Origin and Meaning

The term "symplectic" in mathematics originates from the Greek word συμπλεκτικός, which poetically translates to "plaiting together" or "intertwining". In the context of symplectic spaces, this term reflects the intricate fusion of algebra and geometry.

Symplectic spaces, in essence, interweave or "plait together" the characteristics of an underlying vector space with the geometric features invoked by the symplectic form, a higher-order structure imposing additional geometric constraints. This dynamic interplay between algebraic and geometric properties creates a more complex framework than that of a simple vector space.

This intertwining is evident in classical mechanics, where a symplectic space, serving as the phase space, seamlessly combines the positions and momenta of a system. Moreover, in quantum mechanics, the so-called "canonical commutation relations" which bind position and momentum operators, mirror this concept of "plaiting together" on a quantum level.

Definition

A symplectic space is a 2n-dimensional vector space $V$ over the real numbers, along with a non-degenerate skew-symmetric bilinear form $\omega$, referred to as the symplectic form.

The symplectic form is a 2-form, which is a map $\omega : V \times V \rightarrow \mathbb{R}$, satisfying two properties:

1. Skew-symmetry: For all $x, y$ in $V$, $\omega(x, y) = -\omega(y, x)$.
2. Non-degeneracy: For all $x$ in $V$, if $\omega(x, y) = 0$ for all $y$ in $V$, then $x = 0$.

Given any pair of vectors $(x, y) \in V$, the symplectic form $\omega(x, y)$ which yields a real number can also be extended to an operator by associating an operator $\Omega_x : V \rightarrow \mathbb{R}$ to each $x \in V$ defined by

$$\Omega_x(y) = \omega(x, y) \forall y \in V$$

The complex structure $J$ in the symplectic space is a linear transformation $J : V \rightarrow V$, such that $J^2 = -I$, where $I$ is the identity operator in $V$.

A Kähler manifold is a manifold equipped with a Riemannian metric $g$, a complex structure $J$, and a symplectic form $\omega$, satisfying a compatibility condition: for all vectors $x, y$ in the tangent space at any point, $g(x, y) = \omega(x, Jy)$.

The connection to holomorphic functions comes through the notion of harmonic conjugates. Given a real-valued function $u$ defined on some open subset of the complex plane, a harmonic conjugate $v$ of $u$ is a function such that $f = u + iv$ is holomorphic. This leads to the Cauchy-Riemann equations, the differential equations that real and imaginary parts of holomorphic functions satisfy.

Let's consider an example. Let $f(z) = z^2$, a holomorphic function in the complex plane $\mathbb{C}$. Its real and imaginary parts are given by $u(x, y) = x^2 - y^2$ and $v(x, y) = 2xy$ respectively, where $z = x + iy$. The complex structure in the plane is given by the matrix

$$ \begin{bmatrix} 0 & -1 \ 1 & 0 \ \end{bmatrix} $$

corresponding to multiplication by $i$.

Then, we can introduce a symplectic structure on $\mathbb{C}$ by defining

$$\omega((x_1, y_1), (x_2, y_2)) = x_1y_2 - x_2y_1$$

One can verify that this $\omega$ is skew-symmetric and non-degenerate, and that

$$g((x_1, y_1), (x_2, y_2)) = \omega((x_1, y_1), J(x_2, y_2))$$

where $g$ is the Euclidean metric, thus $\mathbb{C}$ with this symplectic form, the given complex structure, and the Euclidean metric is a Kähler manifold.

Here, the holomorphic function $f$, along with its real and imaginary parts $u$ and $v$, encode the geometric structure of the manifold and the interplay of the symplectic and complex structures. This example is, in essence, the simplest non-trivial case of a K"ahler manifold, which is a central object in complex geometry and theoretical physics.

Symplectomorphism and Canonical Transformations

In the context of classical mechanics, the symplectic space is commonly referred to as the phase space of a system. A symplectomorphism represents a transformation of this phase space that is both volume-preserving and preserves the symplectic structure of the phase space.

In other words, if $(V, \omega)$ and $(V', \omega')$ are symplectic spaces and $\phi: V \rightarrow V'$ is a bijection such that

$\phi^{\ast}\omega' = \omega$, then $\phi$ is a symplectomorphism. Here, $\phi^*$ is the pullback of $\phi$, which is the map induced by $\phi$ on the cotangent spaces of $V$ and $V'$.

In the language of classical mechanics, these transformations are referred to as canonical transformations because the adjective canonical means that something is the simplest and most significant form possible without the loss of its general applicability.

Canonical transformations are integral to Hamilton's formulation of classical mechanics and have deep connections with conservation laws via Noether's theorem. They provide the coordinate systems that describe changes in the choice of canonical coordinates (the perspective) from which dynamical systems are observed, without changing the underlying dynamics of the system. They play a crucial role in simplifying and solving problems in mechanics. Notably, canonical transformations in classical mechanics correspond to unitary transformations in quantum mechanics, thereby forming a bridge between these two paradigms of physics.