Spinors in $\mathbb{K}$-Hilbert Spaces

Abstract

We consider a structure of the $\mathbb{K}$-Hilbert space, where $\mathbb{K}\simeq\mathbb{R}$ is a field of real numbers, $\mathbb{K}\simeq\mathbb{C}$ is a field of complex numbers, $\mathbb{K}\simeq\mathbb{H}$ is a quaternion algebra, within the framework of division rings of Clifford algebras. The $\mathbb{K}$-Hilbert space is generated by the Gelfand-Naimark-Segal construction, while the generating $C^\ast$-algebra consists of the energy operator $H$ and the generators of the group $SU(2,2)$ attached to $H$. The cyclic vectors of the $\mathbb{K}$-Hilbert space corresponding to the tensor products of quaternionic algebras define the pure separable states of the operator algebra. Depending on the division ring $\mathbb{K}$, all states of the operator algebra are divided into three classes: 1) charged states with $\mathbb{K}\simeq\mathbb{C}$; 2) neutral states with $\mathbb{K}\simeq\mathbb{H}$; 3) truly neutral states with $\mathbb{K}\simeq\mathbb{R}$. For pure separable states that define the fermionic and bosonic states of the energy spectrum, the fusion, doubling (complexification) and annihilation operations are determined.