# Thermal density matrix QFT

The density matrix of a system at finite temperature is give by $$\langle\psi_1|\rho|\psi_2\rangle=\frac{1}{Z}\langle\psi_1|e^{-\beta H}|\psi_2\rangle,$$ where $$Z$$ is a normalization constant. We can then express this matrix element as the Euclidean path integral $$\frac{1}{Z}\langle\psi_1|e^{-\beta H}|\psi_2\rangle= \int_{\psi(t_E=0)=\psi_1}^{\psi(t_E=\beta)=\psi_2}\mathcal D\psi~e^{i \int_0^\beta dt_E~\int d^3 x ~\mathcal L_E[\phi]} ,$$ where $$t_E$$ is the Euclidean time and $$\mathcal L_E$$ is the Euclidean Lagrangian.

Question: If I want to describe a system at finite temperature with chemical potential $$\mu$$ corresponding to some $$U(1)$$ conserved charge $$Q$$ so that the density matrix is now given by $$\langle\psi_1|\rho|\psi_2\rangle=\frac{1}{Z}\langle\psi_1|e^{-\beta (H-\mu Q)}|\psi_2\rangle,$$ is there a path-integral representation for this matrix element?

It is the same. Just add $$- \mu Q$$ to your Lagrangian $$L_E$$. Treat $$H - \mu Q$$ as your new Hamiltonian.