In last week's lecture, we started our discussion of the density matrix. Three properties allowed us to evaluate this object at low temperature:

Convolution




Trotter decomposition (high temperature limit)



The expression of the density matrix in the free case





In the present lecture, we first introduce the Bloch equation for the density matrix



and discuss the relation between the density matrix and the time evolution in classical diffusion. Then, we compute the exact density matrix for a quantum particle in an harmonic well.
Finally we discuss a direct sampling approach to the free path integral, one of the most beautiful solutions
of a complicated sampling problem.