Coherent states in Bernoulli noise functionals. (English) Zbl 1220.60041

Wang, Caishi; Han, Qi

Bull. Aust. Math. Soc. 84, No. 1, 116-126 (2011).

Summary: 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 \(\varphi \) from \(l^2(\mathbb N)\) to \(\mathbb L^2(\Omega,{\mathcal F},\mathbb P)\) which is called the coherent mapping. We prove that \(\varphi\) has the continuity property and other properties of operation. We then define functionals of the form \(\varphi (f)\) 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 \(\varphi\) can be used to factorize \({\mathcal L}^2(\Omega,{\mathcal F},\mathbb P)\). Finally, we give an application of the coherent states to the calculus of quantum Bernoulli noise.

MSC:

60H40	White noise theory
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
81S25	Quantum stochastic calculus

Keywords:

Bernoulli noise; coherent state; factorization of functional space

References:

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