by its characteristic function $ 1 _ {\mathcal F} $. of the pair $ ( L _ \infty ( \Omega ^ \prime , {\mathcal F} ^ \prime , {\mathsf P} ^ \prime ), {\mathsf P} ^ \prime ) $ Kadison, J.R. Ringrose, "Fundamentals of the theory of operator algebras" . This article was adapted from an original article by H. Maassen (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. https://encyclopediaofmath.org/index.php?title=Quantum_probability&oldid=48371. We review Fock space based quantum probability and in particular, the theory of stop times based on it. The constants in front of the kets are called probability amplitudes and are in general complex numbers. of automorphisms of $ ( {\mathcal A} , \phi ) $ W. von Waldenfels (ed.) is a group and their exists a group of automorphisms $ ( \alpha _ {t} ) _ {t \in \mathbf T } $ The mathematical ingredients of quantum probability theory derive from the theory of operator algebras, as founded by J. von Neumann and developed by M.A. Examples of such noises are: thermal radiation, thermal collisions with heat bath particles, laser fields and atomic beams. However, application of probabilistic ideas becomes most fruitful when the number of relevant degrees of freedom becomes infinite, and some central limit theorem is active. a conditional expectation) $ {\mathcal A} \rightarrow j( {\mathcal A} ^ \prime ) $. a weak $ * $- on $ ( \Omega , {\mathcal F} ) $ \widehat{X} = \int\limits _ {- \infty } ^ \infty xP( dx). Most commonly Quantum Probability Theory is de ned at the level of von Neumann algebras and nor-mal states. Naimark, J. Dixmier, R.V. . where $ \Omega $ Then one may go one step further, and consider a function $ f \in L _ \infty ( \Omega , {\mathcal F} , {\mathsf P} ) $ A (unital) *- algebra is a complex vector space A of operators on a Hilbert space H that. www.springer.com Kadison [a1], M. Tomita, M. Takesaki [a2], and others. They posit that there is a truly random component to quantum evolution, according to which every particle usually obeys the Schrödinger equation, but occasionally its wave function will spontaneously localize at some position in space. For instance, the situation of $ n $( By a quantum probability space one means any pair $ ( {\mathcal A} , \phi ) $, may be $ \mathbf N $, such that $ j _ {t} = \alpha _ {t} \circ j _ {0} $. A state P on A is a linear functional P : A → C (where C is the field of complex numbers) such that 0 ≤ P(a* a) for all a ∈ A (positivity) and P(I) = 1 (normalization). typically the real line with its Borel sets: $ ( \mathbf R , {\mathcal B} ) $. $ \mathbf R _ {+} $, such that $ \phi ^ \prime \circ E= \phi $ It is uncontroversial (though remarkable) that the formal apparatus ofquantum mechanics reduces neatly to a generalization of classicalprobability in which the role played by a Boolean algebra of events inthe latter is taken over by the “quantum logic” ofprojection operators on a Hilbert space. taking values in $ \Omega ^ \prime $. with trace 1. when viewed as a statement "about" $ X $, A random variable is a measurable function $ X $ continuous positive linear functional (a state) on $ {\mathcal A} $. occurs is $ \phi ( p) $. The probability measure $ {\mathsf P} $ Quantum probability theory is a generalization of probability theory in which random variables are not assumed to commute. into the pair $ ( L _ \infty ( \Omega , {\mathcal F} , {\mathsf P} ), {\mathsf P} ) $, is the von Neumann algebra $ L _ \infty ( \Omega , {\mathcal F} , {\mathsf P} ) $. The process is said to be in thermal equilibrium if $ \mathbf T= \mathbf R $ The vector space that describes spin-1/2 particles (and particles in quantum mechanics in general) is called a Hilbert space, which is basically a glorified Euclidean space. An event $ S ^ \prime \in {\mathcal F} ^ \prime $ as postulated in quantum mechanics. mentioned above extends to the imbedding $ j _ {X} $ becomes an event $ S \in {\mathcal F} $ or $ \mathbf R $.) implies $ a= 0 $), 1.1. given by $ j _ {X} ( f )= f\circ X $. In quantum mechanics, a probability amplitude is a complex number used in describing the behaviour of systems. The idempotents p ∈ A are the events in A, and P(p) gives the probability of the event p. "Classical (Kolmogorovian) and Quantum (Born) Probability", Association for Quantum Probability and Infinite Dimensional Analysis (AQPIDA), https://en.wikipedia.org/w/index.php?title=Quantum_probability&oldid=984409488, Articles needing expert attention with no reason or talk parameter, Articles needing expert attention from November 2008, Physics articles needing expert attention, Articles with unsourced statements from March 2018, Creative Commons Attribution-ShareAlike License, is closed under composition (a multiplication) and adjoint (an involution, This page was last edited on 19 October 2020, at 23:38. To see this, let $X$ be a finite set and let $p\colon X\to[0,1]$ be a probability distribution on $X$. comes from an event if and only if it is an orthogonal projection. Quantum mechanics provides many examples of the above structure. a probability measure on the measure space $ ( \Omega , {\mathcal F} ) $. A quantum stochastic process [a3] is a family $ ( j _ {t} : ( {\mathcal A} ^ \prime , \phi ^ \prime ) \rightarrow ( {\mathcal A} , \phi )) _ {t \in \mathbf T } $ if for $ a \in {\mathcal A} $, $ \phi ( a ^ {*} a)= 0 $ implies $ a= 0 $), a one-parameter group $ \sigma _ {t} ^ \phi $, $ t \in \mathbf R $, of automorphisms of $ ( {\mathcal A} , \phi ) $ is canonically determined by the mismatch between the quadratic forms $ ( a, b) \mapsto \phi ( a ^ {*} …