zbMATH — the first resource for mathematics

Quantum Poisson processes and dilations of dynamical semigroups. (English) Zbl 0684.60081
Summary: The notion of a quantum Poisson process over a quantum measure space is introduced. This process is used to construct new quantum Markov processes on the matrix algebra \(M_ n\) with stationary faithful state \(\phi\). If (\({\mathcal M},\mu)\) is the quantum measure space in question (\({\mathcal M}^ a \)von Neumann algebra and \(\mu\) a faithful normal weight), then the semigroup \(e^{tL}\) of transition operators on \((M_ n,\phi)\) has generator \[ L: M_ n\to M_ n: a\to i[h,a]+(id\otimes \mu)(u^*(a\otimes\mathbf{1})u-a\otimes\text\textbf{1}), \] where u is an arbitrary unitary element of the centraliser of \((M_ n\otimes {\mathcal M},\phi \otimes \mu)\).

60K35 Interacting random processes; statistical mechanics type models; percolation theory
81P20 Stochastic mechanics (including stochastic electrodynamics)
Full Text: DOI
[1] Accardi, L., Parthasarathy, K.R.: Stochastic calculus on local algebras. (Lect. Notes Math., vol. 1136, pp. 9–23) Berlin Heidelberg New York: Springer 1985 · Zbl 0584.60075
[2] Appelbaum, D., Hudson, R.L.: Fermion Itô’s formula and stochastic evolutions. Commun. Math. Phys.96, 473–496 (1984) · Zbl 0572.60052
[3] Araki, H.: Factorizable representations of current algebra. Publ. RIMS Kyoto Univ.5, 361–442 (1970) · Zbl 0238.22014
[4] Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics II. Berlin Heidelberg New York: springer 1981 · Zbl 0463.46052
[5] Barnett, C., Streater, R.F., Wilde, I.F.: The Itô-Clifford integral. J. Funct. Anal.48, 172–212 (1982) · Zbl 0492.46051
[6] Davies, E.B.: A model of heat conduction. J. Stat. Phys.16, 161–170 (1978)
[7] Dümcke, R.: The low-density limit for anN-level system interacting with a free Bose or Fermi gas. Commun. Math. Phys.97, 331–359 (1985) · Zbl 0614.46069
[8] Evans, M., Hudson, R.L.: Multidimensional quantum diffusions. (Lect. Notes. Math., vol. 1303, pp. 69–88) Berlin Heidelberg New York: Springer 1988 · Zbl 0648.46056
[9] Frigerio, A., Gorini, V.: Markov dilations and quantum detailed balance. Commun. Math. Phys.93, 517 (1984) · Zbl 0553.46044
[10] Frigerio, A.: Covariant Markov dilations of quantum dynamical semigroups. Publ. RIMS Kyoto Univ.21, 657–675 (1985) · Zbl 0593.60092
[11] Frigerio, A.: Construction of stationary quantum Markov process through quantum stochastic calculus. (Lect. Notes Math., vol. 1136, pp. 207–222) Berlin Heidelberg New York: Springer 1985
[12] Frigerio, A.: Quantum Poisson processes: physical motivations and applications. (Lect. Notes Math., vol. 1303, pp. 107–127) Berlin Heidelberg New York: Springer 1988 · Zbl 0635.60095
[13] Hudson, R.L., Lindsay, J.M.: Use of non-Fock quantum Brownian motion and a quantum martingale representation theorem. (Lect. Notes Math., vol. 1136, pp. 276–305) Berlin Heidelberg New York: Springer 1985 · Zbl 0569.60055
[14] Hudson, R.L., Parthasarathy, K.R.: Quantum Itô’s formula and stochastic evolutions. Commun. Math. Phys.93, 301–323 (1984) · Zbl 0546.60058
[15] Hudson, R.L., Parthasarathy, K.R.: Stochastic dilations of uniformly continuous completely positive semigroups. Acta Appl. Math.2, 353–378 (1984) · Zbl 0541.60066
[16] Kossakowski, A., Frigerio, A., Gorini, V., Verri, M.: Quantum detailed balance and KMS condition. Commun. Math. Phys.57, 97–110 (1977) · Zbl 0374.46060
[17] Kümmerer, B.: Markov dilations onW *-algebras. J. Funct. Anal.63, 139–177 (1985) · Zbl 0601.46062
[18] Kümmerer, B.: Markov dilations and non-commutative Poisson processes; preprint, Tübingen 1986. See also: Kümmerer, B.: Survey on a Theory of non-commutative stationary Markov processes. (Lect. Notes Math., vol. 1303, pp. 154–182) Berlin Heidelberg New York: Springer 1988
[19] Kümmerer, B.: On the structure of Markov dilations onW *-algebras. (Lect. Notes Math., vol. 1136, pp. 318–331) Berlin Heidelberg New York: Springer 1985
[20] Kümmerer, B., Maassen, H. The essentially commutative dilations of dynamical semigroups onM n . Commun. Math. Phys.109, 1–22 (1987) · Zbl 0627.60015
[21] Lindsay, J.M., Maassen, H.: The stochastic calculus of Bose noise. Preprint, Nijmegen 1988 · Zbl 0652.60068
[22] Lindsay, J.M.: Ph.D. thesis, Nottingham 1985
[23] Maassen, H.: Quantum Markov processes on Fock space described by integral kernels. (Lect. Notes Math., vol. 1136, pp. 361–374) Berlin Heidelberg New York: Springer 1985
[24] Streater, R.F., Wulfsohn, A.: Continuous tensor products of Hilbert space and generalized random fields, Il Nuovo Cimento57B, 330–339 (1968) · Zbl 0162.21302
[25] Takesaki, M.: Conditional expectations in von Neumann algebras. J. Funct. Anal.9, 306–321 (1971) · Zbl 0245.46089
[26] Takesaki, M.: Tomita’s theory of modular Hilbert algebras and its applications. (Lect. Notes Math., vol. 128) Berlin Heidelberg New York: Springer 1970 · Zbl 0193.42502
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.