Quadratic fock space calculus (I): some results on quadratic creation and preservation operators

This paper is a fundamental exploration of quantum theory within the quadratic Fock space in consistency with the quadratic quantization program, with a particular focus on two sets of operators that hold immense significance: the quadratic creation and preservation operators. In this paper, we highlight a critical contribution to the quadratic quantization program. In which we prove that when the argument f is a real valued function, the quadratic preservation operator is symmetric and even essentially self-adjoint. This mathematical confirmation not only solidifies the foundations of quantum theory but also amplifies its practical applicability in real-world scenarios. In consistency with previous result, we give the exponential action of the creation operator on the domain of quadratic exponential vectors. This is an expansion of what obtained in Accardi, Ouerdiane, Rebei (Infin Dimens Anal Quantum Probab Relat Top 13(4):551–587, 2010) for the one-mode case.