Renormalized square of white noise quantum time shift

Abstract

Quantum time shifts are a very special class of continuous time random walk type additive processes on Lie algebras whose exponentiation gives rise to quantum Markov processes whose infinitesimal generators possess a total set of unitary eigen-operators: such a class of quantum Markov gen- erators was not known in the literature. This property generalizes the known fact that trigonometric exponentials are eigenvectors of the usual classical Laplacian and was first noticed in connection with the quantum Laplacian in (5). In that case the eigen-operators turn out to be the Weyl operators, which are known to be non-commutative extensions of the trigonometric ex- ponentials. A generalization of these results, from the Weyl algebra to the harmonic oscillator algebra was obtained in (6). In the present paper we extend the above results to the renormalized square of white noise algebra. This provides among other things, a quantum extension of the Markov gen- erators of the Meixner processes. To our knowledge such an extension was not previously known.