Cotangent phase data, observables and action dimensions

Assume countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$: every sequence of nonempty sets has a choice function (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $Q$ be a finite-dimensional, Hausdorff, second-countable smooth manifold without boundary, of dimension $n$. Its cotangent bundle is

$T^{\ast }Q={\coprod }_{q\in Q}{T}_q^{\ast }Q,\pi :T^{\ast }Q\to Q,\pi (q,p)=q,$

with the canonical smooth structure and induced cotangent charts supplied by the manifold construction in Tautological one-form on a cotangent bundle. The tautological one-form is ${\theta }_{(q,p)}(\xi )=p(d{\pi }_{(q,p)}\xi )$, and the canonical cotangent two-form is $\omega =-d\theta$. The exact theorem The canonical cotangent two-form is symplectic makes $(T^{\ast }Q,\omega )$ a symplectic manifold in the sense of Symplectic form and symplectic manifold. We call this pair the cotangent phase space of $Q$.

In an induced chart $(q^1,\dots ,q^n,p_1,\dots ,p_n)$, $\theta =p_{i}d{q}^i$ and $\omega =d{q}^i\wedge d{p}_i$. A mathematical observable on this phase space means a smooth function $F:T^{\ast }Q\to \mathbb{R}$; this definition does not assert that a measurement has been performed or that every such function is physically measurable. For $G\in C^{\mathrm{\infty }}(T^{\ast }Q)$, let $X_G$ be the unique vector field satisfying ${\iota }_{X_G}\omega =dG$, which exists because $\omega$ is nondegenerate, and define the canonical bracket by $\{F,G\}:=dF(X_G)$. In the induced chart, contraction gives $X_G=G_{p_i}{\mathrm{\partial }}_{q^i}-G_{q^i}{\mathrm{\partial }}_{p_i}$, so

$\{F,G\}=F_{q^i}{G}_{p_i}-F_{p_i}{G}_{q^i}\mathrm{.}$

For an interpreted particle model, assign coordinates $q^i$ their declared units and assign the conjugate covector components $p_i$ units of action divided by the units of $q^i$. Thus $\theta$ and $\omega$ have action units, conventionally joule-seconds, and the defined canonical bracket of quantities with units $[F]$ and $[G]$ has units $[F][G]\mathrm{/}(\text{action})$. These are dimensional interpretations of the model, not additional mathematical premises. The use of ${\mathrm{A}\mathrm{C}}_{\omega }$ is exactly the use in the smooth cotangent-bundle suppliers; no full Axiom of Choice is used.