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**States, effects, and operations. Fundamental notions of quantum theory. Lectures in mathematical physics at the University of Texas at Austin. Ed. by A. Böhm, J. D. Dollard and W. H. Wootters.**
*(English)*
Zbl 0545.46049

Lecture Notes in Physics, 190. Berlin etc.: Springer-Verlag. IX, 151 p. DM 21.00; $ 8.20 (1983).

These notes contain the essential modern results as developed during the last two decades, on the formal description of measurement processes in Hilbert space quantum theory. This development was initiated in connection with G. Ludwig’s approach to a realistic foundation of quantum physics but it can be understood without this background as well. The presentation presupposes only some knowledge on conventional quantum theory and is well suited for students after a first course on quantum theory. The central point, from which most ideas are developed here, is the concept of operation which, in a sense, also is the starting point of the conventional measuring theory. After a short introductory chapter (§ 1) about states and effects, in which the usual basic concepts are reminded and related to the point of view in Ludwig’s approach, the author introduces operations (§ 2) and describes in a lucid manner their abstract but convenient representation by completely positive linear operators on the complex space of trace class operators. A rigorous formulation and proof of this ’first representation theorem’ is given (§ 3). A chapter on composite systems (§ 4) prepares to the ’second representation theorem’ (§ 5) which relates the complete positive linear mappings to interaction processes between objects and apparatuses as far as a description by scattering operators makes sense. The last chapter is devoted to the notions of coexistence and commensurability (§ 6). The formal possibility of measuring processes for combined production of coexistent effects is demonstrated and the construction of observables is considered. All the formal assumptions and derivations are accompagnied by broad physical considerations thereby warning of possible misinterpretations. Because the author has confined himself only to Hilbert space quantum theory the presentation is particularly lucid and easy to read. This booklet can be best recommended to anyone who wants to go without too great effort beyond the everyday understanding of quantum measurements.

Reviewer: K.-E.Hellwig

### MSC:

46L60 | Applications of selfadjoint operator algebras to physics |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

81P05 | General and philosophical questions in quantum theory |

46H10 | Ideals and subalgebras |

46L51 | Noncommutative measure and integration |

46L53 | Noncommutative probability and statistics |

46L54 | Free probability and free operator algebras |

### Keywords:

measurement processes; Hilbert space quantum theory; operation; states; effects; completely positive linear operators; trace class operators; composite systems; coexistence; commensurability; construction of observables### References:

[1] | Ludwig, G.: Foundations of quantum mechanics.. (1983) · Zbl 0509.46057 |

[2] | Jammer, M.: The Philosophy of Quantum Mechanics, Wiley, New York · Zbl 0625.01001 |

[3] | Kraus, K.: Ann. phys.. 64, 311-335 (1971) |

[4] | Aerts, D.: Found. phys.. 12, 1131 (1982) |

[5] | Jauch, J. M.; Piron, C.: Helv. phys. Acta. 40, 559-570 (1967) |

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