# How does the quantum partition function of a many body system relate to that of a single body system

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So from what I understand, if we have a quantum system, described by the Hilbert space $$\mathscr{H}$$, in thermal equilibrium with a large environment, then the partition function $$Z$$ is given by $$Z=\tr {e^{-\beta H}}$$.

But if we have a many-body system consisting of fermions/bosons so that the Hilbert space describing the system is the Fock space, how is the partition function $$Z$$ of the many-body system related to that of the single particle $$Z_1$$? We may assume that they are non-interacting if necessary.

I have seen classically that $$Z=Z_1^N/N!$$ due to the fact that the particles are indistinguishable and non-interacting, but how would it look for a quantum system with no constraint on the total number of particles?

EDIT: To be a little more specific, I came upon this question when I was reading Giamarchi, Quantum Physics in One Dimension, Appendix B.2 and trying to understand how he derived the total partition function (equations B.13 to B.19) for the fermionic and bosonic system, OR EQUIVALENTLY, equations 3.36a,b in F.D.M. Haldane's 1981 paper on Luttinger liquid theory. If anyone is familiar with the derivation, I would appreciate it if you could elaborate.

• Are you hoping to consider the single particle as a subsystem of a larger system of fermions or bosons? That's problematic, because in that type of system, observables are not tied to particles. Observables are tied to regions of space. Or is the goal to compare the partition function for a single particle to the partition function for a separate system of fermions or bosons, without trying to regard the former as a subsystem of the latter? – Chiral Anomaly Jun 26 at 4:18