[Paper Review] Noncommutative sedeons and their application in field theory

В. Л. Миронов, С. В. Миронов|arXiv (Cornell University)|Nov 17, 2011

Quantum Mechanics and Applications77 references3 citations

TL;DR

This paper introduces sixteen-component noncommutative sedeons to construct an associative noncommutative space-time algebra, proposing generalized second- and first-order relativistic wave equations using sedeonic wave functions and operators. The framework enables a unified description of massive fields via second- and first-order equations for sedeonic potentials, offering a novel algebraic approach to relativistic field theory.

ABSTRACT

We present sixteen-component values "sedeons", generating associative noncommutative space-time algebra. The generalized second-order and first-order equations of relativistic quantum mechanics based on sedeonic wave function and sedeonic space-time operators are proposed. We also discuss the description of fields with massive quantum on the basis of second-order and first-order equations for sedeonic potentials.

Motivation & Objective

• To develop a noncommutative algebraic structure for space-time using sixteen-component sedeons.
• To formulate generalized second- and first-order relativistic wave equations based on sedeonic wave functions.
• To describe massive quantum fields using sedeonic potentials within a consistent algebraic framework.
• To establish an associative noncommutative space-time algebra that extends traditional relativistic field theory.
• To unify the description of massive fields through first- and second-order equations in the sedeonic formalism.

Proposed method

• Introduces sedeons as sixteen-component mathematical objects forming an associative noncommutative algebra over space-time.
• Constructs space-time operators using sedeonic algebra to generalize relativistic wave equations.
• Derives second-order and first-order equations of motion for sedeonic wave functions in noncommutative space-time.
• Applies the formalism to describe fields with massive quanta through sedeonic potential fields.
• Uses algebraic structures to unify the description of spin-0 and spin-1/2 fields in a single formalism.
• Employs a noncommutative algebraic framework to generalize the Dirac and Klein-Gordon equations in a sedeonic representation.

Experimental results

Research questions

• RQ1 Can a noncommutative algebraic structure based on sixteen-component sedeons unify relativistic wave equations?
• RQ2 How can sedeonic wave functions and operators be used to generalize second- and first-order relativistic equations?
• RQ3 Can massive quantum fields be consistently described using sedeonic potentials within this noncommutative framework?
• RQ4 What algebraic properties ensure associativity and closure in the space-time algebra defined by sedeons?
• RQ5 How does the sedeonic formalism compare to standard relativistic field theories in describing massive fields?

Key findings

• The paper constructs a noncommutative, associative algebra of space-time using sixteen-component sedeons.
• Generalized second-order and first-order relativistic wave equations are derived for sedeonic wave functions.
• The formalism allows for a unified description of massive fields through sedeonic potential equations.
• The algebraic structure supports both spin-0 and spin-1/2 field descriptions within a single mathematical framework.
• The noncommutative nature of sedeons enables a novel algebraic unification of relativistic field equations.
• The framework provides a consistent mathematical setting for describing massive quantum fields using first- and second-order equations in sedeonic form.