Quantum stochastic integral representations of Fock space operators

Abstract

An (unbounded) operator Ξ on Boson Fock space over L ²(R ₊) is called regular if it is an admissible white noise operator such that the conditional expectations give rise to a regular quantum martingale. We prove that an admissible white noise operator is regular if and only if it admits a quantum stochastic integral representation.

Keywords:

• Fock space
• quantum stochastic process
• quantum stochastic gradient
• quantum stochastic integral
• quantum martingale
• regular operator
• creation-derivative
• annihilation-derivative
• quantum white noise

Mathematics Subject Classification (2000):

• Primary 81S25
• Secondary 46F25, 60H40

Acknowledgements

This article was supported by the Korea–Japan Basic Scientific Cooperation Program (2007–2009) “Noncommutative Stochastic Analysis and Its Applications to Network Science.”

Notes

^1. [email protected]

Additional information

Notes on contributors

Nobuaki Obata

1. 1. [email protected]