Symplectic Geometry, Revisited

We are now in a position to re-introduce symplectic manifolds, only this time more formally, and equipped with some more machinery.

A symplectic manifold is an even-dimensional manifold $M$ (we’ll see why shortly!) equipped with an $\omega\in\Omega^{2}(M)$ called the symplectic $ 2 $-form that satisfies two conditions. The first is that it is closed: \begin{equation} \mathrm{d}\omega=0 \ . \label{eq:symplectic-closed} \end{equation} The second is that it is non-degenerate, meaning if a vector $X$ satisfies \begin{equation} \omega(X,Y)=0 \quad \text{for every} \quad Y\in TM \ , \end{equation} then it must be that $X=0$. Equivalently, the map \begin{equation} TM\longrightarrow T^{\star}M \quad \text{that sends} \quad X\longmapsto\iota _ {X}\omega \ , \label{eq:symplectic-identification} \end{equation} is invertible. Here $\iota _ {X}\omega$ denotes the one-form obtained by inserting $X$ into the first slot of $\omega$; we will define this “contraction” operation more generally in the following section.

It is useful to rephrase the above conditions in terms of coordinates. The symplectic $ 2 $-form in some choice of local coordinates $ y ^ {\mathsf{a} } $ is: \begin{equation} \omega=\frac{1}{2}\omega _ {\mathsf{ab}}(y)\, \mathrm{d}y^{\mathsf{a}}\wedge\mathrm{d}y^{\mathsf{b}} \ . \label{eq:symplectic-form-components} \end{equation} Since the wedge product is antisymmetric, we must have $\omega _ {\mathsf{ab}}=-\omega _ {\mathsf{ba}}$. The non-degeneracy of the symplectic form implies that the matrix $\omega _ {\mathsf{ab}}$ is invertible. Finally, closure gives \begin{equation} \partial _ {\mathsf{a}}\omega _ {\mathsf{bc}} +\partial _ {\mathsf{b}}\omega _ {\mathsf{ca}} +\partial _ {\mathsf{c}}\omega _ {\mathsf{ab}}=0 \ . \end{equation} In the above equation, we have used the definition of the exterior derivative, and the aforementioned antisymmetry of $ \omega _ {\mathsf{ab} } $. An invertible antisymmetric matrix can only exist in even dimensions so every symplectic manifold has dimension $2N$.

The inverse of $\omega _ {\mathsf{ab}}$ will be denoted $\omega^{\mathsf{ab}}$: \begin{equation} \omega^{\mathsf{ab}}\omega _ {\mathsf{bc}} =\delta^{\mathsf{a}} _ {\mathsf{c}} \ . \end{equation} It defines the Poisson bracket of two functions $f,g\in\Omega^{0}(M)$, \begin{equation} \left\lbrace f,g\right\rbrace =\omega^{\mathsf{ab}}\partial _ {\mathsf{a}}f\, \partial _ {\mathsf{b}}g \ . \label{eq:poisson-symplectic} \end{equation} The antisymmetry of this bracket follows from the antisymmetry of $\omega^{\mathsf{ab}}$, while its Jacobi identity is equivalent to $\mathrm{d}\omega=0$.

These are precisely the properties encountered when we were motivating the introduction of symplectic geometry, so although it is mildly repetitive, I hope it is clear how symplectic geometry is the “natural” home for classical dynamics. The fact that the corresponding home for quantum mechanics is a Hilbert space, and the question of how one goes over into the other, is a question that will occupy us for some time during the latter part of these notes.

As we have mentioned before, a theorem due to Darboux states that in a sufficiently small neighbourhood of every point, coordinates $(q _ {a},p^{a})$ can be chosen such that \begin{equation} \omega=\mathrm{d}p^{a}\wedge\mathrm{d}q _ {a} \ . \label{eq:symplectic-darboux} \end{equation} In these coordinates, $\omega$ clearly has constant components — as we have seen before — and its inverse reproduces \begin{equation} \left\lbrace q _ {a},p^{b}\right\rbrace =\delta _ {a}{}^{b} \ . \end{equation} The conclusion we draw from this is rather striking: symplectic manifolds, in a small neighbourhood around every point, always look like ordinary phase space. The interesting distinction between symplectic manifolds (of the same dimension, naturally) is in their \emph{global} structure.

In the Darboux coordinates (meaning locally) it is useful to introduce what is called the canonical one-form \begin{equation} \theta=p^{a}\,\mathrm{d}q _ {a} \ , \label{eq:canonical-one-form} \end{equation} for which $\omega=\mathrm{d}\theta$. A symplectic form is always locally of this form, but a globally defined $\theta$ need not exist. Thus, every exact symplectic form is closed, but a closed symplectic form need not be globally exact. We saw a version of this statement earlier, for forms in general, and here it is specialised to the case of symplectic manifolds.

A diffeomorphism $\Phi:M\rightarrow M$ is called a canonical transformation (or, in mathematics, a symplectomorphism) if it preserves the symplectic form: \begin{equation} \Phi^{\star}\omega=\omega \ . \label{eq:symplectomorphism} \end{equation} In Darboux coordinates, this is nothing but the coordinate-independent definition of a canonical transformation. Indeed, since the Poisson brackets are defined using the inverse of the symplectic form, \eqref{eq:symplectomorphism} guarantees that Poisson brackets are preserved under canonical transformations: \begin{equation} \Phi^{\star}\left\lbrace f,g\right\rbrace =\left\lbrace \Phi^{\star}f,\Phi^{\star}g\right\rbrace \ . \end{equation} We see in geometric language what we saw earlier: that what we call position and momenta depends on a choice of local coordinates, but the physics of a classical dynamical system is invariant under canonical transformations that exchange or otherwise mix up such labels.

This invariant definition also recovers the generating functions of canonical transformations introduced earlier. In a Darboux patch where $\omega=\mathrm{d}\theta$, a symplectomorphism satisfies \begin{equation} \mathrm{d}\left(\Phi^{\star}\theta-\theta\right) =\Phi^{\star}\omega-\omega=0 \ . \end{equation} The difference $\Phi^{\star}\vartheta-\vartheta$ is therefore locally exact, and hence \begin{equation} \Phi^{\star}\theta=\theta+\mathrm{d}G \end{equation} for some function $G$, which is precisely the generator of canonical transformations. Thus, the statement that the canonical one-form changes by a total derivative is the local counterpart of the invariant condition $\Phi^{\star}\omega=\omega$.

The Symplectic Volume Form

On a $2N$-dimensional symplectic manifold, the highest non-vanishing wedge power of $\omega$ is \begin{equation} \operatorname{vol} _ {\omega} =\frac{1}{N!}\underbrace{\omega\wedge\cdots\wedge\omega}_{N\ \text{factors}} =\frac{1}{N!}\omega^{N} \ . \label{eq:symplectic-volume} \end{equation} Now, the non-degeneracy of $\omega$ guarantees that $\operatorname{vol} _ {\omega}$ is nowhere vanishing. It is therefore a volume form and, in particular, supplies every symplectic manifold with a natural orientation. In Darboux coordinates, \begin{equation} \operatorname{vol} _ {\omega} =\mathrm{d}p^{1}\wedge\mathrm{d}q _ {1}\wedge\cdots\wedge \mathrm{d}p^{N}\wedge\mathrm{d}q _ {N} \ , \end{equation} This is the invariant form of the measure on phase space.

Canonical transformations preserve this volume form, since they preserve the symplectic form: \begin{equation} \Phi^{\star}\operatorname{vol} _ {\omega} =\operatorname{vol} _ {\omega} \ . \label{eq:canonical-volume-invariance} \end{equation} This is the geometric version of Liouville’s theorem, which states that infinitesimal volumes in phase space (measured by the above volume form) are invariant under canonical transformations in general, and since time evolution is a canonical transformation generated by the Hamiltonian, under time evolution in particular.