Introduction to Random Time and Quantum Randomness

The following sections are included:

• Monographs of the Portuguese Mathematical Society

• Monografias da Sociedade Portuguesa de Matemática

• Guide

• Contents

Part 1 originated in a series of lectures given in May 1997 at the Grupo de Física Matemática in Lisboa, by invitation of Professor Zambrini. My notes were later considerably expanded and portions of them were presented in seminars at the Scuola Normale Superiore at Pisa, University of California at San Diego, and Korteweg-DeVries Institute at Amsterdam. The announced title for the last occasion was: "Il Tempo Va Meglio Per Caso" [Time Better Be Random], which gives an idea of my intention. Diverse kinds of RANDOM TIME in various probability schemes, some ancient and some modernistic, are revealed and studied, and their often tremendous impact on the impending process illustrated. These random times sometimes behave like nonrandom (constant) ones and sometimes radically differently. Two of them, the first exit time, and the last exit time of the Brownian paths from a conductor, have successfully served in the mathematical theory of the Green-Gauss-Schrödinger-Feynman "Ideenkreis" (see reference [7] at the end of Part 1). Other kinds of random time and their significance remain to be discovered. With the introduction of probability into quantum physics, there would seem to be a fertile field of exploitation and cultivation. As my former colleague Dick Feynman (c. 1949) might put it: here is a bag of tricks for the smart cookies to pick up. At least such is the hope of both authors of this small volume.

The following sections are included:

• Prologue

• Stopping Time

• Martingale Stopped

• Random Past and Future

• Other Times

• From First to Last

• Gapless Time

• Markov Chain in Continuum Time

• The Trouble with the Infinite

• References

After the series of lectures given in 1997 by Professor Kai Lai Chung at the Group of Mathematical Physics in Lisbon, where he described many kinds of random time, we started to talk more generally about the strange status of probability theory in quantum mechanics. The subject has been, to say the least, much debated from the very beginning of quantum theory. A number of peremptory "solutions of the problem" have been announced in mathematical physics, then discreetly withdrawn…

The following sections are included:

• Classical Prologue

• Standard Quantum Mechanics

• Probabilities in Standard Quantum Mechanics

• Feynman's Approach to Quantum Probabilities
  • Lagrangian mechanics
  • Feynman's space-time reinterpretation of quantum mechanics

• Schrödinger's Euclidean Quantum Mechanics
  • A probabilistic interpretation of Feynman's approach
  • Feynman's results revisited

• Beyond Feynman's Approach
  • More quantum symmetries
  • Introduction to functional calculus

• Time for a Dialogue

• References

The following sections are included:

• Index