##
**Multiquantum systems and point processes. I: Generating functionals and nonlinear semigroups.**
*(English)*
Zbl 0731.60110

Summary: An algebraic approach to representation theory and the description of multicomponent quantum systems is considered. A generating multiquantum state functional and nonlinear completely positive map are introduced and a dilation theorem giving a nonlinear extension of GNS and Stinenspring theorem is proved. A number particle operator-valued weight and an empirical weight operator generating a macroscopic inductive algebra are defined, and asymptotic commutativity of this algebra is proved. A canonical multiquantum stochastic process called quasi-Poissonian is constructed and the general structure of the generator for infinite divisible multi-quantum states as well as multiquantum semigroups is found. An existence theorem extending the Lindblad theorem to unbounded generators as well as nonlinear generators is proved. The class of quasi- free quantum point stochastic processes is introduced to describe Markovian dynamics of non-interacting quantum particles and corresponding birth, branching and current nonlinear semigroups and their generators are studied.

### MSC:

60K40 | Other physical applications of random processes |

81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |

81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |

60J99 | Markov processes |

### Keywords:

multicomponent quantum systems; dilation theorem; asymptotic commutativity; quasi-free quantum point stochastic processes### References:

[1] | Stinenspring, W. F., Proc. Amer. Math. Soc., 6, 211-216 (1955) · Zbl 0064.36703 |

[2] | Evans, D. E.; Lewis, J. T., Comm. Dublin Institute for Advanced Studies, 24, 104 (1977) · Zbl 0365.46059 |

[3] | Lindblad, G., Comm. Math. Phys., 48, 119-130 (1976) · Zbl 0343.47031 |

[4] | Belavkin, V. P.; Maslov, V. P., Theor. and Math. Phys., 33, 852-862 (1977) |

[5] | Belavkin, V. P., Mathematical models of statistical physics, ((1982), Tumen), 3-12, (in Russian) |

[6] | Belavkin, V. P., Dokl. Akad. Nauk SSSR., 293, 18-21 (1987) |

[7] | Belavkin, V. P., Theor. and Math. Phys., 62, 275-289 (1985) |

[8] | Accardi, L.; Frigerio, A.; Lewis, J. T., Publ. RIMS Kyoto Univ., 18, 97-133 (1982) · Zbl 0498.60099 |

[9] | Belavkin, V. P., Dokl. Akad. Nauk SSSR, 301 (1988) |

[10] | Belavkin, V. P.; Staszewski, P., Rept. Math. Phys., 20, 373-384 (1984) · Zbl 0587.46056 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.