Boson fock representations of stochastic processes

J. Martin Lindsay, Ivan F. Wilde

Yuanbao Kang, Caishi Wang

Nobuaki Obata

L. Accardi, Takeyuki Hida, WIN WIN HTAY

A, classification theory of quantum stationary processes similar to the corresponding theory for classical stationary processes is presented, Our main result is the classification of those pairs of classical stationary processes that admit a joint boson Fock canonical representation.

Stanley Gudder

Radhakrishnan Balu

The building blocks of Hudson-Parthasarathy quantum stochastic calculus start with Weyl operators on a symmetric Fock space. To realize a relativistically covariant version of the calculus we construct representations of Poincare group in terms of Weyl operators on suitably constructed, Bosonic or Fermionic based on the mass and spin of the fundamental particle, Fock spaces. We proceed by describing the orbits of homogeneous Lorentz group on R4 and build fiber bundle representations of Poincaré group induced from the stabilizer subgroups (little groups) and build the Boson Fock space of the Hilbert space formed from the sections of the bundle. Our Weyl operators are constructed on symmetric Fock space of this space and the corresponding annihilation, creation, and conservation operators are synthesized in the usual fashion in relativistic theories for space-like, time-like, and light-like fields. We achieve this by constructing transitive systems of imprimitivity (second-quantized SI), which are dynamical systems with trajectories dense in the configuration space, by induced representations. We provide the details of the field operators for the case of massive Bosons as the rest are similar in construction and indicate the ways to construct adapted processes paving way for building covariant quantum stochastic calculus.