Why is the phase space a symplectic manifold rather than a manifold with a metric?

Question

Why does phase space require a symplectic geometry rather than a metric? Is there some scenario where a metric is unable to describe the notion of length in phase space, specifically in relation to the uncertainty principle?

Answer 1

1. A metric structure $g$ and
2. a symplectic structure $\omega$

are two very different structures, although sometimes they can co-exist in a compatible way.

Unlike a symplectic structure, there are no Jacobi-like identity and no Darboux-like theorem for a metric structure.

There exists a unique torsionfree metric connection $\nabla$ on a pseudo-Riemannian manifold $(M,g)$, known as the Levi-Civita connection, while there exist infinitely many torsionfree symplectic connections $\nabla$ on a symplectic manifold $(M,\omega)$.

Unlike a pseudo-Riemannian manifold $(M,g)$, there are no scalar curvature, or local invariants on a symplectic manifold $(M,\omega)$.

The group of local symplectomorphisms on a symplectic manifold $(M,\omega)$ is infinite dimensional, while the group of local Killing symmetries on a pseudo-Riemannian manifold $(M,g)$ is typically finite dimensional.

A metric structure on spacetime is important for general relativity (GR), but irrelevant for the Hamiltonian formalism.$^1$ If we artificially assign a metric on phase space, it would typically not be preserved under symplectomorphisms.

On the other hand, note that in the Hamiltonian formalism, the cotangent bundle of the configuration space is born with a canonical symplectic structure, cf. e.g. this Phys.SE post.

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$^1$ Note in particular that in the Hamiltonian formulation of GR, known as the ADM formulation, the pertinent phase space is different from spacetime itself, and also different from the configuration space of all possible spacetime metrics. The configuration space of all possible spacetime metrics is endowed with the DeWitt metric.

Answer 2

So, in classical mechanics, we know nothing of this strange "spacetime" and its metric. We know only that systems are described by $n$ continuous generalized coordinates $q^i$ with a certain range, and we take the manifold $\mathcal{M}$ consisting of all possible different $\vec q$ as our starting point. Note that there is no metric, no form, nothing on this.

We now include the generalized momenta $p_i$ at any point as elements of the cotangent space at that point, and henceforth call the cotangent bundle our phase space $\mathcal{P} := T^*\mathcal{M}$.

Now, on the cotangent bundle, there is the tautological one-form $\theta : \mathcal{P} \to T^*\mathcal{P}$. It is uniquely characterized to be the form that undoes the pullback $\beta^*$ by $\beta : \mathcal{M} \to T^*\mathcal{M}$ on the coordinate manifold by $ \beta^*\theta = \beta$. It is natural in the sense that it is characterized by an universal property and always exists uniquely. In coordinates, it is $\theta = p_i \mathrm{d}q^i$.

From this, the symplectic form simply arises as $\omega := \mathrm{d}\theta$. As $\theta$ was natural, so is $\omega$, and the proof that $\omega$ is symplectic is simply that it is locally given by $\omega = \mathrm{d}p_i \wedge \mathrm{d}q^i$, which is obviously a symplectic form on $\mathbb{R}^n \times \mathbb{R}^n$

So, we do not choose a symplectic form, there is one, and only one, natural candidate. Note that there is no such candidate for a metric.

If we are now given an "energy function" - the Hamiltonian $H$ - on $\mathcal{P}$, it turns out that demanding that $\theta(X_H)$ be the action reproduces Hamilton's equations, which, by their equivalence to Lagrangian/Newtonian mechanics, are the correct equations of motions. Furthermore, the symplectic form gives the correct way to implement the Poisson bracket (which is antisymmetric, and which a metric could never do) by

$$ \{f,g\} := \omega(X_f,X_g)$$

where $X_f$ is the vector field defined by a differentiable function $f : \mathcal{P} \to \mathbb{R}$ through $\mathrm{d}f \overset{!}{=} \omega(X_f,\dot{})$.

One can also show now that this is really a Poisson bracket and derive the time evolution equation on the phase space for observables, but I believe this is sufficient to be convincing that the symplectic structure is indeed the natural setting for Hamiltonian mechanics.

Lastly, and somewhat unrelated, we note that, on Riemannian manifolds, it is quite possible to consider the geodesics as a Hamiltonian flow, where the Hamiltonian is defined by the metric, but we still use the symplectic cotangent bundle to define the flow on.