Spectral families of quantum stochastic integrals

Yu.M. Berezanski, N. V. Zhernakov, Georgi F. Us

David Applebaum

A concept of quantum martingale measure is introduced and examples are constructed as quantum stochastic spectral integrals in Fock space. These are then utilized as space–time noise to drive a parabolic stochastic partial differential equation (spde). We establish the existence and uniqueness of the solutions as families of densely defined closable operators in Fock space that are jointly continuous in time and space variables and satisfy a Markov property.

Tomaž Prosen

A family of reversible deterministic cellular automata, including the rules 54 and 201 of [Bobenko et al., Commun. Math. Phys. 158, 127 (1993)] as well as their kinetically constrained quantum (unitary) or stochastic deformations, is shown to correspond to integrable Floquet circuit models with local interactions round-a-face. Using inhomogeneous solutions of the star-triangle relation with a one or two dimensional spectral parameter, changing their functional form depending on the orientation, we provide an explicit construction of the transfer matrix and establish its conservation law and involutivity properties. Integrability is independently demonstrated by numerically exploring the spectral statistics via the Berry-Tabor conjecture. Curiously, we find that the deformed rule 54 model generically possesses no other local conserved quantities besides the net soliton current.

MARTIN BROOKS

Atsuo Kuniba, Masato Okado, Satoshi W. Watanabe

We present a brief review on integrability of multispecies zero range process in one dimension introduced recently.The topics range over stochastic R matrices of quantum affine algebra U q A (1) n , matrix product construction of stationary states for periodic systems, q-boson representation of Zamolodchikov-Faddeev algebra, etc.We also introduce new commuting Markov transfer matrices having a mixed boundary condition and prove the factorization of a family of R matrices associated with the tetrahedron equation and generalized quantum groups at a special point of the spectral parameter.

Andrzej Łuczak