Local incompatibility of the microlocal spectrum condition with the KMS property along spacelike directions in quantum field theory on curved spacetime

  • Nicola Pinamonti
  • Ko SandersEmail author
  • Rainer Verch


States of a generic quantum field theory on a curved spacetime are considered which satisfy the KMS condition with respect to an evolution associated with a complete (Killing) vector field. It is shown that at any point where the vector field is spacelike, such states cannot satisfy a certain microlocal condition which is weaker than the microlocal spectrum condition in the case of asymptotically free fields.


Quantum field theory KMS condition Microlocal spectrum condition 

Mathematics Subject Classification




NP thanks the Institute for Theoretical Physics of the University of Leipzig for the kind hospitality during the preparation of this work and the DAAD for supporting this visit with the program “Research Stays for Academics 2017.” KS thanks Christian Gérard for bringing his notes on this result to the attention of NP and RV


  1. 1.
    Bisognano, J.J., Wichmann, E.H.: On the duality condition for quantum fields. J. Math. Phys. 17, 303–321 (1976)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Borchers, H.J.: Theory of Local Observables and KMS Condition. Institut für Theoretische Physik, University of Göttingen. (1994)
  3. 3.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, vols. 1 and 2. Springer, Berlin (1987, 1997)Google Scholar
  4. 4.
    Brunetti, R., Fredenhagen, K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys. 208, 623 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Buchholz, D., Verch, R.: Macroscopic aspects of the Unruh effect. Class. Quantum Gravity 32, 245004 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fewster, C.J., Verch, R.: Stability of quantum systems at three scales: passivity, quantum weak energy inequalities and the microlocal spectrum condition. Commun. Math. Phys. 240, 329–375 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fewster, C.J., Verch, R.: The necessity of the Hadamard condition. Class. Quantum Gravity 30, 235027 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fredenhagen, K., Haag, R.: Generally covariant quantum field theory and scaling limits. Commun. Math. Phys. 108, 91 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gransee, M., Pinamonti, N., Verch, R.: KMS-like properties of local equilibrium states in quantum field theory. J. Geom. Phys. 117, 15–35 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Haag, R.: Local Quantum Physics, 2nd edn. Springer, Berlin (1992). ISBN: 3-540-61451-6Google Scholar
  12. 12.
    Haag, R., Hugenholtz, N., Winnink, M.: On the equilibrium state in quantum statistical mechanics. Commun. Math. Phys. 5, 215–236 (1967)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hollands, S., Wald, R.M.: Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223, 289 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hollands, S., Wald, R.M.: Existence of local covariant time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators I, 2nd edn. Springer, Berlin (1989). ISBN: 3-540-00662-1Google Scholar
  16. 16.
    Kay, B.S.: The double wedge algebra for quantum fields on Schwarzschild and Minkowski space-times. Commun. Math. Phys. 100, 57 (1985)ADSCrossRefzbMATHGoogle Scholar
  17. 17.
    Kay, B.S., Wald, R.M.: Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon. Phys. Rep. 207, 49–136 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pedersen, G.K.: \(C^*\)-Algebras and their Automorphism Groups. Lecture Notes in Mathematics. Academic Press, London (1979)Google Scholar
  19. 19.
    Radzikowski, M.J.: Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179, 529 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sahlmann, H., Verch, R.: Passivity and microlocal spectrum condition. Commun. Math. Phys. 214, 705–731 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sahlmann, H., Verch, R.: Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime. Rev. Math. Phys. 13, 1203–1246 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sanders, K.: Equivalence of the (generalised) hadamard and microlocal spectrum condition for (generalised) free fields in curved spacetime. Commun. Math. Phys. 295, 485–501 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sanders, K.: The locally covariant Dirac field. Rev. Math. Phys. 22, 381–430 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sanders, K.: Thermal equilibrium states of a linear scalar quantum field in stationary spacetimes. Int. J. Mod. Phys. A 28, 1330010 (2013)ADSCrossRefzbMATHGoogle Scholar
  25. 25.
    Sanders, K.: On the construction of Hartle-Hawking-Israel states across a static bifurcate Killing horizon. Lett. Math. Phys. 105, 575–640 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Strohmaier, A., Verch, R., Wollenberg, M.: Microlocal analysis of quantum fields on curved space-times: analytic wave front sets and Reeh–Schlieder theorems. J. Math. Phys. 43, 5514–5530 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Trebels, S.: Über die geometrische Wirkung modularer Automorphismen: Analyse in Algebraischer Quantenfeldtheorie. Ph.D. thesis, Department of Physics, University of Göttingen (1997)Google Scholar
  28. 28.
    Verch, R.: Wavefront sets in algebraic quantum field theory. Commun. Math. Phys. 205, 337–367 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Wald, R.M.: General Relativity. University of Chicago Press, Chicago (1984)CrossRefzbMATHGoogle Scholar
  30. 30.
    Wald, R.M.: Quantum Field Theory in Curved Space-Time and Black Hole Thermodynamics. University of Chicago Press, Chicago (1995)Google Scholar

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Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di GenovaGenoaItaly
  2. 2.Istituto Nazionale di Fisica Nucleare - Sezione di GenovaGenoaItaly
  3. 3.School of Mathematical Sciences and Centre for Astrophysics and RelativityDublin City UniversityDublin 9Ireland
  4. 4.Institute for Theoretical PhysicsUniversity of LeipzigLeipzigGermany

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