COHERENT STATES IN BERNOULLI NOISE FUNCTIONALS

Abstract

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $Z=(Z_K)_{k\in \mathbb{N}}$ a Bernoulli noise on $(\Omega,\mathcal{F},\mathbb{P})$, which has the chaotic representation property. In this paper, we investigate a special family of functionals of $Z$, which we call the coherent states. First, with the help of $Z$, we construct a mapping $\phi$ from $l^2(\mathbb{N})$ to $\mathcal{L}^2(\Omega,\mathcal{F},\mathbb{P})$, which is called the coherent mapping. We prove that $\phi$ has continuity property and other properties of operation. We then define functionals of the form $\phi(f)$ with $f\in l^2(\mathbb{N})$ as the coherent states and prove that all the coherent states are total in $\mathcal{L}^2(\Omega,\mathcal{F},\mathbb{P})$. We also show that $\phi$ can be used to factorize $\mathcal{L}^2(\Omega,\mathcal{F},\mathbb{P})$. Finally we give an application of the coherent states to calculus of quantum Bernoulli noise.