Past exam of the theoretical physics II course of the University of Cambridge / 2013 / Paper 1 / 4

There is a state-assumption issue. For a number-conserving Gaussian quantum state with no anomalous contractions, put ${\psi }_i=\psi ({\mathbf{r}}_i)$, $n_i=⟨{\psi }_i^{†}{\psi }_i⟩$ and the one-body density matrix $g_{ij}=⟨{\psi }_i^{†}{\psi }_j⟩$. Wick's theorem pairs each creation operator with an annihilation operator. The direct pairing gives $n_1{n}_2$; the exchanged pairing gives $ϵg_{12}{g}_{21}$, with $ϵ=+1$ for bosons and $-1$ for fermions. Hencenormal ordering removes the equal-position delta-function self-counting term. Fermionic occupation-number product states are Slater determinants and obey this Wick theorem. Independent thermal bosonic modes also obey it, with $N_{\alpha }$ understood as mean occupations.

A bosonic Fock product state does not generally obey the asserted identity. The bosonic number-state corrections to Gaussian density correlations follow from $⟨a_{\alpha }^{†2}{a}_{\alpha }^2⟩=N_{\alpha }(N_{\alpha }-1)$ and giveFor one occupied mode this is $N(N-1)|{\phi }_1{|}^2|{\phi }_2{|}^2$, whereas the uncorrected Gaussian expression gives $2N^2|{\phi }_1{|}^2|{\phi }_2{|}^2$. This counterexample disproves the asserted identity for arbitrary bosonic number-product states. The formulas below give both the Gaussian result, with its additional state assumption, and the exact bosonic number-state alternative. The ensemble distinction is also explicit in a later primary treatment of ideal-gas correlations.^[ref]

Under the Gaussian assumption, $g(\mathbf{r},\mathbf{r})=n(\mathbf{r})$, so the coincident normally ordered density correlation is $(1+ϵ)n(\mathbf{r}{)}^2$. For a homogeneous nonzero density $n$, the large-separation limit is $n^2$, givingThese coincident-point bunching factors describe enhanced bosonic coincidence and excluded same-component fermionic coincidence. More generally the density-normalized ratio $C_{12}/(n_1{n}_2)$ tends to one at large separation. The unnormalized ratio printed in the paper additionally needs matching nonzero local and remote densities; $g_{12}\to 0$ alone does not ensure this. For a bosonic Fock product state, the local normalized value instead is $2-{\sum }_{\alpha }{N}_{\alpha }(N_{\alpha }+1)|{\phi }_{\alpha }(\mathbf{r}){|}^4/n(\mathbf{r}{)}^2$, and even a large-separation factorization must be checked separately.

The three-point density correlation of an ideal quantum gas follows first for a number-conserving Gaussian quantum state. There are six pairings, one for each permutation of three annihilation operators. The identity gives the product of densities; each transposition has sign $ϵ$; both three-cycles have positive sign. ThusThe last two terms are complex conjugates. They equal $2\mathrm{Re}(g_{12}{g}_{23}{g}_{31})$, not an additional signed pair term. This is the permanent of a matrix $(g_{ij})$ for bosons and its determinant for fermions. Wick's theorem supplies the six terms only with the state assumption stated in part (a).

For completeness an exact bosonic Fock-state alternative can be given without that assumption. Let $\mathbf{\ell }$ be a mode-occupation decrement with ${\sum }_{\alpha }{\ell }_{\alpha }=3$. Define $A_{\mathbf{\ell }}(1,2,3)$ as the sum of ${\prod }_{j=1}^3{\phi }_{{\alpha }_j}({\mathbf{r}}_j)$ over all distinct ordered triples having these mode multiplicities. Acting three annihilation fields on the number state leaves mutually orthogonal states for different $\mathbf{\ell }$. ThereforeHere $N^{\underline{\ell }}=N!/(N-\ell )!$, with zero when $\ell >N$. This expression includes repeated-mode occupation corrections absent from the Gaussian permanent; for a single mode it reduces to $N(N-1)(N-2){\prod }_j|\phi ({\mathbf{r}}_j){|}^2$.

At coincident points the Gaussian expression has $g_{ij}=n$. Its six contributions giveWhen all mutual separations become large and the density stays the same nonzero $n$, only $n_1{n}_2{n}_3=n^3$ remains. ConsequentlyThe bosonic value is $3!$ from Gaussian exchange pairings; the fermionic determinant vanishes when its rows coincide. Density-normalized correlations avoid the additional homogeneity assumption. For a specified bosonic number product, part (c)'s occupation sum must replace the six-pairing formula, so neither the universal factor six nor a universal printed ratio follows from that specification alone.