TY - JOUR

T1 - Integral Kernel Operators on Fock Space - Generalizations and Applications to Quantum Dynamics

AU - Obata, Nobuaki

PY - 1997

Y1 - 1997

N2 - 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 ap]proach 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.

AB - 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 ap]proach 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.

KW - Fock space

KW - Integral kernel operator

KW - Quantum Langevin equation

KW - Quantum stochastic integral

KW - White noise distribution theory

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U2 - 10.1023/A:1005777929658

DO - 10.1023/A:1005777929658

M3 - Article

AN - SCOPUS:0002059859

SN - 0167-8019

VL - 47

SP - 49

EP - 77

JO - Acta Applicandae Mathematicae

JF - Acta Applicandae Mathematicae

IS - 1

ER -