A symplectic bilinear form is a mapping ω : V × V → F that is
If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric form, but not vice versa.
Working in a fixed basis, ω can be represented by a matrix. The conditions above are equivalent to this matrix being skew-symmetric, nonsingular, and hollow (all diagonal entries are zero). This should not be confused with a symplectic matrix, which represents a symplectic transformation of the space. If V is finite-dimensional, then its dimension must necessarily be even since every skew-symmetric, hollow matrix of odd size has determinant zero. Notice that 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 symmetric form, for example, the scalar product on Euclidean vector spaces.
where In is the n × n identity matrix. In terms of basis vectors (x1, ..., xn, y1, ..., yn):
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 dual space. Now consider the direct sum W = V ⊕ V∗ of these spaces equipped with the following form:
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,
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.
Just as every symplectic structure is isomorphic to one of the form V ⊕ V∗, every complex structure on a vector space is isomorphic to one of the form V ⊕ V. Using these structures, the tangent bundle of an n-manifold, considered as a 2n-manifold, has an almost complex structure, and the cotangent bundle 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 real subspace, a subspace whose complexification is the whole space: W = V ⊕ J V. As can be seen from the standard symplectic form above, every symplectic form on R2n is isomorphic to the imaginary part of the standard complex (Hermitian) inner product on Cn (with the convention of the first argument being anti-linear).
Let ω be an alternating bilinear form 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 volume form. 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
By reordering, one can write
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 alternating product contains a factor of n! or not. The volume form defines an orientation on the symplectic vector space (V, ω).
Suppose that (V, ω) and (W, ρ) are symplectic vector spaces. Then a linear map f : V → W is called a symplectic map if the pullback 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.
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 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.
Let W be a linear subspace of V. Define the symplectic complement of W to be the subspace
The symplectic complement satisfies:
However, unlike orthogonal complements, W⊥ ∩ W need not be 0. We distinguish four cases:
Referring to the canonical vector space R2n above,
A vector space can be thought of as a commutative Lie group (under addition), or equivalently as a commutative Lie algebra, meaning with trivial Lie bracket. The Heisenberg group is a central extension of such a commutative Lie group/algebra: the symplectic form defines the commutation, analogously to the canonical commutation relations (CCR), and a Darboux basis corresponds to canonical coordinates – in physics terms, to momentum operators and position operators.
Indeed, by the Stone–von Neumann theorem, 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 group algebra of (the dual to) a vector space is the symmetric algebra, and the group algebra of the Heisenberg group (of the dual) is the Weyl algebra: one can think of the central extension as corresponding to quantization or deformation.
Formally, the symmetric algebra of a vector space V over a field F is the group algebra of the dual, Sym(V) := F[V∗], and the Weyl algebra is the group algebra of the (dual) Heisenberg group W(V) = F[H(V∗)]. Since passing to group algebras is a contravariant functor, the central extension map H(V) → V becomes an inclusion Sym(V) → W(V).