Skip to main content
SpringerLink
Log in

Irreversible dynamics of infinite fermion systems

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We investigate the irreversible dynamics of infinite systems as specified by completely positive, strongly continuous, one-parameter semigroups on a suitableC*-algebra. Having shown how to construct such a semigroup from a fairly general evolution equation we determine when the semigroup is spatial with respect to a given representation of the algebra. A special class of exactly soluble evolution equations on the CAR algebra is studied in detail in order to test conjectured extensions of the theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alfsen, E.M.: Borel structure on a metrizable Choquet simplex and on its extreme boundary. Math. Scand.19, 161–171 (1966)

    Google Scholar 

  2. Alfsen, E.M.: Compact convex sets and boundary integrals. Berlin-Heidelberg-New York: Springer 1971

    Google Scholar 

  3. Araki, H., Sewell, G.L.: KMS conditions and local thermodynamic stability of quantum lattice systems. Commun. math. Phys.52, 103–109 (1977)

    Google Scholar 

  4. Araki, H., Wyss, W.: Representations of canonical anticommutation relations. Helv. Phys. Acta37, 136–159 (1964)

    Google Scholar 

  5. Balslev, E., Verbeure, A.: States on Clifford algebras. Commun. math. Phys.7, 55–76 (1968)

    Google Scholar 

  6. Bratteli, O., Robinson, D.W.: Unbounded derivations in operator algebras. Commun. math. Phys.42, 253–268 (1975)

    Google Scholar 

  7. Chernoff, P.R.: Note on product formulas for operator semigroups. J. Funct. Anal.2, 238–242 (1968)

    Google Scholar 

  8. Davies, E.B.: Markovian master equations. Commun. math. Phys.39, 91–110 (1974)

    Google Scholar 

  9. Davies, E.B.: Quantum dynamical semigroups and the neutron diffusion equation. Rept. math. Phys.11, 169–188 (1977)

    Google Scholar 

  10. Davies, E.B.: First and second quantised neutron diffusion equations. Commun. math. Phys.52, 111–126 (1977)

    Google Scholar 

  11. Davies, E.B.: Dilations of completely positive maps. To appear

  12. Davies, E.B.: Quantum theory of open systems. New York: Academic Press 1976

    Google Scholar 

  13. Davies, E.B., Eckmann, J.-P.: Time decay for fermion systems with persistent vacuum. Helv. Phys. Acta48, 731–742 (1975)

    Google Scholar 

  14. Dell'Antonio, G.F.: Structure of the algebras of some free systems. Commun. math. Phys.9, 81–117 (1968)

    Google Scholar 

  15. Dixmier, J.: Les algèbres d'operateurs dans l'espace hilbertien. 2nd ed. Paris: Gauthier-Villars 1969

    Google Scholar 

  16. Doplicher, S., Kadison, R.V., Kastler, D., Robinson, D.W.: Asymptotically abelian systems. Commun. math. Phys.6, 101–120 (1967)

    Google Scholar 

  17. Doplicher, S., Powers, R.T.: On the simplicity of the even CAR algebra and free field models. Commun. math. Phys.7, 77–79 (1968)

    Google Scholar 

  18. Dubin, D.: Solvable models in algebraic statistical mechanics. Oxford: Univ. Press 1974

    Google Scholar 

  19. Emch, G.G.: Algebraic methods in statistical mechanics and quantum field theory. New York: Wiley-Interscience 1972

    Google Scholar 

  20. Evans, D.E.: Positive linear maps on operator algebras. Commun. math. Phys.48, 15–22 (1976)

    Google Scholar 

  21. Evans, D.E., Lewis, J.T.: Completely positive maps on someC*-algebras. Preprint

  22. Evans, D.E., Lewis, J.T.: Dilations of dynamical semigroups. Commun. math. Phys.50, 219–227 (1976)

    Google Scholar 

  23. Gorini, V., Kossakowki, A.: N-level systems in contact with a singular reservoir. J. Math. Phys.17, 1298–1305 (1976)

    Google Scholar 

  24. Haag, R., Hugenholtz, N.M., Winnink, M.: On the equilibrium states in quantum statistical mechanics. Commun. math. Phys.5, 215–236 (1967)

    Google Scholar 

  25. Haag, R., Kadison, R.V., Kastler, D.: Nets ofC*-algebras and classification of states. Commun. math. Phys.16, 81–104 (1970)

    Google Scholar 

  26. Haag, R., Kastler, D., Trych-Pohlmeyer, E.B.: Stability and equilibrium states. Commun. math. Phys.38, 173–193 (1974)

    Google Scholar 

  27. Haake, F.: Statistical treatment of open systems by generalised master equations. Springer tracts in modern physics, Vol. 66. Berlin-Heidelberg-New York: Springer 1973

    Google Scholar 

  28. Hoegh-Krohn, R.: Asymptotic fields in some models of quantum field theory II. J. Math. Phys.10, 639–643 (1969)

    Google Scholar 

  29. Hugenholtz, N.M., Kadison, R.V.: Automorphisms and quasi-free states on the CAR algebra. Commun. math. Phys.43, 181–197 (1975)

    Google Scholar 

  30. Kadison, R.V.: Transformations of states in operator theory and dynamics. Topology3 (Suppl. 2), 177–198 (1965)

    Google Scholar 

  31. Kato, T.: Perturbation theory for linear operators. Berlin-Heidelberg-New York: Springer 1966

    Google Scholar 

  32. Kraus, K.: General state changes in quantum theory. Ann. Phys.64, 311–335 (1971)

    Google Scholar 

  33. Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. math. Phys.48, 119–130 (1976)

    Google Scholar 

  34. Manuceau, J., Rocca, F., Testard, D.: On the product form of quasi-free states. Commun. math. Phys.12, 43–57 (1969)

    Google Scholar 

  35. Palmer, P.F.: The singular coupling and weak coupling limits. J. Math. Phys.18, 527–529 (1977)

    Google Scholar 

  36. Powers, R.T., Sakai, S.: Existence of ground states and KMS states for approximately inner dynamics. Commun. math. Phys.39, 273–288 (1975)

    Google Scholar 

  37. Powers, R.T., Størmer, E.: Free states of the canonical anticommutation relations. Commun. math. Phys.16, 1–33 (1970)

    Google Scholar 

  38. Pulè, J.: The Bloch equations. Commun. math. Phys.38, 241–256 (1974)

    Google Scholar 

  39. Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. 2. New York: Academic Press 1975

    Google Scholar 

  40. Robinson, D.W.: Statistical mechanics of quantum spin systems II. Commun. math. Phys.7, 21–32 (1970)

    Google Scholar 

  41. Robinson, D.W.: Return to equilibrium. Commun. math. Phys.31, 171–189 (1973)

    Google Scholar 

  42. Ruelle, D.: Statistical mechanics. New York: W.A. Benjamin Inc. 1969

    Google Scholar 

  43. Schatten, R.: A theory of cross-spaces. Princeton: Univ. Press 1950

    Google Scholar 

  44. Schrader, R., Uhlenbrock, D.A.: Markov structures on Clifford algebras. J. Funct. Anal.18, 369–413 (1975)

    Google Scholar 

  45. Shale, D., Stinespring, W.F.: States of the Clifford algebra. Ann. Math.80, 365–381 (1964)

    Google Scholar 

  46. Smithies, F.: Integral equations. Cambridge: Univ. Press 1962

    Google Scholar 

  47. Stinespring, W.F.: Positive functions onC*-algebras. Proc. Amer. Math. Soc.6, 211–216 (1955)

    Google Scholar 

  48. Størmer, E.: The even CAR algebra. Commun. math. Phys.16, 136–137 (1970)

    Google Scholar 

  49. Streater, R.F.: On certain non-relativistic quantised fields. Commun. math. Phys.7, 93–98 (1968)

    Google Scholar 

  50. Streater, R.F., Wilde, I.F.: The time evolution of quantised fields with bounded quasi-local interaction density. Commun. math. Phys.17, 21–32 (1970)

    Google Scholar 

  51. Takesaki, M.: Tomita's theory of modular Hilbert algebras and its applications. Lecture notes in math. Vol. 128. Berlin-Heidelberg-New York: Springer 1970

    Google Scholar 

  52. Yosida, K.: Functional analysis. Berlin-Heidelberg-New York: Springer 1965

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. John's College, OX1 3JP, Oxford, England

    E. B. Davies

Authors
  1. E. B. Davies
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by J.L. Lebowitz

Rights and permissions

Reprints and permissions

About this article

Cite this article

Davies, E.B. Irreversible dynamics of infinite fermion systems. Commun.Math. Phys. 55, 231–258 (1977). https://doi.org/10.1007/BF01614549

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01614549

Keywords

Search

Navigation