  Quantum chaotic states over a noncommutative monoid, a unitalization of a
noncommutative Ito algebra parametrizing a quantum stochastic Levy process, are
described in terms of their infinitely divisible generating functionals over
the monoid-valued processes on an atomless `space-time' set. A canonical
decomposition of the logarithmic conditionally posive-definite generating
functional is constructed in a pseudo-Euclidean space, given by a quadruple
defining the monoid triangular operator representation and a cyclic zero
pseudo-norm state in this space. It is shown that the exponential
representation in the corresponding pseudo-Fock space yields the
infinitely-divisible generating functional with respect to the exponential
state vector, and its compression to the Fock space defines the cyclic
infinitly-divisible representation associated with the Fock vacuum state. The
structure of states on an arbitrary Ito algebra is studied with two canonical
examples of quantum Wiener and Poisson states. A generalized quantum stochastic
nonadapted multiple integral is explicitly defined in Fock scale, its
continuity and quantum stochastic differentiability is proved. A unified
non-adapted and functional quantum Ito formula is discovered and established
both in weak and strong sense, and the multiplication formula on the
exponential Ito algebra is found for the relatively bounded kernel-operators in
Fock scale. The unitarity and projectivity properties of nonadapted quantum
stochastic linear differential equations are studied, and their solution is
constructed for the locally bounded nonadapted generators in terms of the
chronological products in the underlying kernel algebra canonically represented
by triangular operators in the pseudo-Fock space.
