Abstract
We introduce the quadratic analog of the tensor Bogolyubov representation of the CCR. Our main result is the determination of the structure of these maps: each of them is uniquely determined by two arbitrary complex-valued Borel functions of modulus 1 and two maps of R-d into itself whose inverses induce transformations that map the Lebesgue measure lambda into measures lambda(c), lambda(s). absolutely continuous with respect to it.. Furthermore, the Radon-Nikodyn derivatives c(2), s(2), of these measures with respect to lambda, must satisfy the relation c(2)(x) - s(2)(x) = 1 for lambda-almost every x is an element of R-d. This makes a surprising bridge with the hyperbolic sine and cosine defining the structure of usual (i.e. first-order) Bogolyubov transformations. The reason of the surprise is that the linear and quadratic commutation relations are completely different.