Reconstruction theorem for a quantum stochastic process

Statistically interpretable axioms are formulated that define a quantum stochastic process (QSP) as a causally ordered field in an arbitrary space-time localization region $T$ of an observable physical system. It is shown that to every QSP described in the weak sense by a self-consistent system of causally ordered correlation kernels there corresponds a unique, up to unitary equivalence, minimal QSP in the strong sense. It is shown that the proposed QSP construction, which reduces in the case of the linearly ordered :r  to the construction of the inductive limit of Lindblad's canonical representations [8], corresponds to Kolmogorov's classical reconstruction [12] if the order on $T=\mathbb Z$ is ignored and leads to Lewis construction [14] if one uses the system of all (not only causal) correlation kernels, regarding this system as lexicographically ordered on $\mathbb Z\times T$. The approach presented encompasses both nonrelativistic and relativistic irreversible dynamics of open quantum systems and fields satisfying the conditions of semigroup eovariance and local commutativity. Also given are necessary and sufficient conditions of dynamicity (conditional Markovness) and regularity, these leading to the properties of complete mixing (relaxation) and ergodicity of the QSP.