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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.

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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.

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

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