Quantum Stochastic Analysis After Four Decades

  • R. L. Hudson
Part of the Lecture Notes in Statistics book series (LNS, volume 128)


This is a personal view of the development of quantum stochastic analysis from early days to the present time, with particular emphasis on quantum stochastic calculus.


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  1. [A]
    D B Applebaum, Stochastic dilations of the Bloch Equation in Boson and Fermion noise, J Phys A 19(1986) 937–959.MathSciNetMATHCrossRefGoogle Scholar
  2. [AFL]
    L Accardi, A Frigerio and Y G Lu, The weak coupling limit as a quantum central limit, Commun Math Phys 131(1991) 537–576.MathSciNetCrossRefGoogle Scholar
  3. [AGL]
    L Accardi, J Gough and Y G Lu, On the stochastic limit for quantum theory, Rep Math Phys 36(1995) 155–187.MathSciNetMATHCrossRefGoogle Scholar
  4. [AL]
    L Accardi and Y G Lu, On the weak coupling limit for quantum electrodynamics, pp 16–22, in Probabilistic methods in mathematical physics, ed F Guerra et al, World Scientific (1992).Google Scholar
  5. [AM]
    S Attal and P-A Meyer, Interpretation probabiliste et extension des integrales stochastiques noncommutatives, Strasbourg preprint (1994).Google Scholar
  6. [AV]
    L Accardi and I Volovich, The stochastic limit of quantum field theory, Rome II preprint (1994).Google Scholar
  7. [BHH]
    V P Belavkin, O Hirota and R L Hudson (eds) Quantum Communication and Measurement, Plenum (1995).Google Scholar
  8. [BL]
    A Barchielli and G Lupieri, Quantum stochastic calculus, operation valued stochastic processes and continual measurements in quantum mechanics, J Math Phys 26(1985) 2222–2230.MathSciNetMATHCrossRefGoogle Scholar
  9. [BL]
    V P Belavkin and P Staszewski, Nondemolition measurement of a free quantum particle, Phys Rev A 45(1992) 1347–1356.Google Scholar
  10. [BS]
    C Barnett, R F Streater and I Wilde, The Ito Clifford integral, J Fund Anal 48(1982) 172–212.Google Scholar
  11. [CC]
    A Connes and J Cuntz, Quasi-homomorphismes, cohomologie cyclique et positivité, Commun Math Phys 114(1988) 515–526.MathSciNetMATHCrossRefGoogle Scholar
  12. [CF]
    A Chebotarev and F Fagnola, Sufficient conditions for conservativity of quantum dynamical semigroups, J Funct Anal 118(1993) 113–153.MathSciNetCrossRefGoogle Scholar
  13. [CH]
    A M Cockroft and R L Hudson, Quantum mechanical Wiener processes, J Multivariate Anal 7(1977) 107–124.MathSciNetMATHCrossRefGoogle Scholar
  14. [CoH]
    P Beazley Cohen and R L Hudson, Generators of quantum stochastic flows, Cuntz morphisms and cyclic cohomology, Nottingham preprint (1994).Google Scholar
  15. [EH1]
    M P Evans and R L Hudson, Perturbations of quantum diffusions, J London Math Soc (2) 41 (1990) 373–384.Google Scholar
  16. [EH2]
    M P Evans and R L Hudson, Multidimensional quantum diffusions, pp 69–88, in Quantum Probability IV, ed L Accardi et al, Springer LNM 1303 (1988).Google Scholar
  17. [Ev]
    M P Evans, Existence of quantum diffusions, Prob Theory and Related Fields 81(1989) 473–483.MATHCrossRefGoogle Scholar
  18. [Ey]
    T M W Eyre, Chaotic expansion for Lie superalgebras in quantum stochastic calculus, Nottingham preprint (1996).Google Scholar
  19. [EyH]
    T M W Eyre and R L Hudson, Representation of Lie superalgebras and generalized Boson-Fermion equivalence in quantum stochastic calculus, Nottingham preprint (1996), to appear in Commun Math Phys. Google Scholar
  20. [Fo]
    V A Fock, Konfigurationsraum und zweite Quantelung, Z Physik 75, 622–647 (1932).MATHCrossRefGoogle Scholar
  21. [H1]
    R L Hudson, The strong Markov property for canonical Wiener processes, J Funct Anal 37 (1980) 68–87.Google Scholar
  22. [H2]
    R L Hudson, Quantum diffusions and cohomology of algebras, pp 479–485, Proceedings of 1st World Congress of Bernoulli Society, Tashkent 1986, Vol 1, ed Yu Prohorov et al, VNU (1987).Google Scholar
  23. [HIP]
    R L Hudson, PDF Ion and K R Parthasarathy, Time orthogonal unitary dilations and noncommutative Feynman Kac formulae, Commun Math Phys 83(1982) 261–280.MathSciNetMATHCrossRefGoogle Scholar
  24. [HLi]
    R L Hudson and J M Lindsay, On characterizing quantum stochastic evolutions, Math Proc Camb Phil Soc 102(1987) 363–369.MathSciNetMATHCrossRefGoogle Scholar
  25. [Ho]
    G Hochschild, On the cohomology groups of an associative algebra, Ann of Math 46(1945) 58–67.MathSciNetMATHCrossRefGoogle Scholar
  26. [HP1.
    R L Hudson and K R Parthasarathy, Quantum diffusions, pp 111–121 in Theory and applications of random fields, proceedings, Bangalore 1982, ed V Kallianpur, Springer LN Control theory and IS 49 (1983).Google Scholar
  27. [HP2]
    R L Hudson and K R Parthasarathy, Quantum Ito’s formula and stochastic evolutions, Commun Math Phys 93 (1984) 301–323.Google Scholar
  28. [HP3]
    R L Hudson and K R Parthasarathy, Unification of Boson and Fermion quantum stochastic calculus, Commun Math Phys 104 (1986) 457–470.Google Scholar
  29. [HP4]
    R L Hudson and K R Parthasarathy, Stochastic dilations of uniformly continuous quantum dynamical semigroups, Acta Applicandae Math 2 (1984) 457–470.Google Scholar
  30. [HPu]
    R L Hudson and S Pulmannova, Chaotic expansion of elements of the universal enveloping algebra of a Lie algebra associated with a quantum stochastic calculus, Nottingham preprint (1996).Google Scholar
  31. [HR1]
    R L Hudson and P Robinson, Quantum diffusions and the noncommutative torus, Lett Math Phys 15 (1988) 47–53.Google Scholar
  32. [HR2]
    R L Hudson and P Robinson, Quantum diffusions on the noncommutative torus and solid state physics, pp 338–345, in Differential geometric methods in solid state physics, Chester, 1988, ed A Solomon, World Scientific 1989.Google Scholar
  33. [HSh]
    R L Hudson and P Shepperson, Stochastic dilations of quantum dynamical semigroups using one-dimensional quantum stochastic calculus, pp 216–218 in Quantum Probability V, ed L Accardi et al, Springer LNM 1442(1990).Google Scholar
  34. [HSj]
    R L Hudson and V R Struleckaja, Nonabelian cohomology and Fermionic flows over Z2-graded *-algebras, Lett Math Phys 38(1996) 13–22.MathSciNetMATHCrossRefGoogle Scholar
  35. [HSt]
    R L Hudson and R F Streater, Noncommutative martingales and stochastic integrals in Fock space, pp 216–227 in Stochastic processes in quantum theory and statistical physics, proceedings, Marseilles 1982, ed Albeverio, Springer LN Physics 173(1983).Google Scholar
  36. [Jo]
    J L Journé, Structure des cocycles Markoviens sur l’espace de Fock, Prob Theor Rel Fields 75(1987) 291–316.MATHCrossRefGoogle Scholar
  37. [Ld]
    G Lindblad, On the generators of quantum dynamical semigroups, Commun Math Phys 48(1976) 119–130.MathSciNetMATHCrossRefGoogle Scholar
  38. [Li]
    J M Lindsay, Independence for quantum stochastic integrators, pp 325–332, in Quantum probability VI, ed L Accardi et al, World Scientific (1991).Google Scholar
  39. [Lu]
    A S Lue, Nonabelian cohomology of associative algebras, Quart Jour Math 19(1968) 159–180.MathSciNetMATHCrossRefGoogle Scholar
  40. [MR]
    H Maassen and P Robinson, Quantum stochastic calculus and the dynamical Stark effect, Rep Math Phys 30(1992) 185–203.MathSciNetGoogle Scholar
  41. [Ne]
    E Nelson, The free Markoff field, J Funct Anal 12(1973) 211–227.MATHCrossRefGoogle Scholar
  42. [PM]
    A Mohari and K R Parthasarathy, On a class of generalized Evans-Hudson flows related to classical Markov processes, pp 221–249, in Quantum Probability and Applications VII, ed L Accardi et al, World Scientific (1992).Google Scholar
  43. [PSc]
    K R Parthasarathy and K Schmidt, Factorizable representations of current groups and the Araki–Woods embedding theorem, Acta Mathematica 128(1972) 53–71.MathSciNetMATHCrossRefGoogle Scholar
  44. [Psi1]
    K R Parthasarathy and K B Sinha, Stop times in Fock space stochastic calculus, pp 495–498, in Proceedings of 1st World Congress of Bernoulli Society, Tashkent 1986 vol 1, ed Yu Prohorov et al, VNU (1987).Google Scholar
  45. [PSi1]
    K R Parthasarathy and K B Sinha, Stochastic integral representations of bounded quantum martingales in Fock space, J Funct Anal 67 (1986) 126–151.Google Scholar
  46. [Se]
    I R Senitzky, Dissipation in quantum mechanics. The harmonic oscillator, Phys Rev 119(1960) 670–679.MathSciNetMATHCrossRefGoogle Scholar
  47. [Sg]
    I E Segal, Tensor algebras over Hilbert spaces, Trans Amer Math Soc 81(1956) 106–134.MathSciNetMATHCrossRefGoogle Scholar
  48. [Sp]
    R Speicher, A new example of ‘Independence’ and ‘White noise’, Prob Theor Rel Fields 84(1990) 141–154.MathSciNetMATHCrossRefGoogle Scholar
  49. [VDN]
    D Voiculescu, K J Dykema and A Nica, Free random variables, American Mathematical Society CRM Monographs (1992).Google Scholar
  50. [Wi]
    N Wiener, The homogeneous chaos, Amer J Math 60(1930) 897–936.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1998

Authors and Affiliations

  • R. L. Hudson
    • 1
  1. 1.University of NottinghamNottinghamUK

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