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

TL;DR

This paper explores the structure of $K$-Hilbert spaces over different division rings within Clifford algebras, classifying states and analyzing fermionic and bosonic spectra through algebraic operations.

Contribution

It introduces a unified framework for $K$-Hilbert spaces over real, complex, and quaternionic fields, classifying states and detailing algebraic operations for fermionic and bosonic states.

Findings

Classifies states into charged, neutral, and truly neutral based on division ring $K$

Defines algebraic operations for fermionic and bosonic states

Connects $K$-Hilbert spaces with Clifford algebra structures

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.