Difference between Green's function and density matrix 
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*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?
 A: A 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.
A: 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 E_n (t-t')/\hbar} $,
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.
