Classical particle counting and physical observables

For a declared classical model with $N_a$ operationally indistinguishable particles of each species $a$, label those particles by $\{(a,i):1\le i\le N_a\}$ on the auxiliary labelled cover, and let $S_{N_a}$ denote the permutation group of that species' finite label set. The product group $G={\prod }_a{S}_{N_a}$ acts on the labelled phase space ${\mathrm{\Gamma }}_N=T^{\ast }{Q}_N$ by the cotangent lifts of within-species permutations. Let ${\mathrm{\Pi }}_N:{\mathrm{\Gamma }}_N\to {\mathrm{\Gamma }}_N\mathrm{/}G$ be its measurable orbit map from Classical configuration, cotangent phase and reference measures. Choose $a_0>0$ as a fixed primitive reference parameter with action units J s; Dimensions and SI unit conventions of the non-relativistic classical mechanics framework supplies its dimension, not its value or the choice of this parameter. Here $d$ is the number of spatial dimensions and $N={\sum }_a{N}_a$. No numerical value or identification of $a_0$ with Planck's constant is imposed.

Use the weighted labelled lift $m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:=\frac{1}{{\prod }_a{N}_a!a_0^{dN}}d{\mathrm{\Lambda }}_N=\frac{1}{{\prod }_a{N}_a!a_0^{dN}}dqdp$ on ${\mathrm{\Gamma }}_N$, where $d{\mathrm{\Lambda }}_N$ is the Liouville density and the last expression is its coordinate density in canonical charts, with coordinates read in SI numerical units. The coordinate volume $dqdp$ has action dimension $(\mathrm{J}\mathrm{s}{)}^{dN}$, so the reference is dimensionless. The physical quotient reference is ${\bar{m}}_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(B):=m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}({\mathrm{\Pi }}_N^{-1}(B))(B\in {\mathcal{F}}_{\mathrm{q}\mathrm{u}\mathrm{o}\mathrm{t}})\mathrm{.}$ Thus $m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ is on the labelled cover ${\mathrm{\Gamma }}_N$, and ${\bar{m}}_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=({\mathrm{\Pi }}_N{)}_{\ast }{m}_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ is on ${\mathrm{\Gamma }}_N\mathrm{/}G$; with $c_N=a_0^{-dN}$, the latter is exactly the normalized quotient reference of Classical configuration, cotangent phase and reference measures.

Physical observables for indistinguishable particles are measurable $G$-invariant functions $A:{\mathrm{\Gamma }}_N\to \mathbb{R}$, equivalently measurable functions $\bar{A}$ on the quotient with $A=\bar{A}\circ {\mathrm{\Pi }}_N$. For every such observable with an integrable expectation, its integral against $m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ equals the quotient integral against ${\bar{m}}_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$. A labelled density represents a physical state on the quotient only if it is invariant under $G$ almost everywhere with respect to $m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$; an invariant representative may then be used. In particular, an indistinguishable-particle Hamiltonian must be invariant under these permutations (or agree almost everywhere with its permuted versions when only its Gibbs density is at issue). A label-specific coordinate statistic on the lift is not itself a quotient observable. Under a permutation-invariant state its marginal law can instead be interpreted as the law of a uniformly selected particle within the same species, or of a uniformly selected coordinate of the same type within that species. Symmetric sums and averages within each species are quotient observables. Calculations of labelled momentum components in Canonical momentum distribution and equipartition use this symmetric labelled lift, and their permutation-invariant kinetic sums and averages are physical observables.

If labels are operationally distinguishable, omit the corresponding permutation factor and use the labelled phase space directly. The convention is part of the preparation and reference model; it is not deduced from Hamiltonian mechanics. Whether a quantity is represented by a gauge-dependent canonical momentum or by a gauge-invariant mechanical momentum is fixed by the stated model and gauge convention. No quantum exchange law, actual preparation mechanism, or empirical indistinguishability test is asserted.