Quaternionic and Octonionic Field Theories
Summary
Quaternionic and octonionic field theories extend the familiar framework of complex-valued fields by employing higher-dimensional hypercomplex algebras. Quaternions, discovered by Hamilton, are four-dimensional extensions of complex numbers that introduce three mutually non-commuting imaginary units. This non-commutativity endows field equations with richer transformation properties, permitting unified descriptions of spin, rotations and more intricate coupling structures. Octonions, as eight-dimensional non-associative numbers, further expand this landscape, offering a novel algebraic setting in which electromagnetic, gravitational and internal gauge fields may be encoded within a single formalism.
In quaternionic formulations, wave functions, potentials and field strengths are represented as quaternion-valued functions on real or complex Hilbert spaces. This approach has led to generalised Dirac–Maxwell equations, alternative representations of angular momentum addition and distinctive scattering phenomena. Octonionic schemes exploit the exceptional algebra of octonions to merge gravitational and electromagnetic interactions through a unified potential, highlighting deep links between colour degrees of freedom, duality symmetries and non-linear sigma models.
Together, these theories promise fresh insights into unification programmes, supersymmetry and higher-dimensional gravity, while offering concrete computational tools for relativistic scattering, magnetohydrodynamics and black-hole solutions. Their global significance lies in potential applications to particle physics, condensed matter and quantum computation, where hypercomplex structures naturally encode multi-component fields and entangled rotational symmetries.
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Quaternionic and Octonionic Field Theories publication trend
The graph below shows the total number of articles in quaternionic and octonionic field theories across all publications each year (not limited to Nature Index journals).
Technical terms
Quaternion: A four-dimensional hypercomplex number comprising one real and three imaginary units, with non-commutative multiplication rules.
Octonion: An eight-dimensional hypercomplex extension of quaternions with seven imaginary units; multiplication is non-associative but alternative.
Non-commutative algebra: An algebraic system in which the product of two elements depends on their order.
Two-spinor formalism: A representation of Lorentz transformations using two-component spinors arising in the description of relativistic spin.
Field strength: A tensor or multicomponent function describing the intensity and orientation of a physical field, derived from its potential.
Bessel functions: Special functions solving differential equations with cylindrical symmetry, often appearing in radial wave solutions.
Hilbert space: A complete vector space equipped with an inner product, foundational for quantum mechanics.
References
- Generalization of adding angular momenta and circular potential in quaternionic quantum mechanics. Heliyon (2024).
- Field Equations in the Complex Quaternion Spaces. Advances in Mathematical Physics (2014).
- Quaternion Electromagnetism and the Relation with Two-Spinor Formalism. Universe (2019).
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