Seminar at SMU Delhi
February 26, 2014 (Wednesday) ,
3:30 PM at Webinar
K. R. Parthasarathy,
Indian Statistical Institute, Delhi
Normal distribution in classical probability and the gaussian state in quantum probability
Abstract of Talk
The multivariate normal distribution is a basic object in classical (commutative) probability. In noncommutative probability there is a similar fundamental object called a gaussian state determined by CCR (canonical commutation relations). We shall describe a structure theorem for the most general gaussian state and the most general pure gaussian wave function in $R^n$ and compare them with the classical normal distribution. An analysis of the covariance matrices of gaussian states shows that the set of all gaussian states in $L^2(R^n)$ is stable under the action of the symplectic group $Sp(2n,R).$ This gives rise to some interesting open problems which we shall describe.