Mean and variance of the cardinality of particles in infinite true polyanalytic Ginibre processes via a coherent states quantization method

Abstract

Abstract We discuss the mean and variance of the number ‘point-particles’ ♯ D R inside a disk D R centered at the origin of the complex plane C and of radius R > 0 with respect to an infinite true polyanalytic process of index m ∈ Z + by quantizing the phase space C via a set of generalized coherent states (CSs) z , m of the harmonic oscillator on L 2 R . By this procedure, the spectrum of the quantum observable representing the indicator function χ D R ofD R (viewed as a classical observable) allows to compute the mean value of ♯ D R . The variance of ♯ D R is obtained as a special eigenvalue of a quantum observable involving the auto-convolution of χ D R . By adopting a CSs quantization approach, we seek to identify classical observables on C , whose quantum counterparts may encode the first cumulants of ♯ D R through spectral properties.