Letters in Mathematical Physics

, Volume 73, Issue 1, pp 1–15 | Cite as

p-Nuclearity in a New Perspective

  • Christopher J. FewsterEmail author
  • Izumi ojima
  • Martin Porrmann


In this Letter we try to settle some confused points concerning the use of the notion of p-nuclearity in the mathematical and physical literature, pointing out that the nuclearity index in the physicists’ sense vanishes for any p> 1. Our discussion of these issues suggests a new perspective, in terms of ε-entropy and operator spaces, which might permit connections to be drawn between phase space criteria and quantum energy inequalities.


algebraic quantum field theory nuclearity conditions operator spaces quantum energy inequalities. 


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  1. 1.
    Akashi, S. 1990The asymptotic behavior of ε-entropy of a compact positive operatorJ. Math. Anal. Appl153250257CrossRefGoogle Scholar
  2. 2.
    Amari, S.-I. and Nagaoka, H.: Methods of Information Geometry, Oxford University Press, Oxford, 2001. See also a remark at the end of a paper by one of the authors: Ojima, I., Temperature as order parameter of broken scale invariance, Publ. RIMS 40 (2004), 731–756.Google Scholar
  3. 3.
    Blecher, D. P. 1996A Generalization of Hilbert modulesJ. Funct. Anal136365421CrossRefGoogle Scholar
  4. 4.
    Buchholz, D. 1996Phase space properties of local observables and structure of scaling limitsAnn. Inst. Henri Poincaré – Physique théorique64433459Google Scholar
  5. 5.
    Buchholz, D., D’Antoni, C. 1995Phase space properties of charged fields in theories of local observablesRev. Math. Phys7527557CrossRefGoogle Scholar
  6. 6.
    Buchholz, D., D’Antoni, C., Longo, R. 1990Nuclear maps and modular structures II: Applications to quantum field theoryCommun. Math. Phys129115138Google Scholar
  7. 7.
    Buchholz, D., Jacobi, P. 1987On the nuclearity condition for massless fieldsLett. Math. Phys13313323CrossRefGoogle Scholar
  8. 8.
    Buchholz, D., Junglas, P. 1989On the existence of equilibrium states in local quantum field theoryCommun. Math. Phys121255270CrossRefGoogle Scholar
  9. 9.
    Buchholz, D., Porrmann, M. 1990How small is the phase space in quantum field theory?Ann Inst Henri Poincaré – Physique Théorique52237257Google Scholar
  10. 10.
    Buchholz, D., Wichmann, E.H. 1986Causal independence and the energy-level density of states in local quantum field theoryCommun Math Phys106321344CrossRefGoogle Scholar
  11. 11.
    D’Antoni, C. and Hollands, S.: Nuclearity, local quasiequivalence and split property for Dirac quantum fields in curved spacetime, 2004.[arXiv:math-ph/0106028 v3]Google Scholar
  12. 12.
    Effros, E., Ruan, Z.-J. 2000Operator SpacesOxford University PressOxfordGoogle Scholar
  13. 13.
    Eveson, S. P., Fewster, C. J., Verch, R. 2005Quantum inequalities in quantum mechanicsAnn. Henri Poincaré6130CrossRefMathSciNetGoogle Scholar
  14. 14.
    Fewster, C. J. and Hollands, S. Quantum energy inequalities in two-dimensional conformal field theory. [arXiv:math-ph/0412028].Google Scholar
  15. 15.
    Fidaleo, F.: Operator space structures and the split property, J. Operator Theory31: 1994), 207–218. See also Fidaleo, F.: On the split property for inclusions of W*-algebras, Proc. Amer. Math. Soc.130 (2002), 121–127.Google Scholar
  16. 16.
    Haag, R. 1996Local Quantum Physics2Springer-VerlagHeidelberg, Berlin, New YorkGoogle Scholar
  17. 17.
    Haag, R., Swieca, J. A. 1965When does a quantum field theory describe particles?Commun. Math. Phys1308320CrossRefGoogle Scholar
  18. 18.
    Jarchow, H.: Locally Convex Spaces, B. G. Teubner, Stuttgart, 1981.Google Scholar
  19. 19.
    Mohrdieck, S. 2002Phase space structure and short distance behavior of local quantum field theoriesJ. Math. Phys4335653574CrossRefGoogle Scholar
  20. 20.
    Pietsch, A. 1972Nuclear Locally Convex SpacesSpringer-VerlagHeidelberg, Berlin, New YorkGoogle Scholar
  21. 21.
    Pisier, G. 2003Introduction to Operator Space TheoryCambridge University PressCambridgeGoogle Scholar
  22. 22.
    Schauder, J. 1927Zur Theorie stetiger Abbildungen in FunktionalräumenMath. Z264765CrossRefGoogle Scholar
  23. 23.
    Schumann, R. 1996Operator ideals and the statistical independence in quantum field theoryLett. Math. Phys37249271Google Scholar
  24. 24.
    Singer, I. 1970Bases in Banach Spaces ISpringer-VerlagBerlin, Heidelberg, New YorkGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Christopher J. Fewster
    • 1
    Email author
  • Izumi ojima
    • 2
  • Martin Porrmann
    • 3
  1. 1.Department of MathematicsUniversity of YorkHeslingtonUnited Kingdom
  2. 2.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan
  3. 3.Institut für Theoretische PhysikUniversität HamburgHamburgGermany

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