Classical and Quantum Probability: The Two Logics of Science

  • Philippe BlanchardEmail author
Part of the On Thinking book series (ONTHINKING, volume 4)


We review and discuss first briefly the algebraic framework of classical and quantum physics and commutative and noncommutative probability theory. After that we propose a mathematical definition of decoherence sufficiently general to accommodate quantum systems with infinitely many degrees of freedom and give an exhaustive list of possible scenarios that can emerge due to decoherence. We conclude with some messages of quantum science.


Classical Probability Quantum Probability Superselection Rule Algebraic Framework Classical Probability Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



In this paper I reported on results obtained together with Mario Hellmich and Robert Olkiewicz. It is a pleasure to express my gratitude to both of them for the joy of a long and friendly collaboration. I also benefited from numerous discussions with my colleague and friend Jürg Fröhlich during the ZIF-Research Group “The message of Quantum Science—Attempts towards a synthesis” (February–May 2012) at the Center for Interdisciplinary Research (ZiF) of Bielefeld University. I am grateful to Mario Hellmich for precious remarks and Hanne Litschewsky for help in preparing the manuscript.


  1. 1.
    Aharonov Y, Bergmann PG, Lebowitz JL (1964) Time symmetry in the quantum process of measurement. Phys Rev 134:B1410CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Allingham M (2002) Choice theory: a very short introduction. Oxford University Press, OxfordCrossRefGoogle Scholar
  3. 3.
    Araki H (1999) Mathematical theory of quantum field. Oxford University Press, OxfordGoogle Scholar
  4. 4.
    Blanchard Ph, Olkiewicz R (2006) Decoherence as irreversible dynamical process. In: Open quantum systems III. Lecture notes in mathematics, vol 1882, pp 117–160Google Scholar
  5. 5.
    Blanchard Ph, Olkiewicz R (2003) Decoherence induced transition from quantum to classical dynamics. Rev Math Phys 15:217–243CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Blanchard Ph, Olkiewicz R (2003) Decoherence-induced continuous pointer states. Phys Rev Lett 96:010403CrossRefGoogle Scholar
  7. 7.
    Blanchard Ph, Olkiewicz R (2003) From quantum to quantum via decoherence. Phys Lett A 314:29–36CrossRefADSzbMATHMathSciNetGoogle Scholar
  8. 8.
    Blanchard Ph, Hellmich M (2012) Decoherence in infinite quantum systems, quantum Africa 2010: theoretical and experimental foundations of recent quantum technology. AIP conference proceedings, vol 1469, pp 2–15Google Scholar
  9. 9.
    Blanchard Ph et al (2000) Decoherence: theoretical, experimental and conceptual problems. Lecture notes in physics, vol 538. Springer, BerlinGoogle Scholar
  10. 10.
    Borel E (1937) Valeur pratique et philosophique des probabilités. Gauthier-Villars, ParisGoogle Scholar
  11. 11.
    Bratelli O, Robinson D (1987) Operator algebras and quantum statistical mechanics I, 2nd edn. Springer, New YorkCrossRefGoogle Scholar
  12. 12.
    Duplantier B et al (2007) Quantum decoherence—poincarè seminar 2005. Progress in mathematical physics. Birkhäuser, BaselzbMATHGoogle Scholar
  13. 13.
    Emch G (1972) Algebraic methods in statistical mechanics and quantum field theory. Wiley, New YorkzbMATHGoogle Scholar
  14. 14.
    Fröhlich J, Schubnel B (2013) Do we understand quantum mechanics—finally? In: Erwin Schrödinger – 50 years after. ESI lectures in mathematics and physics. European Mathematical Society, Zürich, pp 37–84CrossRefGoogle Scholar
  15. 15.
    Giulini D et al (1996) Decoherence and the appearance of a classical world in quantum theory. Springer, BerlinCrossRefzbMATHGoogle Scholar
  16. 16.
    Haag E (1996) Local quantum physics, 2nd edn. Springer, BerlinCrossRefzbMATHGoogle Scholar
  17. 17.
    Haag R, Kastler D (1964) An algebraic approach to quantum field theory. J Math Phys 5:848–861CrossRefADSzbMATHMathSciNetGoogle Scholar
  18. 18.
    Haroche S, Raimond JM (2006) Exploring the quantum: atoms, cavities and photons. In: Oxford graduate texts. Oxford University Press, OxfordCrossRefGoogle Scholar
  19. 19.
    Hellmich M (2011) Quantum dynamical semigroups and decoherence. Adv Math Phys 2011:625978CrossRefMathSciNetGoogle Scholar
  20. 20.
    Laloë F (2012) Do we really understand quantum mechanics. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  21. 21.
    Lugiewicz P, Olkiewicz R (2003) Classical properties of infinite quantum open systems. Commun Math Phys 208:245–265MathSciNetGoogle Scholar
  22. 22.
    Omnés R (1999) Understanding quantum mechanics. Princeton University Press, PrincetonzbMATHGoogle Scholar
  23. 23.
    Penrose R (1997) The large, the small and the human mind. Cambridge University Press, CambridgezbMATHGoogle Scholar
  24. 24.
    Rauch H, Werner A (2000) Neutron interferometry: lessons in experimental quantum mechanics. Oxford University Press, OxfordGoogle Scholar
  25. 25.
    Sakai S (1991) Operator algebras in dynamical systems. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  26. 26.
    Segal IE (1947) Postulates for general quantum mechanics. Ann Math 48:930–948CrossRefzbMATHGoogle Scholar
  27. 27.
    Strocchi F (2005) An introduction to the mathematical structure of quantum mechanics. World Scientific, SingaporeCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Mathematical Physics and BiBoSUniversity of BielefeldBielefeldGermany

Personalised recommendations