##
**Introduction to random time and quantum randomness.
new edition.**
*(English)*
Zbl 1041.81074

Monographs of the Portuguese Mathematical Society 1. River Edge, NJ: World Scientific (ISBN 981-238-415-4/pbk; 981-238-388-3/hbk). xii, 211 p. (2003).

The volume is composed of two scientific essays: “Introduction to random time” by K. L. Chung and “Introduction to quantum randomness” by J.-C. Zambrini. I recommed reading them to all readers who take pleasure from an informal style of presentation, where an introductory description to rather advanced probabilistic concepts is ornamented with personal memories, anecdotes and reflections plus harsh comments on competing or outdated viewpoints.

The common theme of both esssays is time, specifically various notions of time which are omnipresent in the theory of stochastic processes and general stochastic analysis. An important topic addressed is this of time reversal for Markov processes. K. L. Chung tells why he believes that probability theory starts where random time appears. J.-C. Zambrini tells why he believes that quantum physics is not a regular probabilistic theory and gives his own explanation of Feynman’s (“path integral” versus Wiener path measures) approach to probabilities in quantum mechanics. Schrödinger’s year 1932 paper on the probabilistic meaning of his wave functions, gave impetus to the Euclidean quantum mechanics proposal developed by Zambrini, which (this is however not mentioned by the author) may be pushed far away from the original quantum motivation and shown to encompass typical diffusion-type processes of nonequilibrium statistical mechanics.

The major mathematical input are here Bernstein processes with their natural forward and backward dynamics. Since an issue of the quantum time “observable” is addressed and reexamined form the author’s personal perspective, one should be aware that in the physics literature various methods were developed to bypass the time operator non-existence problem and nonetheless to infer experimentally verifiable data on various times (like e.g. the time of arrival, the time of escape from a bounded area, the transit or tunneling time etc.) from the standard formalism of quantum theory. See e.g. the volume edited by J. G. Muga et al [Time in quantum mechanics (2002; Zbl 1055.81002)].

The common theme of both esssays is time, specifically various notions of time which are omnipresent in the theory of stochastic processes and general stochastic analysis. An important topic addressed is this of time reversal for Markov processes. K. L. Chung tells why he believes that probability theory starts where random time appears. J.-C. Zambrini tells why he believes that quantum physics is not a regular probabilistic theory and gives his own explanation of Feynman’s (“path integral” versus Wiener path measures) approach to probabilities in quantum mechanics. Schrödinger’s year 1932 paper on the probabilistic meaning of his wave functions, gave impetus to the Euclidean quantum mechanics proposal developed by Zambrini, which (this is however not mentioned by the author) may be pushed far away from the original quantum motivation and shown to encompass typical diffusion-type processes of nonequilibrium statistical mechanics.

The major mathematical input are here Bernstein processes with their natural forward and backward dynamics. Since an issue of the quantum time “observable” is addressed and reexamined form the author’s personal perspective, one should be aware that in the physics literature various methods were developed to bypass the time operator non-existence problem and nonetheless to infer experimentally verifiable data on various times (like e.g. the time of arrival, the time of escape from a bounded area, the transit or tunneling time etc.) from the standard formalism of quantum theory. See e.g. the volume edited by J. G. Muga et al [Time in quantum mechanics (2002; Zbl 1055.81002)].

Reviewer: Piotr Garbaczewski (Zielona Góra)

### MSC:

81S20 | Stochastic quantization |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

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

60J55 | Local time and additive functionals |

60G40 | Stopping times; optimal stopping problems; gambling theory |

81S40 | Path integrals in quantum mechanics |

60G17 | Sample path properties |

60H07 | Stochastic calculus of variations and the Malliavin calculus |