Abstract
We show that using the thermodynamic limit, one can give a simple and natural construction of noncommutative\(\mathbb{L}_p \) spaces for quantum systems on a lattice. Within this framework, we discuss the construction and ergodicity properties of stochastic dynamics of spin flip and diffusion type.
Similar content being viewed by others
References
HaagerupU.:L p-spaces associated with an arbitrary von Neumann algebra, inAlgèbres d'opérateurs et leurs applications en physique mathématique, Colloques internationaux du CNRS, No. 274, Marseille 20–24 juin 1977, Editions du CNRS, Paris, 1979, pp. 175–184.
Terp, M.:L p-spaces associated with von Neumann algebras, Københavns Universitet, Mathematisk Institut, Rapport No. 3 (1981).
KosakiH.: Application of the complex interpolation method to a von Neumann algebra (Non-commutativeL p-spaces),J. Funct. Anal. 56 (1984), 29–78.
NelsonE.: Notes on non-commutative integration,J. Funct. Anal. 15 (1974), 103–116.
StragierG., QuaegebeurJ., and VerbeureA.: Quantum detailed balance,Ann. Inst. Henri Poincaré 41 (1984), 25–36.
LiebE. H. and RobinsonD. W.: The finite group velocity of quantum spin systems,Comm. Math. Phys. 28 (1972), 251–257.
Majewski, A. W. and Zegarlinski, B.: Quantum stochastic dynamics I: Spin systems on a lattice, in preparation.
BratteliO. and RobinsonD. W.:Operator Algebras and Quantum Statistical Mechanics, Springer-Verlag, New York, Heidelberg, Berlin, vol. I (1979), vol. II (1981).
DixmierJ.: Formes linéaires sur un anneau d'opérateurs,Bull. Soc. Math. France. 81 (1953), 9–39.
SegalI. E.: A non-commutative extension of abstract integration,Ann. of Math. 57 (1953), 401–457.
HaagR.:Local Quantum Physics, Springer-Verlag, Berlin, Heidelberg, New York, 1992.
PowersR.: Representation of uniformly hyperfinite algebras and their associated von Neumann rings,Ann. Math. 86 (1967), 138–171.
KadisonR. V. and RingroseJ. R.:Fundamentals of the Theory of Operator Algebras, Academic Press, New York, vol. I (1983), vol. II (1986).
AccardiL.: Topics in quantum probability,Phys. Rep. 77 (1981), 169–192.
StroockD. W. and ZegarlinskiB.: The equivalence of the logarithmic Sobolev inequality and Dobrushin-Shlosman mixing condition,Comm. Math. Phys. 144 (1992), 303–323.
AizenmanM. and HolleyR.: Rapid convergence to equilibrium of stochastic Ising Models in the Dobrushin-Shlosman régime, in H.Kesten (ed),Percolation Theory and Ergodic Theory of Infinite Particle Systems, IMS Vol. Math. Appl., vol. 8, Springer-Verlag, Berlin, Heidelberg, New York, 1987, pp. 1–11.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Majewski, A.W., Zegarlinski, B. On quantum stochastic dynamics and noncommutative\(\mathbb{L}_p \) spaces. Lett Math Phys 36, 337–349 (1996). https://doi.org/10.1007/BF00714401
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00714401