Skip to main content
SpringerLink
Log in

On the Stability of KMS States in Perturbative Algebraic Quantum Field Theories

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

Abstract

We analyze the stability properties shown by KMS states for interacting massive scalar fields propagating over Minkowski spacetime, recently constructed in the framework of perturbative algebraic quantum field theories by Fredenhagen and Lindner (Commun Math Phys 332:895, 2014). In particular, we prove the validity of the return to equilibrium property when the interaction Lagrangian has compact spatial support. Surprisingly, this does not hold anymore, if the adiabatic limit is considered, namely when the interaction Lagrangian is invariant under spatial translations. Consequently, an equilibrium state under the adiabatic limit for a perturbative interacting theory evolved with the free dynamics does not converge anymore to the free equilibrium state. Actually, we show that its ergodic mean converges to a non-equilibrium steady state for the free 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. Altherr, T.: Infrared problem in \({g\varphi^4}\) theory at finite temperature. Phys. Lett. B 238(24), 360–366 (1990)

  2. Araki H.: Relative Hamiltonian for faithful normal states of a von Neumann algebra. Publ. Res. Inst. Math. Sci. 9(1), 165–209 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bros J., Buchholz D.: Asymptotic dynamics of thermal quantum fields. Nucl. Phys. B 627, 289 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Bratteli O., Kishimoto A., Robinson D.W.: Stability properties and the KMS condition. Commun. Math. Phys. 61, 209–238 (1978)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Bratteli O., Robinson D.W.: Operator algebras and quantum statistical mechanics 2. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  6. Brunetti R., Duetsch M., Fredenhagen K.: Perturbative algebraic quantum field theory and the renormalization groups. Adv. Theor. Math. Phys. 13, 1541–1599 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brunetti R., Fredenhagen K.: Micrological analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys. 208, 623 (2000)

    Article  ADS  MATH  Google Scholar 

  8. Brunetti, R., Fredenhagen, K.: Quantum Field Theory on Curved Backgrounds, in Lecture Notes in Physics 786, ed. Springer (2009), pp. 129–155, Chapter 5.

  9. Brunetti R., Fredenhagen K., Köhler M.: The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes. Commun. Math. Phys. 180, 633 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Brunetti R., Fredenhagen K., Verch R.: The generally covariant locality principle: a new paradigm for local quantum physics. Commun. Math. Phys. 237, 31 (2003)

    Article  ADS  MATH  Google Scholar 

  11. Chilian B., Fredenhagen K.: The time-slice axiom in perturbative quantum field theory on globally hyperbolic spacetimes. Commun. Math. Phys. 287, 513–522 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Drago N., Hack T.-P., Pinamonti N.: The generalised principle of perturbative agreement and the thermal mass. Ann. Henri. Poincaré 18(3), 807–868 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Dütsch M., Fredenhagen K.: Causal perturbation theory in terms of retarded products, and a proof of the action ward identity. Rev. Math. Phys. 16, 1291 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Epstein, H., Glaser V.: The role of locality in perturbation theory. Ann. Inst. Henri Poincaré Section A, vol. XIX, n.3, 211 (1973)

  15. Fredenhagen K., Lindner F.: Construction of KMS states in perturbative QFT and renormalized Hamiltonian dynamics. Commun. Math. Phys. 332, 895 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Fredenhagen, K., Rejzner, K.: Perturbative algebraic quantum field theory, arXiv:1208.1428 [math-ph] (2012)

  17. Fredenhagen K., Rejzner K.: QFT on curved spacetimes: axiomatic framework and examples. J. Math. Phys. 57, 031101 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Gradshteyn I.S., Ryzhik I.M.: Table of Integrals, Series, and Products, Seventh Edition. Academic Press, Cambridge (2007)

    MATH  Google Scholar 

  19. Haag R.: Local Quantum Physics. Fields Particles Algebras. Text and Monographs in Physics. Springer, Berlin (1992)

    MATH  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Hollands S., Wald R.M.: Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223, 289 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Hollands S., Wald R.M.: Existence of local covariant time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Hollands S., Wald R.M.: On the renormalization group in curved spacetime. Commun. Math. Phys. 237, 123–160 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Jakšić V., Pillet C.-A.: On entropy production in quantum statistical mechanic. Commun. Math. Phys. 217, 285 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Jakšić V., Pillet C.-A.: Non-equilibrium steady states of finite quantum systems coupled to thermal reservoirs. Commun. Math. Phys. 226, 131–162 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Landsman N.P., van Weert C.G.: Real and imaginary time field theory at finite temperature and density. Phys. Rep. 145, 141 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  28. Le Bellac M.: Thermal Field Theory. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  29. Lindner, F.: Perturbative Algebraic Quantum Field Theory at Finite Temperature. Ph.D. thesis, University of Hamburg (2013).

  30. Ojima I., Hasegawa H., Ichiyanagi M.: Entropy production and its positivity in nonlinear response theory of quantum dynamical systems. J. Stat. Phys. 50, 633 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Ojima I.: Entropy production and non-equilibrium stationarity in quantum dynamical systems: physical meaning of van Hove limit. J. Stat. Phys. 56, 203 (1989)

    Article  ADS  Google Scholar 

  32. Ojima, I.: Entropy production and non-equilibrium stationarity in quantum dynamical systems. In: Proceedings of International Workshop on Quantum Aspects of Optical Communications. Lecture Notes in Physics 378, 164. Berlin: Springer (1991)

  33. Radzikowski M.J.: Micro-local approach To The Hadamard condition in quantum field theory on curved Space-time. Commun. Math. Phys. 179, 529 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Ruelle D.: Natural nonequilibrium states in quantum statistical mechanics. J. Stat. Phys. 98, 57–75 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  36. Steinmann, O.: Perturbation Expansions in Axiomatic Field Theory, Lect. Notes in Phys. 11. Berlin: Springer (1971)

  37. Steinmann O.: Perturbative quantum field theory at positive temperature: an axiomatic approach. Commun. Math. Phys. 170, 405–416 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Genova, Via Dodecaneso, 35, 16146, Genoa, Italy

    Nicolò Drago, Federico Faldino & Nicola Pinamonti

  2. Istituto Nazionale di Fisica Nucleare - Sezione di Genova, Via Dodecaneso, 33, 16146, Genoa, Italy

    Nicolò Drago & Nicola Pinamonti

Authors
  1. Nicolò Drago
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Federico Faldino
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nicola Pinamonti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicola Pinamonti.

Additional information

Communicated by D. Buchholz, K. Fredenhagen, Y. Kawahigashi

Dedicated to the memory of Rudolf Haag

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Drago, N., Faldino, F. & Pinamonti, N. On the Stability of KMS States in Perturbative Algebraic Quantum Field Theories. Commun. Math. Phys. 357, 267–293 (2018). https://doi.org/10.1007/s00220-017-2975-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-017-2975-x

Search

Navigation