Kalman filter and quantization

We give an interpretation of the problem of filtering of diffusion processes as a quantization problem. Based on this, we show that the classical Kalman-Bucy linear filter describes a flow of automorphisms of the Heisenberg algebra. We obtain new formulas for the unnormalized conditional density in the linear case, a new interpretation of the Mehler formula for the fundamental solution of the Schrödinger operator for a harmonic oscillator, and formulas for a regularized determinant of a Sturm-Liouville operator.