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,
• Nondegenerate: if ω(u, v) = 0 for all v ∈ V then u = 0.

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

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, which is a different concept discussed below).

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 inner product g, we have g(v,v) > 0 for all nonzero vectors v, whereas a symplectic form ω satisfies ω(v,v) = 0.

Contents

1 Standard symplectic space 2 Volume form 3 Symplectic map 4 Symplectic group 5 Subspaces 6 See also 7 References

Standard symplectic space

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

<math>\omega = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix}<math>

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

<math>(x_1, \ldots, x_n, y_1, \ldots, y_n)<math>:

<math>\omega(x_i, y_j) = -\omega(y_j, x_i) = \delta_{ij}\,<math>

<math>\omega(x_i, x_j) = \omega(y_i, y_j) = 0\,<math>.

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 <math>W := V \oplus V^*<math> of these spaces equipped with the following form:

<math>\omega((x,\eta),(y,\xi)) = \xi(x) - \eta(y)<math>

Now choose any basis <math>(v_1, \ldots, v_n)<math> of V and consider its dual basis

<math>(v^*_1, \ldots, v^*_n)<math>.

We can interpret the basis vectors as lying in W if we write <math>x_i=(v_i, 0)<math> and <math>y_i=(0, v^*_i)<math>. Taken together, these form a complete basis of W,

<math>(x_1, \ldots, x_n, y_1, \ldots, y_n)<math>.

The form <math>\omega<math> defined here can be shown to have the same properties as in the beginning of this section.

Volume form

Let ω be any alternating form on a finite-dimensional real vector space, that is, <math>\omega \in \mbox{Alt}^2(V)<math> for a n-dimensional real vector space V. Then ω is non-degenerate if and only if n is even, and <math>\omega^n=\omega\wedge\ldots\wedge\omega<math> is a volume form. A volume form on a n-dimensional vector space V is a multiple of the (unique) n-form <math>e^*_1\wedge\ldots\wedge e^*_n<math> where the <math>e_i<math> are standard basis vectors on V.

Note that for the standard basis defined in the previous section, we have

<math>\omega^n=(-1)^{n/2} x^*_1\wedge\ldots \wedge x^*_n

\wedge y^*_1\wedge \ldots \wedge y^*_n<math>.

Note that by reordering, one can write

<math>\omega^n= x^*_1\wedge y^*_1\wedge \ldots \wedge x^*_n

\wedge y^*_n<math>.

Authors variously define <math>\omega^n<math> or <math>(-1)^{n/2}\omega^n<math> as the standard volume form. Note that an occasional factor of n! may also appear, depending on whether the definition of the alternating product contains a factor of n! or not. Note that the volume form defines an orientation on the symplectic vector space <math>(V,\omega)<math>.

Symplectic map

Suppose that <math>(V,\omega)<math> and <math>(W,\rho)<math> are symplectic vector spaces. Then a linear map <math>f:V\rightarrow W<math> is called a symplectic map if and only if the pullback <math>f^*<math> preserves the symplectic form, that is, if <math>f^*\rho=\omega<math>. The pullback form is defined by

<math>f^*\rho(u,v)=\rho(f(u),f(v))<math>

and thus f is a symplectic map iff

<math>\rho(f(u),f(v))=\omega(u,v)<math>

for all u and v in V. In particular, symplectic maps are volume-preserving, orientation-preserving, and are isomorphisms.

Symplectic group

If V = W, then a symplectic map is called a linear symplectic transformation of V. In particular, in this case one has that

<math>\omega(f(u),f(v)) = \omega(u,v)<math>,

and so the linear transformation f preserves the symplectic form. The set of all symplectic transformations forms a group and in particular a Lie group, called the symplectic group and denoted by Sp(V) or sometimes Sp(V,ω). 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

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

The symplectic complement satisfies

<math>(W^{\perp})^{\perp} = W<math>

and

<math>\dim W + \dim W^\perp = \dim V<math>

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

• W is symplectic if <math>W^\perp \cap W = \{0\}<math>. 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 <math>W \sube W^\perp<math>. This is true iff ω restricts to 0 on W. Any one-dimensional subspace is isotropic.
• W is coisotropic if <math>W^\perp\sube W<math>. W is coisotropic if and only if ω descends to a nondegenerate form on the quotient space <math>W/W^\perp<math>. Equivalently W is coisotropic iff <math>W^\perp<math> is isotropic. Any codimension-one subspace is coisotropic.
• W is Lagrangian if <math>W = W^\perp<math>. 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.

Referring to the canonical vector space R²ⁿ above,

• the subspace spanned by <math>\{x_1,y_1\}<math> is symplectic
• the subspace spanned by <math>\{x_1,x_2\}<math> is isotropic
• the subspace spanned by <math>\{x_1,x_2,\ldots,x_n,y_1\}<math> is coisotropic
• the subspace spanned by <math>\{x_1,x_2,\ldots,x_n\}<math> is Lagrangian.

See also

• A symplectic manifold is a smooth manifold with a smoothly-varying closed symplectic form on each tangent space.
• A symplectic representation is a group representation where each group element acts as a symplectic transformation.

References

• Ralph Abraham and Jarrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See chapter 3.