Classical and Quantum Probability

Abstract

We survey the development of probability from 1900, starting with Bachelier's theory of speculation. Fisher information appears in the theory of estimation. We touch on Brownian motion, and the Wiener integral. The Ito calculus, and its relation to to the heat equation, is mentioned. Quantum theory is introduced as a generalisation of probability, rather than of mechanics. The weakness of attempts to describe quantum theory in terms of hidden variables is explained, by a simple proof of Bell's inequality. Quantum versions of the Langevin equation are discussed, and the theory of continuous tensor products is used to give a possible quantum version. The quantum stochastic calculus of Barnett, Wilde and the author, as well as that of Parthasarathy and Hudson, is introduced.

68 pages, 5.5 in wide and 8 in high

References in corpus (3)

• Quantum Stochastic Positive Evolutions: Characterization, Construction, Dilation
• On Stochastic Generators of Completely Positive Cocycles
• The information manifold for relatively bounded forms

Cited by in corpus (21)

• Geometric Quantum Mechanics
• Interpretations of Negative Probabilities
• Quantum processes on phase space
• Geometry of nonadiabatic quantum hydrodynamics
• The Madelung Picture as a Foundation of Geometric Quantum Theory
• The Nature of Information in Quantum Mechanics
• On the Small Mass Limit of Quantum Brownian Motion with Inhomogeneous Damping and Diffusion
• Algorithmic Information Theoretic Issues in Quantum Mechanics
• Lévy-Khintchine decompositions for generating functionals on algebras associated to universal compact quantum groups
• Quantum theory: the role of microsystems and macrosystems
• Quantum Stochastic Processes and the Modelling of Quantum Noise
• Extended Probabilities: Mathematical Foundations
• From the Heisenberg to the Schrödinger Picture: Quantum Stochastic Processes and Process Tensors
• Stochasticity, topology, and spin
• On the applicability of Kolmogorov's theory of probability to the description of quantum phenomena. Part I: foundations
• Quantum Logic and Non-Commutative Geometry
• Non-Markovian processes in quantum theory
• Relating the wave-function collapse with Euler's formula, with applications to Classical Statistical Field Theory
• Einstein's lifts and topologies: topological investigations on the Principle of Equivalence
• Relativity and EPR Entanglement: Comments
• Static Quantum Games Revisited