Classical configuration, cotangent phase and reference measures

Assume the Axiom of Countable Choice, the exact choice strength required by Tautological one-form on a cotangent bundle to construct the smooth cotangent manifold. Let $d,N\ge 1$, let $Q$ be a smooth $d$-manifold, and choose a countable smooth-chart subcover of its atlas. Such a subcover is obtained from a countable basis of the second-countable topology by choosing, for each basis element contained in a chart domain, one such chart domain; this countable selection is covered by the stated Axiom of Countable Choice. Give $Q_N:=Q^N$ the product topology and the product smooth structure generated by the charts $(U_1\times \cdots \times U_N,(q_1,\dots ,q_N)\mapsto ({\varphi }_1(q_1),\dots ,{\varphi }_N(q_N))),$ where each $(U_i,{\varphi }_i)$ belongs to the chosen countable subcover of charts of $Q$. These images are open in ${\mathbb{R}}^{dN}$, and their transition maps are componentwise smooth, so they form a smooth atlas of dimension $n=dN$; the finite product of the countable chart family is countable. Its generated maximal atlas is the product smooth structure in the convention of Smooth manifolds and their smooth charts. The induced cotangent charts are indexed by these base charts and form a countable atlas of ${\mathrm{\Gamma }}_N$. Its configuration space is $Q_N$. Its phase space is the cotangent bundle ${\mathrm{\Gamma }}_N:=T^{\ast }{Q}_N,$ whose points are pairs $(q,p)$ with $p\in T_q^{\ast }{Q}_N$. Let ${\pi }_N:T^{\ast }{Q}_N\to Q_N$ be the cotangent-bundle projection. The tautological one-form and canonical symplectic form are ${\lambda }_{(q,p)}(\xi ):=p(d{\pi }_N\xi ),{\omega }_N:=-d\lambda \mathrm{.}$ Thus in an induced cotangent chart $(q^1,\dots ,q^n,p_1,\dots ,p_n)$, $\lambda ={\sum }_{j=1}^n{p}_{j}d{q}^j,{\omega }_N={\sum }_{j=1}^{n}d{q}^j\wedge d{p}_j\mathrm{.}$ The cotangent supplier proves ${\omega }_N$ closed and nondegenerate under the same Countable Choice premise. On an open phase chart this is exactly the standard form of A phase domain, its canonical one-form and its standard symplectic two-form; The standard two-form of a phase domain is closed and nondegenerate, with contraction ${\iota }_v\omega ={\sum }_i{a}_{i}d{p}_i-{\sum }_i{b}_{i}d{q}^i$ gives its closedness, contraction and nondegeneracy coordinate calculations.

Let ${\mathcal{F}}_N:=\mathcal{B}({\mathrm{\Gamma }}_N)$ be the Borel sigma-algebra generated by the topology of ${\mathrm{\Gamma }}_N$ (The Borel sigma-algebra of a topological space). The Liouville density, denoted $d{\mathrm{\Lambda }}_N:=\mathrm{\mid }{\omega }_N^{\wedge n}\mathrm{\mid }\mathrm{/}n!,$ is the positive density whose coordinate density in every induced cotangent chart is $d{q}^1\cdots d{q}^{n}d{p}_1\cdots d{p}_n$. Define its Borel measure ${\mathrm{\Lambda }}_N$ explicitly as follows. If ${\mathrm{\Gamma }}_N=\mathrm{\varnothing }$, set ${\mathrm{\Lambda }}_N(\mathrm{\varnothing })=0$. Otherwise choose an at-most-countable induced cotangent atlas $(V_k,{\psi }_k{)}_{k\in J}$ indexed by $J\subseteq \mathbb{N}$, order $J$ by the usual order, and set $E_k:=V_k\setminus {\bigcup }_{j\in J,j<k}{V}_j(k\in J)\mathrm{.}$ For $A\in {\mathcal{F}}_N$, put ${\mathrm{\Lambda }}_N(A):={\sum }_{k\in J}{\lambda }_{2n}({\psi }_k(A\cap E_k)),$ where ${\lambda }_{2n}$ is Lebesgue measure on the chart image. Each $E_k$ is Borel, and each chart map is a homeomorphism, so the terms are defined on Borel sets. The coordinate transitions preserve ${\omega }_N$ because ${\omega }_N=-d\lambda$ is intrinsic; hence they preserve ${\omega }_N^n\mathrm{/}n!$ and have absolute Jacobian one. For two atlases, the intersections of their disjoint chart pieces form a countable common Borel refinement. On each nonempty refinement piece the coordinate transition is a $C^1$ diffeomorphism with absolute Jacobian one, so the change-of-variables theorem applied to its indicator A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions gives the same measure from either atlas. Summing over the refinement proves atlas independence. Countable additivity follows because the chart images of disjoint Borel sets remain disjoint within each $E_k$ and Lebesgue measure is countably additive. For the resulting nonnegative array $a_{k\mathrm{\ell }}$, both ${\sum }_k{\sum }_{\mathrm{\ell }}{a}_{k\mathrm{\ell }}$ and ${\sum }_{\mathrm{\ell }}{\sum }_k{a}_{k\mathrm{\ell }}$ are the supremum of the same finite subsums of $\{a_{k\mathrm{\ell }}\}$, using the nonnegative extended-sum convention of Measures on sigma-algebras. Hence the sums reorder and ${\mathrm{\Lambda }}_N$ is a measure with the displayed coordinate density. Given any finite constant $c_N>0$, define the labelled reference measure $m_N:=c_N{\mathrm{\Lambda }}_N$ on $({\mathrm{\Gamma }}_N,{\mathcal{F}}_N)$. It is sigma-finite: $Q_N$ is countable-atlased as just constructed, so it is covered by countably many cotangent chart images. Each chart image is open in ${\mathbb{R}}^{2n}$ and is covered by the countable family of bounded open boxes with rational endpoints whose closures lie in that chart image: every point of an open set lies in one such box. Each box has finite Euclidean Lebesgue measure, and its chart preimage therefore has finite $m_N$-measure because the Liouville coordinate density is the constant $c_N$. The countably many chart-box preimages cover ${\mathrm{\Gamma }}_N$ and establish sigma-finiteness directly, without a sigma-compactness supplier. The constant $c_N$ is mathematical normalization data; this definition assigns no physical unit to it.

For a partition of the particle labels into species classes of sizes $N_1,\dots ,N_s$, let $G:=S_{N_1}\times \cdots \times S_{N_s}$ act on $Q_N=Q^N$ by permuting factors within each class and on $T^{\ast }{Q}_N$ by the cotangent lift. In canonical coordinates each lift permutes corresponding $q$ and $p$ coordinates, so it preserves ${\omega }_N$, ${\mathrm{\Lambda }}_N$ and $m_N$. Let ${\mathrm{\Pi }}_N:{\mathrm{\Gamma }}_N\to {\mathrm{\Gamma }}_N\mathrm{/}G$ be the orbit map, equip the orbit set with the quotient sigma-algebra ${\mathcal{F}}_{\mathrm{q}\mathrm{u}\mathrm{o}\mathrm{t}}:=\{B\subseteq {\mathrm{\Gamma }}_N\mathrm{/}G:{\mathrm{\Pi }}_N^{-1}(B)\text{ is Borel in }{\mathrm{\Gamma }}_N\},$ and define the quotient reference measure by ${\bar{m}}_N(B):=\frac{1}{{\prod }_{a=1}^s{N}_a!}{m}_N({\mathrm{\Pi }}_N^{-1}(B))\mathrm{.}$ This is a measure because inverse images preserve complements and countable disjoint unions and $m_N$ is countably additive. The stated invariance makes it the normalized quotient pushforward. It is a measurable quotient; no smooth quotient structure at configurations with stabilizers is asserted. For one species the factor is $1\mathrm{/}N!$.

The quotient factor here is mathematical counting data. A physical postulate determines when labels are operationally indistinguishable and how the reference scale is interpreted; neither follows from the cotangent geometry.