Rules of probability in quantum mechanics (original) (raw)

Quantum probability: New perspectives for the laws of chance

The main philosophical successes of quantum probability is the discovery that all the so-called quantum paradoxes have the same conceptual root and that such root is of probabilistic nature. This discovery marks the birth of quantum probability not as a purely mathematical (noncommutative) generalization of a classical theory, but as a conceptual turning point on the laws of chance, made necessary by experimental results.

Quantum Probability: An Historical Survey

What is quantum probability Quantum Probability (QP) is a new branch of mathematics interconnecting classical probability, functional analysis, pure algebra, quantum physics and information and communication engineering. The mid seventies is the period that marks the beginning of QP as an autonomous discipline. Since then this cross disciplinary nature has accompanied the development of QP and still now it is one of its points of strength, making it an original new trend in contemporary mathematics as well as one of the earliest pioneers of non-commutative mathematics: a field now flourishing with the more recent development of quantum groups, non-commutative geometry, quantum computer, . . .

On the Fundamental Nature of the Quantum Mechanical Probability Function

2012

The probability of occurrence of an event or that of the existence of a physical state has no relative existence in the sense that motion is strongly believed to only exist in the relative sense. If the probability of occurrence of an event or that of the existence of a physical state is known by one observer, this probability must be measured to have the same numerical value by any other observer anywhere in the Universe. If we accept this bare fact, then, the probability function can only be a scalar. Consequently, from this fact alone, we argue that the quantum mechanical wavefunction can not be a scalar function as is assumed for the Schrödinger and the Klein-Gordon wavefunctions. This has fundamental implications on the nature of the wavefunction insofar as translations from one reference system to the other is concerned.

A discussion on the origin of quantum probabilities.

We study the origin of quantum probabilities as arising from non-boolean propositionaloperational structures. We apply the method developed by Cox to non distributive lattices and develop an alternative formulation of non-Kolmogorvian probability measures for quantum mechanics. By generalizing the method presented in previous works, we outline a general framework for the deduction of probabilities in general propositional structures represented by lattices (including the non-distributive case).

QUANTUM MECHANICS AS GENERALIZED THEORY OF PROBABILITIES

It is argued that quantum mechanics does not have merely a predictive function like other physical theories; it consists in a formalisation of the conditions of possibility of any prediction bearing upon phenomena whose circumstances of detection are also conditions of production. This is enough to explain its probabilistic status and theoretical structure. Published in: Collapse, 8, 87-121, 2014

An outline of quantum probability

1 INDEX Introduction (1a.) Foundations of quantum theory (1b.) Quantum probability and the paradoxes of quantum theory (1c.) Von Neumann' s measurement theory (1d.) Contemporary measurement theory (1e.) Open systems and quantum noise (1f.) Stochastic calculus (1g.) Laws of large numbers and central limit theorems (1h.) Conditioning PART I: ALGEBRAIC PROBABILITY THEORY (2.) Algebraic probability spaces (3.) Algebraic random variables (4.) Stochastic Processes (5.) The local algebras of a stochastic process (6.) Independence (7.) Example: quantum spin systems (8.) A combinatorial lemma (9.) The Boson law of large numbers for independent random variables (10.) The central limit theorem for product maps (11.) Boson and Fermion Gaussian maps (12.) The quantum commutation relations as GNS representations (13.) The quantum commutation relations (14.) De Finetti' s theorem (15.) Conditioning: expected subalgebras (16.) Conditional amplitudes on B(H o ) (17.) Transition expectations and Markovian operators (18.) Markov chains, stationarity, ergodicity (19.) Conditional density amplitudes, potentials and invariant weights (20.) Multiplicative functionals and the discrete Feynman integral (21.) Quantum Markov chains and high temperature superconductivity models (22.) Kümmerer's Markov chains (23.) The algebraic states of Fannes, Nachtergaele and Slegers (24.) 1-dependence and the Ibragimov-Linnik conjecture (25.) 1-dependent quantum Markov chains 2 (26.) Commuting conditional density amplitudes (27.) Diagonalizable states (28.) A nonlinear chain of harmonic oscillators (29.) Generalized random walks (30.) The diffusion limit of the coherent chain (31.) Cecchini' s Markov chains PART II : STOCHASTIC CALCULUS (32.) Simple stochastic integrals (33.) Semimartingales and integrators (34.) Forward derivatives (35.) The o(dt)-notation (36.) Stochastic differential equations (37.) Meyer brackets and Ito tables (38.) The weak Itô formula (39.) The unitarity conditions (40.) The Boson Lévy theorem PART III : CONDITIONING (41.) The standard space of a von Neumann algebra (42.) The ϕ-conditional expectation

On the strangeness of quantum probabilities

Quantum Studies: Mathematics and Foundations

Here we continue with the ideas expressed in "On the strangeness of quantum mechanics" [1] aiming to demonstrate more concretely how this philosophical outlook might be used as a key for resolving the measurement problem. We will address in detail the problem of determining how the concept of undecidability leads to substantial changes to classical theory of probability by showing how such changes produce a theory that coincides with the principles underlying quantum mechanics.

Critique of “Elements of Quantum Probability”

Quantum Probability Communications, 2001

We analyse the thesis of that classical probability is unable to model the the stochastic nature of the Aspect experiment, in which violation of Bell's inequality was experimentally demonstrated. According to these authors the experiment shows the need to introduce the extension of classical probability known as Quantum Probability. We show that their argument depends on hidden assumptions and a highly restrictive view of the scope of classical probability. A careful probabilistic analysis shows, on the contrary, that it is classical deterministic physical thinking which cannot cope with the Aspect experiment and therefore needs revision. The ulterior aim of the paper is to help mathematical statisticians and probabilists to find their way into the fascinating world of quantum probability (thus: the same aim as that of Kümmerer and Maassen) by dismantling the bamboo curtain between ordinary and quantum probability which over the years has been built up as physicists and pure mathematicians have repeated to one another Feynman's famous dictum 'quantum probability is a different kind of probability'.