Density matrix of quantum fluids

An elaborate cluster analysis of the single-particle occupation probability $n_{\hat{q}}$ and associated one-body density matrix $n\left(r\right)\mathrm{}$ is performed for a Fermi system described by a Jastrow wave function. A diagrammatic formalism rooted in Ursell-Mayer theory facilitates the analysis. It is conjectured, and demonstrated to convincingly high cluster order, that $n_{\hat{q}}$ may be written as $n\left[N\right(q)+N_1\mathrm{}(q\left)\right]$, where $n$ is a strength factor independent of wave number $q$ and the quantities $N\left(q\right)\mathrm{}$ and $N_1\mathrm{}\left(q\right)\mathrm{}$ may be expressed as series of irreducible cluster contributions. The strength factor $n$ has the form $n=e^Q$, where $Q$ may also be expressed as a series of irreducible cluster contributions. Massive partial summations on the latter series yield a compact expression for $Q$ in terms of the spatial distribution functions corresponding to the Jastrow wave function. Working with the Fourier inverse of $n_{\hat{q}}$, it is further demonstrated that $n\left(r\right)\mathrm{}$ may be cast in the form $\rho n\left[N_1\mathrm{}\right(r)+N_2\mathrm{}(r\left)\right]\exp \mathcal{Q}\mathrm{}\left(r\right)\mathrm{}$, where $\rho$ is the particle density and the functions $N_1\mathrm{}\left(r\right)\mathrm{}$, $N_2\mathrm{}\left(r\right)\mathrm{}$, and $\mathcal{Q}\mathrm{}\left(r\right)\mathrm{}$ are all given by irreducible cluster series. Massive partial summations are executed in the $\mathcal{Q}\mathrm{}\left(r\right)\mathrm{}$ series to achieve a compact expression of this quantity in terms of the aforementioned spatial distribution functions. One has $\mathcal{Q}\mathrm{}\left(0\right)=Q$. The leading diagrams necessary for a quantitative evaluation of the momentum distribution of liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and nuclear matter are displayed. Specialization to infinite degeneracy of the single-particle levels, while shrinking the Fermi wave number to zero (Bose limit), allows liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ to be treated as well. In this limit off-diagonal long-range order appears, the condensate fraction ${\rho }^{-1\mathrm{}}n\left(\infty \right)=n_c$ being just the strength factor $n$. It may also be shown (under certain reasonable assumptions) that...