Quantum Ω-Semimartingales and Stochastic Evolutions

Un Cig Ji

R. L. Hudson, Rajeeva Laxman Karandikar, Kalyanapuram Rangachari Parthasarathy

Alexander C. R. Belton

We give details of a *‐linear bijection between adapted (in the sense of Hudson and Parthasarathy) and vacuum‐adapted quantum stochastic integrals. This provides new insight into Attal's remarkable transformation of quantum semimartingales, by showing that it factorizes in a natural manner. The Banach *‐algebras of regular quantum and Ω‐semimartingales are consequently isomorphic, and an intrinsic characterisation of Ω‐semimartingales is obtained as an application of this fact. Various formulae occurring in quantum stochastic calculus are shown to have a more natural appearance in the vacuum‐adapted framework. We finish by providing the full generalization of this theory to higher dimensions.

G. O. S. Ekhaguere

Jean-Claude Zambrini

Within the framework of stochastic calculus of variations for time-symmetric semimartingales X(t,ω), we consider two different stochastic versions of Maupertuis’ least action principle, in Lagrangian and Hamiltonian terms. The general results are applied to classical statistical mechanics, where they coincide with those of classical calculus of variations, and to Nelson’s stochastic mechanics, an approach to quantum mechanics where a time-symmetric semimartingale represents the position of a particle and the dynamics is expressed by a stochastic version of Hamilton’s principle of least action. Some historic examples of old quantum theory are discussed.

Ivan F. Wilde