Noncommutative Probability Theory Research Articles

Memorial: Nikolai Nikolaevitch Chentsov

Memorial: Nikolai Nikolaevitch Chentsov

Additivity of vector gleason measures

We study the degree of additivity of orthogonal Hilbert-space-valued measures on the latticeL(H) of all projections acting on a Hilbert spaceH. We present criteria for such measures to be completely additive and we establish the connection between the additivity of orthogonal measures and the size of almost disjoint families on dimH. [For example, we show that everyH-valued orthogonal measure is weaklyσ-additive iff (dimH)ω > dim H]. As a corollary we see that finitely additive orthogonal measures distinguish dimensions of Hilbert spaces (this can be viewed as a generalization of a theorem by Kruszynski). As a further corollary, we obtain that, for cardinalsκ, ν with κ>ν,κ≥3, there is no Jordan homomorphism from a typeI κ -factor into a typeI λ -factor. Finally, we show that every latticeL(H) with (dimH)ω = dimH admits a nonzero free orthogonal measure with values inH. Our results contribute to the noncommutative probability theory and also may find applications in the theory of the representation ofC *-algebras.

Two examples in noncommutative probability

A simple noncommutative probability theory is presented, and two examples for the difference between that theory and the classical theory are shown. The first example is the well-known formulation of the Heisenberg uncertainty principle in terms of a variance inequality and the second example is an interpretatio of the Bell paradox in terms of noncommuntative probability.

Operator algebras in statistical mechanics and noncommutative probability theory

The fundamental notions of statistical mechanics of quantum spin systems are introduced. A survey of the main properties of the states satisfying the Kubo-Martin-Schwinger boundary conditions is given. The problem of describing the invariant states and the first integrals for the multidimensional Heisenberg model is solved. A central limit theorem of noncommutative probability theory and a noncommutative analog of the individual ergodic theorem are formulated and proved. The asymptotics of the distribution of the eigenvalues of the multiparticle Schrodinger operator is studied.

Algebraic properties of complex fuzzy events in classical and in quantum information systems

Baer ∗-semigroups are regarded as the main abstract structures for an algebraic analysis of complex fuzzy events in generalized probability theory. This assumption is verified in the case of classical probability theory in the framework of measure and integration theory. The corresponding fuzzy language is extended to the non-commutative probability theory based on operators in Hilbert space. Starting from a quantum information system a quantum probability space is constructed, which is naturally embedded in a classical information system. In this last both exact than fuzzy quantum events are represented as classical fuzzy events. Lastly, the classical fuzzy events which correspond to exact quantum events are characterized by some minimality properties.

Sufficiency and strong commutants in quantum probability theory

A probability algebra (A, *, ω) consisting of a*algebraA with a faithful state ω provides a framework for an unbounded noncommutative probability theory. A characterization of symmetric probability algebra is obtained in terms of an unbounded strong commutant of the left regular representation ofA. Existence of coarse-graining is established for states that are absolutely continuous or continuous in the induced topology. Sufficiency of a*subalgebra relative to a family of states is discussed in terms of noncommutative Radon-Nikodym derivatives (a form of Halmos-Savage theorem), and is applied to couple of examples (including the canonical algebra of one degree of freedom for Heisenberg commutation relation) to obtain unbounded analogues of sufficiency results known in probability theory over a von Neumann algebra.

Conditional probabilities on orthomodular lattices

A definition of generalized probability on an orthomodular lattice which includes as particular cases the classical probability space and non-commutative probability theory on a von Neumann algebra is proposed. In this generalized structure the problem of conditioning with respect to Boolean σ-subalgebras is examined.

Phase-space distributions which are probability distributions

The article deals with non-commutative (boson) probability theory. The phase space considered is a linear symplectic space built up from a Hilbert space. Generating functionals for representations of the canonical commutation relations are related to characteristic functionals for random distributions on the underlying Hilbert space. An attempt is made to characterise those Levy-Khinchin characteristic functionals on phase space which are generating functionals by means of a generalised Heisenberg inequality.

Time-orthogonal unitary dilations and noncommutative Feynman-Kac formulae

An analysis of Feynman-Kac formulae reveals that, typically, the unperturbed semigroup is expressed as the expectation of a random unitary evolution and the perturbed semigroup is the expectation of a perturbation of this evolution in which the latter perturbation is effected by a cocycle with certain covariance properties with respect to the group of translations and reflections of the line. We consider generalisations of the classical commutative formalism in which the probabilistic properties are described in terms of non-commutative probability theory based on von Neumann algebras. Examples of this type are generated, by means of second quantisation, from a unitary dilation of a given self-adjoint contraction semigroup, called the time orthogonal unitary dilation, whose key feature is that the dilation operators corresponding to disjoint time intervals act nontrivially only in mutually orthogonal supplementary Hilbert spaces.

Markov fields in noncommutative probability theory on W* algebras

Many important results in ordinary probability theory do not have extensions in noncommutative probability theory. Here we consider Markov fields, and we prove that new Markov fields may be generated from old ones by means of multiplicative measurable operators. There is an analog of this result in ordinary probability theory involving multiplicative functionals. It is envisaged that the result established here would be of use in the study of interacting Fermion quantum fields.

Stochastic independence in non-commutative probability theory

AbstractFor a family {Xα} of random variables over a probability space , stochastic independence can be formulated in terms of factorization properties of characteristic functions. This idea is reformulated for a family {Aα} of selfadjoint operators over a probability gage space and is shown to be inappropriate as a non-commutative generalization. Indeed, such factorization properties imply that the {Aα} mutually commute and are versions of independent random variables in the usual sense.

Causal independence of free quantum fields

Causal independence of local observable algebras for free Dirac field, free electromagnetic field and free scalar massless field is proved. A method suggested for the proof is based on the noncommutative probability theory technique.

A Noncommutative Probability Theory

A noncommutative probability theory is developed in which no boundedness, finiteness, or “tracial” conditions are imposed. The underlying structure of the theory is a “probability algebra” $(\mathcal {a},\omega )$ where $\mathcal {a}$ is a *-algebra and $\omega$ is a faithful state on $\mathcal {a}$. Conditional expectations and coarse-graining are discussed. The bounded and unbounded commutants are considered and commutation theorems are proved. Two classes of probability algebras, which we call closable and symmetric probability algebras are shown to have important regularity properties. The canonical algebra of quantum mechanics is considered in some detail and a strong commutation theorem is proven for this case. Moreover, in this case, isotropic normal states, KMS states, and stable states are defined and characterized.

A noncommutative probability theory

A noncommutative probability theory is developed in which no boundedness, finiteness, or “tracial” conditions are imposed. The underlying structure of the theory is a “probability algebra” ( a , ω ) (\mathcal {a},\omega ) where a \mathcal {a} is a *-algebra and ω \omega is a faithful state on a \mathcal {a} . Conditional expectations and coarse-graining are discussed. The bounded and unbounded commutants are considered and commutation theorems are proved. Two classes of probability algebras, which we call closable and symmetric probability algebras are shown to have important regularity properties. The canonical algebra of quantum mechanics is considered in some detail and a strong commutation theorem is proven for this case. Moreover, in this case, isotropic normal states, KMS states, and stable states are defined and characterized.

Conditional expectations on von Neumann algebras: A new approach

In the framework of non-commutative probability theory on von Neumann algebras the concept of conditional expectation is redefined in such a way that it exists relative to arbitrary a priori distributions and subalgebras. We derive explicit expressions for the expectations relative to Abelian subalgebras and establish their connection with the concept of coarse-graining.

Markovian representations of current algebras

Hegerfeldt's (1974) concept of T-positivity in Euclidean random fields is generalised to non-commutative probability theory, that is, to Euclidean Fermi fields and to current algebra with possible Schwinger terms. The axioms imply the Wightman axioms. A non-Abelian form of Markovicity is introduced, and is shown to imply T-positivity if a reflection property holds. The investigation suggests a generalization of Nelson-Symanzik positivity, which might be valid in cases when the extension of the Schwinger functions to coinciding arguments is not expected to maintain both commutativity and positivity (or anti-commutativity and positivity).