Quaternionic scalar field in the real Hilbert space

TL;DR

This paper develops a quaternionic scalar field theory in the real Hilbert space, revealing richer state structures and non-associative algebraic features not present in the traditional complex scalar field models.

Contribution

It introduces a quaternionic quantization scheme with two options and uncovers novel non-associative algebraic structures within the quaternionic scalar field.

Findings

Two quantization schemes for quaternionic fields identified

Richer state structure compared to complex scalar fields

Presence of non-associative algebraic structures in the theory

Abstract

Using the complex Klein-Gordon field as a model, we quantize the quaternionic scalar field in the real Hilbert space. The lagrangian formulation has accordingly been obtained, as well as the hamiltonian formulation, and the energy and charge operators. Conversely to the complex case, the quaternionic quantization admits two quantization schemes, concerning either two or four components. Therefore, the quaternionic field permits a richer structure of states, if compared to the complex scalar field case. Moreover, the quaternionic theory admits as a further novel feature a non-associative algebraic structure in their complex components, something not observed in the complex case.