Quaternionic Quantization Principle in General Relativity and Supergravity

TL;DR

This paper introduces a quaternionic-based generalized quantization principle applicable to quantum mechanics, quantum gravity, and supergravity, leading to new algebraic structures and relations among quantum variables across different space-time directions.

Contribution

It develops a novel quaternionic quantization framework and applies it to quantum mechanics, quantum gravity, and supergravity, extending traditional commutation relations and operator algebra.

Findings

Formulated quaternionic Dirac equation and plane waves

Derived generalized algebraic relations for gravitational observables

Extended quantization to supergravity with Dirac brackets

Abstract

A generalized quantization principle is considered, which incorporates nontrivial commutation relations of the components of the variables of the quantized theory with the components of the corresponding canonical conjugated momenta referring to other space-time directions. The corresponding commutation relations are formulated by using quaternions. At the beginning, this extended quantization concept is applied to the variables of quantum mechanics. The resulting Dirac equation and the corresponding generalized expression for plane waves are formulated and some consequences for quantum field theory are considered. Later, the quaternionic quantization principle is transferred to canonical quantum gravity. Within quantum geometrodynamics as well as the Ashtekar formalism the generalized algebraic properties of the operators describing the gravitational observables and the corresponding quantum constraints implied by the generalized representations of these operators are determined. The generalized algebra also induces commutation relations of the several components of the quantized variables with each other. Finally, the quaternionic quantization procedure is also transferred to $\mathcal{N}=1$ supergravity. Accordingly, the quantization principle has to be generalized to be compatible with Dirac brackets, which appear in canonical quantum supergravity.