Stochastic integral representations of second quantization operators. (Q1428455)

The author proves that the second quantization \(\Gamma (h)\) of a bounded operator \(h\) on \(L^2 ({\mathbb R}_+)\) can be represented as a quantum stochastic integral if and only if \(h\) admits a decomposition as the sum of a Hilbert-Schmidt operator \(K\) and an operator \(M\) of multiplication by an essentially bounded function. A similar result is obtained for differential second quantizations, and explicit formulas for the integrands are provided in terms of \(K\) and \(M\). The counter-example of \textit{J. L. Journé} and \textit{P. A. Meyer} to representability of operators on Fock space as quantum stochastic integrals, in [Probabilités, XX, 1984/85, Lect. Notes Math. 1204, 313--316 (1986; Zbl 0611.60048)] is detailed in this context.