In Pathria and Beale Statistical Mechanics section 5.5, the book tries to compute the Partition function of a system of noninteracting, indistinguishable particles confined to a cubical box of volume $V$. The book claims that the density matrix is given by $$\langle \boldsymbol{r}_1,...,\boldsymbol{r}_N|\hat{\rho}|\boldsymbol{r}_1',...,\boldsymbol{r}_N'\rangle$$ and attempts to compute this quantity by inserting a $\sum_{\boldsymbol{k}}|\boldsymbol{k}\rangle\langle\boldsymbol{k}|$ between $\hat{\rho}$ and $|\boldsymbol{r}_1',...,\boldsymbol{r}_N'\rangle$. Here $\boldsymbol{k}$ forms a complete orthonormal basis of the $N$ particle Fock spaces. The book then claims that the partition function is given by $$\int \langle \boldsymbol{r}_1,...,\boldsymbol{r}_N|\hat{\rho}|\boldsymbol{r}_1,...,\boldsymbol{r}_N\rangle\,d^{3N}r.$$
My question is: what is the Hilbert space that $\hat{\rho}$ acts on? My understanding is that the Hilbert space should be the $N$ particle Fock space, but then the question is that when taking the trace to compute the partition function, we can't do $d^{3N}r$ because $d^{3N}r$ includes coordinates that are not symmetrized/antisymmetrized.
However, if the Hilbert space is just the product space, then $|\boldsymbol{k}\rangle$ would not be a complete basis of the Hilbert space.
Add: it appears that Pathria and Beale is just being sloppy. Greiner's thermodynamics and statistical mechanics chapters 10 and 11 discuss this in greater detail, although even Greiner takes some shortcut with the Boson case. After comparing with Greiner, Pathria and Beale definitely should have used the properly (anti)symmetrized position states. It looks like just a coincidence that Pathria and Beale obtains the correct partition function as in Greiner using this sloppy approach.
