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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.