  We study bosonisation in the massive Thirring and sine-Gordon models at
finite temperature T and nonzero fermion chemical potential $\mu$. For that
purpose we use both canonical operator and path integral approaches, paying
particular attention to the issues of thermal normal ordering and
renormalisation. At T>0 and $\mu=0$, the massive Thirring model bosonises to
the sine-Gordon model with the same T=0 identification between coupling
constants. We prove that not only the partition functions of the two models
coincide, as was recently shown, but also that thermal averages of zero-charge
operators can be identified. In particular, analysis of the point-split
regularised fermion current then leads to the thermal equivalence between
sine-Gordon kinks and Thirring fermions. At $\mu \neq 0$, T>0 and working in
perturbation theory about the massless Thirring model, we show that the
bosonised theory is the sine-Gordon model plus an additional topological term
which accounts for the existence of net fermion charge excitations (the
fermions or the kinks) in the thermal bath. This result generalises one
recently obtained for the massless case, and it is the two-dimensional version
of the low-energy QCD chiral Lagrangian at finite baryon density.
