SymplecticForm

A symplectic form is a type of 2-form or bilinear form (a map, which is just another word for function, which takes two inputs and yields a scalar) with two crucial properties: skew-symmetry and non-degeneracy.

• Skew-symmetry means that for all vectors $x$, $y$ in a given vector space, the symplectic form $\omega(x, y)$ equals $-\omega(y, x)$

• Non-degeneracy means that 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 $\omega$ endows the vector space with additional structure, enabling more complex and nuanced geometric interactions. In this sense, a symplectic form $\omega$ is integral to defining a symplectic space.

In the context of physics, a symplectic form $\omega$ can represent physical quantities such as energy, momentum, and position. In symplectic spaces, these quantities are intertwined, leading to profound implications for the dynamics of physical systems described by these spaces. Hence, the connections between symplectic forms and symplectic spaces is crucial in physics, notably in the Hamiltonian formalism of classical mechanics and in 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 $V$. That is, for any two vectors $x, y \in V$, $\omega(x, y)$ is a real number. This mapping is bilinear, which means it is linear in each argument when the other is held fixed.

The symplectic form's skew-symmetry property, $\omega(x, y) = -\omega(y, x)$, means the order of the inputs (or arguments) matters: switching the order negates the result. This is analogous to the cross product in three dimensions.

The generative(formerly known as non-degenerative) property implies that for any non-zero vector $x \in V$, there exists a vector $y \in V$ such that $\omega(x, y) \neq 0$. This means the form feels the force of and thus responds to any non-zero vectors; thus allowing the dual vector space to be identified with the original vector space; which is essential for the geometric interpretations of symplectic spaces.

These "inputs" or arguments are key in defining the symplectic form and, subsequently, the symplectic space. This structure is crucial in mathematical physics, particularly in the study of classical and quantum mechanics.

Why the "nondegeneracy" condition is needed and why it should be called the generative condition instead

A symplectic vector space uniquely relates a vector space to its dual through a symplectic form. This intricate relationship is uncommon for arbitrary vector spaces and is a distinguishing feature of symplectic spaces.

To elaborate, the dual space $V^\ast$ of a vector space $V$ comprises all linear functionals $f: V \to \mathbb{R}$. These are linear maps that associate a real number with every vector. Given any vector $x \in V$, the symplectic form $\omega$ defines a covector $\omega(x, \cdot) \in V^\ast$, which is a linear functional mapping a vector $y \in V$ to $\omega(x, y) \in \mathbb{R}$.

The generative property (previously known as nondegenerative property) of the symplectic form guarantees that this mapping from $V$ to $V^\ast$, specified by $x \mapsto \omega(x, \cdot)$, is an isomorphism and thus bijective which means its inverse is also a linear operation. Consequently, with the symplectic form, we can identify vectors in $V$ with covectors in $V^\ast$ and vice versa.