. A symplectic subspace with the restricted form is a symplectic vector space in its own right.
W is isotropic if $W \sube W^\perp$ . This is true iff ω restricts to 0 on W. Any one-dimensional subspace is isotropic.
W is coisotropic if $W^\perp\sube W$ . W is coisotropic if and only if ω descends to a nondegenerate form on the quotient space $W/W^\perp$ . Equivalently W is coisotropic iff $W^\perp$ is isotropic. Any codimension-one subspace is coisotropic.
W is Lagrangian if $W = W^\perp$ . A subspace is Lagragian iff it is both isotropic and coisotropic. In a finite-dimensional vector space, a Lagrangian subspace is an isotropic one whose dimension is half that of V. Every isotropic subspace can be extended to a Lagrangian one.