Stochastic integral representations of second quantization operators

We give a necessary and sufficient condition for the second quantization operator Γ(h) of a bounded operator h on L2(R+), or for its differential second quantization operator λ(h), to have a representation as a quantum stochastic integral. This condition is exactly that h writes as the sum of a Hilbert–Schmidt operator and a multiplication operator. We then explore several extensions of this result. We also examine the famous counterexample due to Journé and Meyer and explain its representability defect.