Tensor products and probability weights.

*(English)*Zbl 0641.46049We study a general tensor product for two collections of related physical operations or observations. This is a free product, subject only to the condition that the operations in the first collection fail to have any influence on the statistics of operations in the second collection and vice versa. In the finite-dimensional case, it is shown that the vector space generated by the probability weights on the general tensor product is the algebraic tensor product of the vector spaces generated by the probability weights on the components. The relationship between the general tensor product and the tensor product of Hilbert spaces is examined in the light of this result. We show how anomalous states arise (states that seem to assign “negative probabilities”), how they can be interpreted, and how they can be dealt with mathematically. For totally finite systems, the main result is used to determine the dimensions of various other products that are related to the general tensor product.

Alexander Wilce (Thesis, University of Massachusetts, 1988) has shown the more general result that (without any finiteness condition), the algebraic tensor product is always point-wise dense in the general tensor product.

Alexander Wilce (Thesis, University of Massachusetts, 1988) has shown the more general result that (without any finiteness condition), the algebraic tensor product is always point-wise dense in the general tensor product.

Reviewer: M.Kläy

##### MSC:

46M05 | Tensor products in functional analysis |

81P05 | General and philosophical questions in quantum theory |

46C99 | Inner product spaces and their generalizations, Hilbert spaces |

46N99 | Miscellaneous applications of functional analysis |

52Bxx | Polytopes and polyhedra |

##### Keywords:

general tensor product for two collections of related physical operations or observations; free product; tensor product of Hilbert spaces; anomalous states
PDF
BibTeX
XML
Cite

\textit{M. Kläy} et al., Int. J. Theor. Phys. 26, 199--219 (1987; Zbl 0641.46049)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Birkhoff, G., and von Neumann, J. (1936). The logic of quantum mechanics,Annals of Mathematics,37, 823-843. · JFM 62.1061.04 |

[2] | Cook, T. (1985). Banach spaces of weights on quasi manuals,International journal of Theoretical Physics,24, 1113-1131. · Zbl 0579.46006 |

[3] | Dirac, P. (1930).The Principles of Quantum Mechanics, Oxford-Clarendon Press, Oxford. · JFM 56.0745.05 |

[4] | Foulis, D., and Randall, C. (1980). Empirical logic and tensor products, inInterpretations and Foundations of Quantum Theory, pp. 9-20, H. Neumann, ed., Wissenschaftsverlag, Bibliographisches Institut, Mannheim. |

[5] | Foulis, D., and Randall, C. (1985). Dirac revisited, inSymposium on the Foundations of Modern Physics, P. Lahti and P. Mittelstaedt, eds., pp. 97-112, World Scientific, Singapore. |

[6] | Foulis, D., Piron, C., and Randall, C. (1983). Realism, operationalism, and quantum mechanics,Foundations of Physics,13, 813-842. |

[7] | Gleason, A. (1957). Measures on the closed subspaces of a Hilbert space,Journal of Mathematics and Mechanics,6, 885-893. · Zbl 0078.28803 |

[8] | Greechie, R., and Miller, F. (1970). Structures Related to States on an Empirical Logic I: Weights on Finite Spaces, Kansas State University Technical Report Number 14, Manhattan, Kansas. |

[9] | Groenewold, H. (1985). The elusive quantal individual,Physics Reports,127, 379-401. |

[10] | Gudder, S. (1986). Logical cover spaces,Annales de l’Institut Henri Poincar?,45, 327-337. · Zbl 0612.03026 |

[11] | Gudder, S., Kl?y, M., and R?ttimann, G. (1987). States on hypergraphs,Demonstratio Mathematica, to appear. |

[12] | Jauch, J. (1968).Foundations of Quantum Mechanics, Addison-Wesley, Reading, Massachusetts. · Zbl 0166.23301 |

[13] | Kalmbach, G. (1983).Orthomodular Lattices, Academic Press, New York. · Zbl 0512.06011 |

[14] | Kl?y, M. (1985). Stochastic Models on Empirical Systems, Empirical Logics and Quantum Logics, and States on Hypergraphs, Ph.D. thesis, University of Bern, Switzerland. · Zbl 0576.03040 |

[15] | Randall, C., and Foulis, D. (1973). Operational statistics II: Manuals of operations and their logics,Journal of Mathematical Physics,14, 1472-1480. · Zbl 0287.60003 |

[16] | Randall, C., and Foulis, D. (1978). The operational approach to quantum mechanics, inThe Logico-Algebraic Approach to Quantum Mechanics, Volume III, C. Hooker, ed., pp. 167-201, D. Reidel, Boston. |

[17] | Randall, C., and Foulis, D. (1980). Operational statistics and tensor products, inInterpretations and Foundations of Quantum Theory, H. Neumann, ed., pp. 21-28, Wissenschaftsverlag, Bibliographisches Institut, Mannheim. |

[18] | Randall, C., and Foulis, D. (1985). Stochastic entities, inRecent Developments in Quantum Logic, P. Mittelstaedt and E. Stachow, eds., pp. 265-284, Wissenschaftsverlag, Bibliographisches Institut, Mannheim. |

[19] | R?ttimann, G. (1985a). Expectation functionals of observables and counters,Reports on Mathematical Physics,21, 213-222. · Zbl 0591.03048 |

[20] | R?ttimann, G. (1985b). Facial sets of probability measures,Probability and Mathematical Statistics,6, 187-215. |

[21] | Schr?dinger, E. (1935). Discussion of probability relations between separated systems,Proceedings of the Cambridge Philosophical Society,31, 555-562. · JFM 61.1561.03 |

[22] | Schr?dinger, E. (1936). Discussion of probability relations between separated systems,Proceedings of the Cambridge Philosophical Society,32, 446-452. · JFM 62.1613.03 |

[23] | Schultz, F. (1984). A characterization of state spaces of orthomodular lattices,Journal of Combinatorial Theory (A),17, 317-328. · Zbl 0317.06007 |

[24] | Von Neumann, J. (1955).Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, New Jersey. · Zbl 0064.21503 |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.