Discrete Approximation of the Free Fock Space

  • Stéphane AttalEmail author
  • Ion Nechita
Part of the Lecture Notes in Mathematics book series (LNM, volume 2006)


We prove that the free Fock space \(F(L^2 (\mathbb{R}^+;\mathbb{C}))\), which is very commonly used in Free Probability Theory, is the continuous free product of copies of the space \(\mathbb{C}^2\). We describe an explicit embedding and approximation of this continuous free product structure by means of a discrete-time approximation: the free toy Fock space, a countable free product of copies of \(\mathbb{C}^2\). We show that the basic creation, annihilation and gauge operators of the free Fock space are also limits of elementary operators on the free toy Fock space. When applying these constructions and results to the probabilistic interpretations of these spaces, we recover some discrete approximations of the semi-circular Brownian motion and of the free Poisson process. All these results are also extended to the higher multiplicity case, that is, \(F(L^2(\mathbb{R}^+;\mathbb{C}^N))\) is the continuous free product of copies of the space \(\mathbb{C}^{N+1}\).

Free probability Free Fock space Toy Fock space Limit theorems 


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We thank the referee for several helpful remarks that improved the presentation of the paper.


  1. 1.
    Attal, S.: Approximating the Fock space with the toy Fock space. Séminaire de Probabilités, XXXVI. Lecture Notes in Mathematics, vol. 1801, pp. 477–491. Springer, Berlin (2003)Google Scholar
  2. 2.
    Attal, S., Émery, M.: Équations de structure pour des martingales vectorielles. Séminaire de Probabilités, XXVIII. Lecture Notes in Mathematics, vol. 1583, pp. 256–278. Springer, Berlin (1994)Google Scholar
  3. 3.
    Attal, S., Pautrat, Y.: From (n + 1)-level atom chains to n-dimensional noises. Ann. Inst. H. Poincaré Probab. Statist. 41(3), 391–407 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Attal, S., Pautrat, Y.: From repeated to continuous quantum interactions. Ann. Henri Poincaré 7(1), 59–104 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Biane, P., Speicher, R.: Stochastic calculus with respect to free Brownian motion and analysis on Wigner space. Probab. Theor. Relat. Field. 112(3), 373–409 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bożejko, M., Speicher, R.: An example of a generalized Brownian motion. Comm. Math. Phys. 137(3), 519–531 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bruneau, L., Joye, A., Merkli, M.: Asymptotics of repeated interaction quantum systems. J. Funct. Anal. 239(1), 310–344 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bruneau, L., Pillet, C.-A.: Thermal relaxation of a QED cavity, preprint available at
  9. 9.
    Glockner, P., Schürmann, M., Speicher, R.: Realization of free white noises. Arch. Math. (Basel) 58(4), 407–416 (1992)Google Scholar
  10. 10.
    Hiai, F., Petz, D.: The semicircle law, free random variables and entropy. Mathematical Surveys and Monographs, vol. 77, 376 pp. American Mathematical Society, Providence, RI (2000)Google Scholar
  11. 11.
    Meyer, P.-A.: Quantum probability for probabilists, 2nd edn. Lecture Notes in Mathematics, vol. 1538, 312 pp. Springer (1995)Google Scholar
  12. 12.
    Nica, A., Speicher, R.: Lectures on the combinatorics of free probability. Cambridge University Press, Cambridge (2006)zbMATHCrossRefGoogle Scholar
  13. 13.
    Speicher, R.: A new example of “independence” and “white noise”, Probab. Theor. Relat. Field. 84(2), 141–159 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Voiculescu, D.V.: Lecture notes on free probability. Lecture Notes in Mathematics, vol. 1738, pp. 279–349, Springer, Berlin (2000)Google Scholar
  15. 15.
    Voiculescu, D.V., Dykema, K., Nica, A.: Free random variables. CRM Monographs Series No. 1. American Mathematical Society, Providence, RI (1992)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Institut Camille JordanUniversité de Lyon, Université de Lyon 1Villeurbanne CedexFrance

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