Quantum Probability: an introduction to some basic ideas and trends

M. Fannes

Lee Rudolph

This chapter has two related purposes: to generate interest in a new and fascinating approach to understanding behavioral measures based on quantum probability principles, and to introduce and provide a tutorial of the basic ideas in a manner that is interesting and easy for social and behavioral scientists to understand.

L. Accardi

Uwe Franz, Adam Skalski

Noncommutative mathematics is a significant new trend of mathematics. Initially motivated by the development of quantum physics, the idea of 'making theory noncommutative' has been extended to many areas of pure and applied mathematics. This book is divided into two parts. The first part provides an introduction to quantum probability, focusing on the notion of independence in quantum probability and on the theory of quantum stochastic processes with independent and stationary increments. The second part provides an introduction to quantum dynamical systems, discussing analogies with fundamental problems studied in classical dynamics. The desire to build an extension of the classical theory provides new, original ways to understand well-known 'commutative' results. On the other hand the richness of the quantum mathematical world presents completely novel phenomena, never encountered in the classical setting. This book will be useful to students and researchers in noncommutative probability, mathematical physics and operator algebras.

Michael Schnabel

This chapter provides a brief introduction to some key concepts of quantum theory that are contrasted with their classical analogs. By means of simple examples, the aim is to develop an intuition for notions such as quantum state, observables, superposition, Born’s rule, collapse of the wave function, Schrödinger’s equation, quantum interference, Heisenberg uncertainty, and incompatibility of observables. The hope is to provide a readable account on these core ideas of quantum theory that is accessible to readers from disciplines other than physics. The mathematical framework is developed from scratch, but some familiarity with basic linear algebra and probability theory will be assumed.

Luigi Accardi