q-Deformed Brownian Motion

Abstract

Brownian motion may be embedded in the Fock space of bosonic free fields in one dimension. Extending this correspondence to a family of creation and annihilation operators satisfying a q-deformed algebra, the notion of q-deformation is carried from the algebra to the domain of stochastic processes. The properties of q-deformed Brownian motion, in particular its non-Gaussian nature and cumulant structure, are established. CERN-TH-6838/93 March 1993 ∗ Permanent/Mailing address: CFMC, Av. Gama Pinto 2, 1699 Lisboa Codex, Portugal The concept of symmetry plays an essential role in the description of physical phenomena. In most cases this symmetry is related to covariance under the transformations induced by a Lie algebra. A generalization of this mathematical structure, the q-deformed (or quantum) algebras, has recently emerged. q-deformed algebras, first discovered in the context of integrable lattice models, were later identified as an underlying mathematical structure in topological field theories and rational conformal field theories. Other attempts to apply the notion of q-deformed algebras cover a wide range of different domains, from space-time symmetries to gauge fields, to quantum chemistry . In view of the actual and potential applications of q-deformation in the context of Lie algebras and superalgebras, it is interesting to ask whether the notion of q-deformation can also be extended to other (non-algebraic) mathematical structures. In this paper we try to extend this notion to stochastic processes. Our starting point is the well-known embedding of Brownian motion in the Fock space of bosonic free fields in one dimension. Extending this correspondence to a time family of creation and annihilation operators satisfying a q-deformed algebra we establish a q-deformation of Brownian motion. q-deformed creation and annihilation operators were defined by several authors. They satisfy the algebra aa − qaa = q (1.a) aa − qaa = q (1.b) where N is the number operator [N, a] = a [N, a] = −a (2) The operators aa may be realized as infinite-dimensional matrices on a vector space by a|n >= √ [n+ 1]|n+1 > a|n >= √ [n]|n−1 > N |n >= n|n > (3) where we used the notation [X] = Xq = sinh(X ln q) sinh(ln q) (4) X being a number or an operator. q-deformation of single boson operators is invariant under the replacement q → q and we write the algebra in an explicitly symmetric form which will be useful later on. aa − 1 2 (q + q)aa = 1 2 (q + q) (5) (Notice that (q + q) = [2]) We now consider a family { aτ , a † τ } of q-deformed operators labelled by a continuous time parameter and a scalar field φτ = aτ + a † τ (6)

Cite this paper

@inproceedings{Manko1993qDeformedBM, title={q-Deformed Brownian Motion}, author={V. I. Man’ko and Petr Nikolaevich Lebedev and R. Vilela Mendes}, year={1993} }