Symplectic form

In, a symplectic vector space is a V over a  F (for example the real numbers R) equipped with a symplectic.

A symplectic bilinear form is a ω : V × V → F that is
 * : in each argument separately,
 * : ω(v, v) = 0 holds for all v ∈ V, and
 * : ω(u, v) = 0 for all v ∈ V implies that u is zero.

If the underlying has  not 2, alternation is equivalent to. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a, but not vice versa. Working in a fixed, ω can be represented by a. The conditions above say that this matrix must be, , and. This is not the same thing as a, which represents a symplectic transformation of the space. If V is, then its dimension must necessarily be since every skew-symmetric, hollow matrix of odd size has  zero. Notice the condition that the matrix be hollow is not redundant if the characteristic of the field is 2. A symplectic form behaves quite differently from a, for example, the scalar product on Euclidean vector spaces.

Standard symplectic space
The standard symplectic space is R2n with the symplectic form given by a,. Typically ω is chosen to be the


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

where In is the n × n. In terms of basis vectors (x1, ..., xn, y1, ..., yn):


 * $$\omega(x_i, y_j) = -\omega(y_j, x_i) = \delta_{ij}\,$$
 * $$\omega(x_i, x_j) = \omega(y_i, y_j) = 0.\,$$

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

There is another way to interpret this standard symplectic form. Since the model space R2n 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. Now consider the W = V ⊕ V∗ of these spaces equipped with the following form:


 * $$\omega(x \oplus \eta, y \oplus \xi) = \xi(x) - \eta(y).$$

Now choose any (v1, ..., vn) of V and consider its


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

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


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

The form ω defined here can be shown to have the same properties as in the beginning of this section. On the other hand, 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, ..., xn) defines a dual basis for a complement, by ω(xi, yj) = δij.

Analogy with complex structures
Just as every symplectic structure is isomorphic to one of the form V ⊕ V∗, every on a vector space is isomorphic to one of the form V ⊕ V. Using these structures, the  of an n-manifold, considered as a 2n-manifold, has an, and the  of an n-manifold, considered as a 2n-manifold, has a symplectic structure: T∗(T∗M)p = Tp(M) ⊕ (Tp(M))∗.

The complex analog to a Lagrangian subspace is a, a subspace whose is the whole space: W = V ⊕ J V. As can be seen from the standard symplectic form above, every symplectic form on $$\mathbb{R}^{2n}$$ is isomorphic to the imaginary part of the standard complex (Hermitian) inner product on $$\mathbb{C}^{n}$$ (with the convention of the first argument being anti-linear).

Volume form
Let ω be an on an n-dimensional real vector space V, ω ∈ Λ2(V). Then ω is non-degenerate if and only if n is even and ωn/2 = ω ∧ ... ∧ ω is a. A volume form on a n-dimensional vector space V is a non-zero multiple of the n-form e1∗ ∧ ... ∧ en∗ where e1, e2, ..., en is a basis of V.

For the standard basis defined in the previous section, we have


 * $$\omega^n=(-1)^{n/2} x^*_1\wedge\ldots \wedge x^*_n \wedge y^*_1\wedge \ldots \wedge y^*_n.$$

By reordering, one can write


 * $$\omega^n= x^*_1\wedge y^*_1\wedge \ldots \wedge x^*_n \wedge y^*_n.$$

Authors variously define ωn or (−1)n/2ωn as the standard volume form. An occasional factor of n! may also appear, depending on whether the definition of the contains a factor of n! or not. The volume form defines an on the symplectic vector space (V, ω).

Symplectic map
Suppose that (V, ω) and (W, ρ) are symplectic vector spaces. Then a f : V → W is called a symplectic map if the  preserves the symplectic form, i.e. f∗ρ = ω, where the pullback form is defined by (f∗ρ)(u, v) = ρ(f(u), f(v)). Symplectic maps are volume- and orientation-preserving.

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 ω(f(u), f(v)) = ω(u, v), and so the f preserves the symplectic form. The set of all symplectic transformations forms a and in particular a, called the  and denoted by Sp(V) or sometimes Sp(V, ω). In matrix form symplectic transformations are given by.

Subspaces
Let W be a 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:


 * $$(W^\perp)^\perp = W$$
 * $$\dim W + \dim W^\perp = \dim V.$$

However, unlike s, W⊥ ∩ W need not be 0. We distinguish four cases:


 * W is symplectic if W⊥ ∩ W = {0}. This is true 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 ⊆ W⊥. This is true if and only if ω restricts to 0 on W. Any one-dimensional subspace is isotropic.
 * W is coisotropic if W⊥ ⊆ W. W is coisotropic if and only if ω descends to a nondegenerate form on the W/W⊥. Equivalently W is coisotropic if and only if W⊥ is isotropic. Any -one subspace is coisotropic.
 * W is Lagrangian if W = W⊥. A subspace is Lagrangian if and only if 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 {x1, y1} is symplectic
 * the subspace spanned by {x1, x2} is isotropic
 * the subspace spanned by {x1, x2, ..., xn, y1} is coisotropic
 * the subspace spanned by {x1, x2, ..., xn} is Lagrangian.

Heisenberg group
A can be defined for any symplectic vector space, and this is the typical way that s arise.

A vector space can be thought of as a commutative Lie group (under addition), or equivalently as a commutative, meaning with trivial Lie bracket. The Heisenberg group is a of such a commutative Lie group/algebra: the symplectic form defines the commutation, analogously to the s (CCR), and a Darboux basis corresponds to s – in physics terms, to s and s.

Indeed, by the, every representation satisfying the CCR (every representation of the Heisenberg group) is of this form, or more properly unitarily conjugate to the standard one.

Further, the of (the dual to) a vector space is the, and the group algebra of the Heisenberg group (of the dual) is the : one can think of the central extension as corresponding to quantization or.

Formally, the symmetric algebra of V is the group algebra of the dual, Sym(V) := K[V∗], and the Weyl algebra is the group algebra of the (dual) Heisenberg group W(V) = K[H(V∗)]. Since passing to group algebras is a, the central extension map H(V) → V becomes an inclusion Sym(V) → W(V).