Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D B Applebaum, Stochastic dilations of the Bloch Equation in Boson and Fermion noise, J Phys A 19(1986) 937–959.
L Accardi, A Frigerio and Y G Lu, The weak coupling limit as a quantum central limit, Commun Math Phys 131(1991) 537–576.
L Accardi, J Gough and Y G Lu, On the stochastic limit for quantum theory, Rep Math Phys 36(1995) 155–187.
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).
S Attal and P-A Meyer, Interpretation probabiliste et extension des integrales stochastiques noncommutatives, Strasbourg preprint (1994).
L Accardi and I Volovich, The stochastic limit of quantum field theory, Rome II preprint (1994).
V P Belavkin, O Hirota and R L Hudson (eds) Quantum Communication and Measurement, Plenum (1995).
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.
V P Belavkin and P Staszewski, Nondemolition measurement of a free quantum particle, Phys Rev A 45(1992) 1347–1356.
C Barnett, R F Streater and I Wilde, The Ito Clifford integral, J Fund Anal 48(1982) 172–212.
A Connes and J Cuntz, Quasi-homomorphismes, cohomologie cyclique et positivité, Commun Math Phys 114(1988) 515–526.
A Chebotarev and F Fagnola, Sufficient conditions for conservativity of quantum dynamical semigroups, J Funct Anal 118(1993) 113–153.
A M Cockroft and R L Hudson, Quantum mechanical Wiener processes, J Multivariate Anal 7(1977) 107–124.
P Beazley Cohen and R L Hudson, Generators of quantum stochastic flows, Cuntz morphisms and cyclic cohomology, Nottingham preprint (1994).
M P Evans and R L Hudson, Perturbations of quantum diffusions, J London Math Soc (2) 41 (1990) 373–384.
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).
M P Evans, Existence of quantum diffusions, Prob Theory and Related Fields 81(1989) 473–483.
T M W Eyre, Chaotic expansion for Lie superalgebras in quantum stochastic calculus, Nottingham preprint (1996).
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.
V A Fock, Konfigurationsraum und zweite Quantelung, Z Physik 75, 622–647 (1932).
R L Hudson, The strong Markov property for canonical Wiener processes, J Funct Anal 37 (1980) 68–87.
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).
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.
R L Hudson and J M Lindsay, On characterizing quantum stochastic evolutions, Math Proc Camb Phil Soc 102(1987) 363–369.
G Hochschild, On the cohomology groups of an associative algebra, Ann of Math 46(1945) 58–67.
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).
R L Hudson and K R Parthasarathy, Quantum Ito’s formula and stochastic evolutions, Commun Math Phys 93 (1984) 301–323.
R L Hudson and K R Parthasarathy, Unification of Boson and Fermion quantum stochastic calculus, Commun Math Phys 104 (1986) 457–470.
R L Hudson and K R Parthasarathy, Stochastic dilations of uniformly continuous quantum dynamical semigroups, Acta Applicandae Math 2 (1984) 457–470.
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).
R L Hudson and P Robinson, Quantum diffusions and the noncommutative torus, Lett Math Phys 15 (1988) 47–53.
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.
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).
R L Hudson and V R Struleckaja, Nonabelian cohomology and Fermionic flows over Z2-graded *-algebras, Lett Math Phys 38(1996) 13–22.
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).
J L Journé, Structure des cocycles Markoviens sur l’espace de Fock, Prob Theor Rel Fields 75(1987) 291–316.
G Lindblad, On the generators of quantum dynamical semigroups, Commun Math Phys 48(1976) 119–130.
J M Lindsay, Independence for quantum stochastic integrators, pp 325–332, in Quantum probability VI, ed L Accardi et al, World Scientific (1991).
A S Lue, Nonabelian cohomology of associative algebras, Quart Jour Math 19(1968) 159–180.
H Maassen and P Robinson, Quantum stochastic calculus and the dynamical Stark effect, Rep Math Phys 30(1992) 185–203.
E Nelson, The free Markoff field, J Funct Anal 12(1973) 211–227.
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).
K R Parthasarathy and K Schmidt, Factorizable representations of current groups and the Araki–Woods embedding theorem, Acta Mathematica 128(1972) 53–71.
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).
K R Parthasarathy and K B Sinha, Stochastic integral representations of bounded quantum martingales in Fock space, J Funct Anal 67 (1986) 126–151.
I R Senitzky, Dissipation in quantum mechanics. The harmonic oscillator, Phys Rev 119(1960) 670–679.
I E Segal, Tensor algebras over Hilbert spaces, Trans Amer Math Soc 81(1956) 106–134.
R Speicher, A new example of ‘Independence’ and ‘White noise’, Prob Theor Rel Fields 84(1990) 141–154.
D Voiculescu, K J Dykema and A Nica, Free random variables, American Mathematical Society CRM Monographs (1992).
N Wiener, The homogeneous chaos, Amer J Math 60(1930) 897–936.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Hudson, R.L. (1998). Quantum Stochastic Analysis After Four Decades. In: Accardi, L., Heyde, C.C. (eds) Probability Towards 2000. Lecture Notes in Statistics, vol 128. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2224-8_12
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2224-8_12
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-98458-2
Online ISBN: 978-1-4612-2224-8
eBook Packages: Springer Book Archive