Strongly supercommuting self-adjoint operators (Q1764270)

For unbounded self-adjoint operators, strong commutativity is defined in terms of commutativity of the corresponding spectral measures. In 1983, the reviewer introduced a concept of strong anticommutativity, also expressed in terms of spectral measures [see Rev. Roum. Math. Pures Appl. 28, 77--91 (1983; Zbl 0525.47017)]. In the present paper, the author constructs a theory which deals simultaneously with commutativity and anticommutativity. To this end, he introduces a notion of strong supercommutativity for self-adjoint operators, using \({\mathbb Z}_2\)-graded Hilbert spaces. In quantum physics, bosons are connected with commutativity and fermions are connected with anticommutativity. For this reason, the supersymmetric quantum theory, dealing simultaneously with bosons and fermions, can be regarded as a theory which unifies commutativity and anticommutativity. Besides the basic properties of the strong supercommutativity, the author presents the theory of super quantization, which is associated with supersymmetric quantum mechanics. Some applications to the theory of quantum physics are also given.