A characterization of the Second Quantization by using the Segal duality transform

The main object of this article is to establish the necessary and sufficient conditions in order to represent the Second Quantization in terms of a certain class of the Fourier–Wiener or the Wiener transforms via the Segal duality transform. We first analyze the finite-dimensional case and derive a matrix expression of this class of the Fourier–Wiener or the Wiener transforms. We then extend this analysis to derive the corresponding results for a certain family of the Fourier–Wiener or the Wiener transforms over Hilbert spaces.