# Difference between green's function and density matrix

What is the basic difference between green's function or propagator of given system and density matrix (in the position basis) of the same system ? Can some one explain the difference between with some example, say one dimensional harmonic oscillator.

The one-particle Green function in the time domain (omitting time ordering) of a non-interacting system can be written in terms of the eigenfunctions of the Hamiltonian $$\psi_n$$

$$G(\mathbf{r},t;\mathbf{r}',t') = \sum_n \psi_n(\mathbf{r}) \, \psi_n^*(\mathbf{r}') \; e^{-i/\hbar E_n (t-t')}$$,

where the exponential at the end is the time evolution operator $$U(t,t')$$ that dictates the evolution of the wavefunction $$\Psi(\mathbf{r}',t')$$ to $$\Psi(\mathbf{r},t)$$ in infinitesimal time intervals.

Compare the previous equation with the first order density matrix of a monodeterminantal wavefunction

$$\gamma_1(\mathbf{r};\mathbf{r}') = \sum_n \psi_n(\mathbf{r}) \, \psi_n^*(\mathbf{r}')$$.

The first order density matrix is obtained from the Green function by contour integration

$$\gamma_1(\mathbf{r};\mathbf{r}') = - \frac{1}{2\pi i} \int_C G(\mathbf{r},t;\mathbf{r}',t) \, \mathrm{d}t$$.

A way to think about the Green function is as a correlation function. While the Green function measures spatio-temporal correlations, the density matrix measures spatial correlations.

• of course you are assuming pures states here... Aug 11 '20 at 13:23

Density matrix can be represented as a Green's function at equal times. E.g., in one-particle case: $$\rho(\mathbf{r},\mathbf{r}',t)=G(\mathbf{r},t; \mathbf{r}',t'=t).$$

Those familiar with diagrammatic expansion might enjoy the review by Rammer (not the better known by Rammer&Smith) where the diagrammatic expansion is obtained for the density matrix.