Symplectic Vector Space - Volume Form

Volume Form

Let ω be a form on a 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∗ ∧ ... ∧ e_n∗ where e₁, e₂, ..., e_n 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).