Annihilation-Derivative,Creation-Derivative and Representation of Quantum Martingales

On the basis of the quantum white noise theory we introduce the notion of creation- and annihilation-derivatives of Fock space operators and study the differentiability of white noise operators. We define the Hitsuda–Skorohod quantum stochastic integrals by the adjoint actions of quantum stochastic gradients and show explicit formulas for their creation- and annihilation-derivatives. As an application, we derive direct formulas for the integrands in the quantum stochastic integral representation of a regular quantum martingale. Work supported by the Korea–Japan Basic Scientific Cooperation Program “Noncommutative Stochastic Analysis and Its Applications to Network Science.”