Abstract
A general theory of operators on Boson Fock space is discussed in terms of the white noise distribution theory on Gaussian space (white noise calculus). An integral kernel operator is generalized from two aspects: (i) The use of an operator-valued distribution as an integral kernel leads us to the Fubini type theorem which allows an iterated integration in an integral kernel operator. As an application a white noise approach to quantum stochastic integrals is discussed and a quantum Hitsuda–Skorokhod integral is introduced. (ii) The use of pointwise derivatives of annihilation and creation operators assures the partial integration in an integral kernel operator. In particular, the particle flux density becomes a distribution with values in continuous operators on white noise functions and yields a representation of a Lie algebra of vector fields by means of such operators.
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References
Belavkin, V. P.: A quantum nonadapted Ito formula and stochastic analysis in Fock scale, J. Funct. Anal. 102 (1991), 414–447.
Bogolubov, N. N., Logunov, A. A., and Todorov, I. T.: Introduction to Axiomatic Quantum Field Theory, Benjamin, New York, 1975.
Bohm, A. and Gadella, M.: Dirac Kets, Gamow Vectors and Gel'fand Triplets, Lect. Notes in Phys. 348, Springer-Verlag, New York, 1989.
Gardiner, C. W.: Quantum Noise, Springer-Verlag, New York, 1991.
Goldin, G. A., Grodnik, J., Powers, R. T., and Sharp, D. H.: Nonrelativistic current algebra in the N/V limit, J. Math. Phys. 15 (1974), 88–100.
Goldin, G. A. and Sharp, D. H.: Particle spin from representations of the diffeomorphism group, Comm. Math. Phys. 92 (1983), 217–228.
Haag, R.: On quantum field theories, Dan. Mat. Fys. Medd. 29(12) (1955), 1–37.
Hida, T.: Analysis of Brownian Functionals, Carleton Math. Lect. Notes 13, Carleton University, Ottawa, 1975.
Hida, T., Kuo, H.-H., Potthoff, J., and Streit, L.: White Noise, Kluwer Academic, Dordrecht, 1993.
Hida, T., Obata, N., and Saitô, K.: Infinite dimensional rotations and Laplacians in terms of white noise calculus, Nagoya Math. J. 128 (1992), 65–93.
Huang, Z.: Quantum white noises — White noise approach to quantum stochastic calculus, Nagoya Math. J. 129 (1993), 23–42.
Krée, P.: Calcul symbolique et seconde quantification des fonctions sesquiholomorphes, CR Acad. Sci. Paris 284A (1977), 25–28.
Krée, P.: La théorie des distributions en dimension quelconque et l'intégration stochastique, in: H. Korezlioglu and A. S. Ustunel (eds), Stochastic Analysis and Related Topics, Lect. Notes in Math. 1316, Springer-Verlag, New York, 1988, pp. 170–233.
Krée, P. and Rączka, R.: Kernels and symbols of operators in quantum field theory, Ann. Inst. Henri Poincaré A 28 (1978), 41–73.
Kubo, I. and Takenaka, S.: Calculus on Gaussian white noise I–IV, Proc. Japan Acad. 56 (1980), 376–380; 411–416; 57 (1981), 433–437; 58 (1982), 186–189.
Lindsay, J. M.: On set convolutions and integral-sum kernel operators, in: B. Grigelionis et al. (eds), Probability Theory and Mathematical Statistics, Vol. 2, Mokslas, Vilnius, 1990, pp. 105–123.
Lindsay, J. M.: Quantum and non-causal stochastic calculus, Probab. Theory Related Fields 97 (1993), 65–80.
Lindsay, J. M. and Maassen, H.: An integral kernel approach to noise, in: L. Accardi and W. von Waldenfels (eds), Quantum Probability and Applications III, Lect. Notes in Math. 1303, Springer-Verlag, New York, 1988, pp. 192–208.
Meyer, P. A.: Distributions, noyaux, symboles d'après Krée, in: J. Azéma et al. (eds), Séminaire de Probabilités XXII, Lect. Notes in Math. 1321, Springer-Verlag, New York, 1988, pp. 467–476.
Meyer, P. A.: Quantum Probability for Probabilists, Lect. Notes in Math. 1538, Springer-Verlag, New York, 1993.
Obata, N.: An analytic characterization of symbols of operators on white noise functionals, J. Math. Soc. Japan 45 (1993), 421–445.
Obata, N.: White Noise Calculus and Fock Space, Lect. Notes in Math. 1577, Springer-Verlag, New York, 1994.
Obata, N.: Operator calculus on vector-valued white noise functionals, J. Funct. Anal. 121 (1994), 185–232.
Obata, N.: Lie algebras containing infinite dimensional Laplacians, in: H. Heyer (ed.), Probability Measures on Groups and Related Structures, World Scientific, Singapore, 1995, pp. 260–273.
Obata, N.: Derivations on white noise functionals, Nagoya Math. J. 139 (1995), 21–36.
Obata, N.: Generalized quantum stochastic processes on Fock space, Publ. RIMS Kyoto Univ. 31 (1995), 667–702.
Parthasarathy, K. R.: An Introduction to Quantum Stochastic Calculus, Birkhäuser, Basel, 1992.
Vershik, A. M., Gel'fand, I. M., and Graev, M. I.: Representations of the group of diffeomorphisms, Russian Math. Surveys 30(6) (1975), 1–50.
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Obata, N. Integral Kernel Operators on Fock Space – Generalizations and Applications to Quantum Dynamics. Acta Applicandae Mathematicae 47, 49–77 (1997). https://doi.org/10.1023/A:1005777929658
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DOI: https://doi.org/10.1023/A:1005777929658