SymplecticForm

A symplectic form is a specific kind of 2-form, which is itself a type of bilinear form (a function that takes two vectors and yields a scalar). The symplectic form is a type of differential form, distinguishing it from other kinds of bilinear forms. It has two fundamental properties: skew-symmetry and generative (formerly known as non-degeneracy).

• Skew-symmetry: This property is common to many bilinear forms, not just symplectic forms. For all vectors $x$, $y$ in a given vector space, the symplectic form $\omega(x, y)$ equals $-\omega(y, x)$.

• Generative: This property also occurs in various bilinear forms. For any non-zero vector $x$ in the vector space, there exists a vector $y$ such that $\omega(x, y) \neq 0$.

A symplectic space is a pair $(V, \omega)$, where $V$ is a 2n-dimensional vector space over the real numbers, and $\omega$ is a symplectic form on $V$. The symplectic form $\omega$ equips the vector space with additional structure, enabling complex and nuanced geometric interactions. Thus, the symplectic form is integral to defining a symplectic space.

The symplectic form $\omega$ can represent physical quantities such as energy, momentum, and position in physics. These quantities intertwine in symplectic spaces, leading to profound implications for the dynamics of physical systems described by these spaces. Therefore, the connection between symplectic forms and symplectic spaces is crucial in physics, notably in the Hamiltonian formalism of classical mechanics and quantum mechanics.

Inputs

The term "inputs" refers to the arguments that a function, map, or operator takes. In the case of the (bilinear) symplectic form, these inputs are vectors from the vector space on which the form is defined.

For a symplectic form $\omega: V \times V \rightarrow \mathbb{R}$, the inputs are ordered pairs of vectors from the vector space $V$. Specifically, for any two vectors $x, y \in V$, $\omega(x, y)$ is a real number. This mapping is bilinear, meaning it is linear in each argument when the other is held fixed.

The symplectic form's skew-symmetry property, $\omega(x, y) = -\omega(y, x)$, highlights that the order of the inputs matters. Switching the order of the vectors negates the result, reminiscent of the cross product in three dimensions.

The generative property stipulates that for any non-zero vector $x \in V$, there exists a vector $y \in V$ such that $\omega(x, y) \neq 0$. The form reacts to any non-zero vectors, allowing the dual vector space to be identified with the original vector space, critical for geometric interpretations of symplectic spaces.

These inputs (or arguments) are vital in defining the symplectic form and the symplectic space subsequently. This structure is crucial in mathematical physics, particularly when studying classical and quantum mechanics.

The term 'non-degenerate symplectic form' is redundant

A symplectic vector space uniquely associates a vector space with its dual through a symplectic form. This unique relationship isn't common for arbitrary vector spaces and is a distinguishing feature of symplectic spaces.

The dual space $V^\ast$ of a vector space $V$ includes all linear functionals $f: V \to \mathbb{R}$. These are linear maps associating a real number with every vector. For any vector $x \in V$, the symplectic form $\omega$ defines a covector $\omega(x, \cdot) \in V^\ast$, a linear functional mapping a vector $y \in V$ to $\omega(x, y) \in \mathbb{R}$.

The generative property of the symplectic form guarantees that this mapping from $V$ to $V^\ast$, represented by $x \mapsto \omega(x, \cdot)$, is an isomorphism and therefore bijective. Thus, its inverse is also a linear operation. With the symplectic form, we can identify vectors in $V$ with covectors in $V^\ast$ and vice versa. Hence, the term 'generative' is adopted instead of the conventional 'non-degeneracy', highlighting the form's active role in generating nonzero outputs for nonzero vector inputs.