R. L. Hudson

Quantum Ito's formula and stochastic evolutions

Using only the Boson canonical commutation relations and the Riemann-Lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes. This leads to a noncommutative Ito product formula, a realisation of the classical Poisson process in Fock space which gives a noncommutative central limit theorem, the construction of solutions of certain noncommutative stochastic differential equations, and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps. The classical Ito product formula for stochastic differentials with respect to Brownian motion and the Poisson process is a special case.

Fermion Ito's Formula and Stochastic Evolutions*

An Ito product formula is proved for stochastic integrals against Fermion Brownian motion, and used to construct unitary processes satisfying stochastic differential equations. As in the corresponding Boson theory [10, 11] these give rise to stochastic dilations of completely positive semigroups.

Quantum mechanical Wiener processes

Quantum mechanical analogues of Wiener processes are defined and their existence proved, in terms of which the field operators of extremal universally invariant representations of the canonical commutation relations are expressible as stochastic integrals. A noncommutative analogue of the Wiener transformations is constructed and shown to have properties analogous to the classical Wiener transformation.

Unification of Fermion and Boson Stochastic Calculus

Fermion annihilation and creation processes are explicitly realised in Boson Fock space as functions of the corresponding Boson processes and second quantisations of reflections. Conversely, Boson annihilation and creation processes can be constructed from the Fermion processes. The existence of unitary stochastic evolutions driven by Fermion and gauge noise is thereby reduced to an equivalent Boson problem, which is then solved.

A non-commutative martingale representation theorem for non-Fock quantum Brownian motion

Abstract A non-commutative theory of stochastic integration is constructed in which the integrators are the components of the quantum Brownian motion with non-unit variance. Unlike the unit variance (Fock) case, there is a Kunita-Watanabe type representation theorem for processes which are martingales with respect to the generated filtration.

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.

Analogs of de Finetti's theorem and interpretative problems of quantum mechanics

It is argued that the characterization of the states of an infinite system of indistinguishable particles satisfying Bose-Einstein statistics which follows from the quantum-mechanical analog of de Finettis theorem(2) can be used to interpret the nonuniqueness of the resolution into a convex combination of pure states of a quantum-mechanical mixed state.

Quantum diffusions and the noncommutative torus

Quantum diffusions driven by the creation and annihilation processes on the noncommutative torus algebra are considered. A cohomological obstruction to the existence of such a diffusion is overcome by constructing a diffusion on a larger algebra. There are implications for models in solid-state physics based on the noncommutative torus.

A quantum-mechanical functional central limit theorem

Continuing an earlier work [4], properties of canonical Wiener processes are investigated. An analog of the sample path continuity property is obtained. A noncommutative counterpart of weak convergence is formulated. Operator processes (Pn, Qn) analogous to the random-walk approximating processes of the Donsker invariance principle are defined in terms of a sequence (pi, qi) of pairs of quantum mechanical canonical observables satisfying hypotheses analogous to those of the classical central limit theorem. It is shown that Pn, Qn) converges weakly to a canonical Wiener process.

Double product integrals and Enriquez quantisation of Lie bialgebras II: The quantum Yang-Baxter equation ∗

AbstractFor a Lie algebra with Lie bracket got by taking commutators in a nonunital associative algebra