Communications in Mathematical Physics

, Volume 357, Issue 1, pp 447–465 | Cite as

Categorial Subsystem Independence as Morphism Co-possibility

  • Zalán Gyenis
  • Miklós Rédei
Open Access


This paper formulates a notion of independence of subobjects of an object in a general (i.e., not necessarily concrete) category. Subobject independence is the categorial generalization of what is known as subsystem independence in the context of algebraic relativistic quantum field theory. The content of subobject independence formulated in this paper is morphism co-possibility: two subobjects of an object will be defined to be independent if any two morphisms on the two subobjects of an object are jointly implementable by a single morphism on the larger object. The paper investigates features of subobject independence in general, and subobject independence in the category of C*-algebras with respect to operations (completely positive unit preserving linear maps on C*-algebras) as morphisms is suggested as a natural subsystem independence axiom to express relativistic locality of the covariant functor in the categorial approach to quantum field theory.


  1. 1.
    Araki, H.: Mathematical Theory of Quantum Fields, volume 101 of International Series of Monograps in Physics. Oxford University Press, Oxford (1999) [Originally published in Japanese by Iwanami Shoten Publishers, Tokyo (1993)] Google Scholar
  2. 2.
    Arveson, W.: Subalgebras of C *-algebras. Acta Math. 123, 141–224 (1969)Google Scholar
  3. 3.
    Awodey S.: Category Theory, 2nd edn. Oxford University Press, Oxford (2010)MATHGoogle Scholar
  4. 4.
    Blackadar B.: Operator Algebras: Theory of C*-Algebras and von Neumann Algebras. Encyclopaedia of Mathematical Sciences, 1st edn. Springer, Berlin (2005)Google Scholar
  5. 5.
    Brunetti, R., Fredenhagen, K.: Algebraic approach to quantum field theory. In: Francoise, J.-P., Naber, G.L., Tsun, T.S. (eds.) Elsevier Encyclopedia of Mathematical Physics, pp. 198–204. Academic Press, Amsterdam (2006). arXiv:math-ph/0411072
  6. 6.
    Brunetti, R., Fredenhagen, K.: Quantum field theory on curved backgrounds. In: Bär, C., Fredenhagen, K. (eds.) Quantum Field Theory on Curved Spacetimes, volume 786 of Lecture Notes in Physics, chapter 5, pp. 129–155. Springer, Heidelberg (2009)Google Scholar
  7. 7.
    Brunetti R., Fredenhagen K., Paniz I., Rejzner K.: The locality axiom in quantum field theory and tensor products of C *-algebras. Rev. Math. Phys. 26, 1450010 (2014). arXiv:1206.5484 [math-ph]MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Brunetti R., Fredenhagen K., Verch R.: The generally covariant locality principle—a new paradigm for local quantum field theory. Commun. Math. Phys. 237, 31–68 (2003) arXiv:math-ph/0112041 ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Buchholz, D.: Algebraic quantum field theory: a status report. In: Grigoryan, A., Fokas, A., Kibble, T., Zegarlinski, B. (eds.) XIIIth International Congress on Mathematical Physics, Imperial College, London, UK, pp. 31–46. International Press of Boston, Sommervile, MA, USA (2001). arXiv:math-ph/0011044
  10. 10.
    Buchholz D., Haag R.: The quest for understanding in relativistic quantum physics. J. Math. Phys. 41, 3674–3697 (2000) arXiv:hep-th/9910243 ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Buchholz D., Summers S.J.: Quantum statistics and locality. Phys. Lett. A 337, 17–21 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Franz, U.: What is stochastic independence? In: Obata, N., Matsui, T., Hora, A., Kenkyūjo, S.K., Daigaku, K. (eds.) Non-commutativity, Infinite-dimensionality and Probability at the Crossroads: Proceedings of the RIMS Workshop on Infinite-Dimensional Analysis and Quantum Probability: Kyoto, Japan, 20–22 November, 2001, QP–PQ Quantum Probability and White Noise Analysis, pp. 254–274. World Scientific (2002). arXiv:math/0206017
  13. 13.
    Fredenhagen K.: Lille 1957: the birth of the concept of local algebras of observables. Eur. Phys. J. H 35, 239–241 (2010)CrossRefGoogle Scholar
  14. 14.
    Fredenhagen K., Reijzner K.: Quantum field theory on curved spacetimes: axiomatic framework and examples. J. Math. Phys. 57, 031101 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fredenhagen, K., Rejzner K.: Local covariance and background independence. In: Finster, F., Müller, O., Nardmann, M., Tolksdorf, J., Zeidler, E. (eds.) Quantum Field Theory and Gravity. Conceptual and Mathematical Advances in the Search for a Unified Framework, pp. 15–24. Birkhäuser, Springer, Basel (2012). arXiv:1102.2376 [math-ph]
  16. 16.
    Haag R.: Local Quantum Physics: Fields, Particles, Algebras. Springer, Berlin (1992)CrossRefMATHGoogle Scholar
  17. 17.
    Haag, R.: Discussion of the ‘axioms’ and the asymptotic properties of a local field theory with composite particles. Eur. Phys. J. H. 35, 243–253 (2010) (English translation and re-publication of a talk given at the international conference on mathematical problems of the quantum theory of fields, Lille, June 1957) Google Scholar
  18. 18.
    Haag R.: Local algebras. A look back at the early years and at some achievements and missed opportunities. Eur. Phys. J. H 35, 255–261 (2010)CrossRefGoogle Scholar
  19. 19.
    Haag R., Kastler D.: An algebraic approach to quantum field theory. J. Math. Phys. 5, 848–861 (1964)ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Horuzhy S.S.: Introduction to Algebraic Quantum Field Theory. Kluwer Academic Publishers, Dordrecht (1990)Google Scholar
  21. 21.
    Kalmbach G.: Orthomodular Lattices. Academic Press, London (1983)MATHGoogle Scholar
  22. 22.
    Kraus K.: States, Effects and Operations, volume 190 of Lecture Notes in Physics. Springer, New York (1983)Google Scholar
  23. 23.
    Maeda F.: Direct sums and normal ideals of lattices. J. Sci. Hiroshima Univ. Ser. A 14, 85–92 (1949)MathSciNetMATHGoogle Scholar
  24. 24.
    Paulsen V.: Completely Bounded Maps and Operator Algebras, volume 78 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  25. 25.
    Pierce, R.S.: Introduction to the Theory of Abstract Algebras. Dover Publications, New York (2014) (Originally published by Holt, Rinehart and Winston, Inc., New York, 1968) Google Scholar
  26. 26.
    Rédei M.: Logical independence in quantum logic. Found. Phys. 25, 411–422 (1995)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Rédei M.: Logically independent von Neumann lattices. Int. J. Theor. Phys. 34, 1711–1718 (1995)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Rédei, M.: Quantum Logic in Algebraic Approach, volume 91 of Fundamental Theories of Physics. Kluwer Academic Publisher, Dordrecht (1998)Google Scholar
  29. 29.
    Rédei M.: Operational independence and operational separability in algebraic quantum mechanics. Found. Phys. 40, 1439–1449 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Rédei M.: A categorial approach to relativistic locality. Stud. Hist. Philos. Mod. Phys. 48, 137–146 (2014)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Rédei, M.: Categorial local quantum physics. In: Butterfield, J., Halvorson, H., Rédei, M., Kitajima, J., Buscemi, F., Ozawa, M. (eds.) Reality and Measurement in Algebraic Quantum Theory, Proceedings in Mathematics and Statistics (PROMS). Springer (2018) (forthcoming) Google Scholar
  32. 32.
    Rédei, M., Summers, S.J.: When are quantum systems operationally independent? Int. J. Theor. Phys. 49, 3250–3261 (2010)Google Scholar
  33. 33.
    Rédei, M., Valente, G.: How local are local operations in local quantum field theory? Stud. Hist. Philos. Mod. Phys. 41, 346–353 (2010)Google Scholar
  34. 34.
    Summers S.J.: On the independence of local algebras in quantum field theory. Rev. Math. Phys. 2, 201–247 (1990)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Summers S.J.: Subsystems and independence in relativistic microphysics. Stud. Hist. Philos. Mod. Phys. 40, 133–141 (2009). arXiv:0812.1517 [quant-ph]CrossRefMATHGoogle Scholar
  36. 36.
    Summers, S.J.: A perspective on constructive quantum field theory (2012). arXiv:1203.3991 [math-ph] (This is an expanded version of an article commissioned for UNESCO’s Encyclopedia of Life Support Systems (EOLSS))
  37. 37.
    Weinberg S.: The Quantum Theory of Fields, vol. 1: Foundations. Cambridge University Press, Cambridge (1995)CrossRefMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of AlgebraBudapest University of Technology and EconomicsBudapestHungary
  2. 2.Department of LogicEötvös Loránd UniversityBudapestHungary
  3. 3.Department of Philosophy, Logic and Scientific MethodLondon School of Economics and Political ScienceLondonUK

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