Abstract:
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.
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