Eight-dimensional Octonion-like but Associative Normed Division Algebra (1908.06172v10)

Abstract: We present an eight-dimensional even sub-algebra of the 2^4 = 16-dimensional associative Clifford algebra Cl(4,0) and show that its eight-dimensional elements denoted as X and Y respect the norm relation ||XY|| = ||X|| ||Y||, thus forming an octonion-like but associative normed division algebra, where the norms are calculated using the fundamental geometric product instead of the usual scalar product so that the underlying coefficient algebra resembles that of split complex numbers instead of reals. The corresponding 7-sphere has a topology that differs from that of octonionic 7-sphere.

• The paper demonstrates that an eight-dimensional subset of Cl₄,₀ supports an associative normed division algebra with octonion-like characteristics.
• It rigorously proves that the algebra satisfies the norm relation ||XY|| = ||X|| ||Y|| while maintaining closure under multiplication.
• The framework offers promising applications in quantum mechanics and computer graphics, potentially streamlining computations and theoretical research.

Essay on "Eight-dimensional Octonion-like but Associative Normed Division Algebra"

The paper by Joy Christian introduces an innovative framework for an eight-dimensional algebra that retains association while exhibiting normed division properties similar to octonions. Fundamentally, the work extends the concept of associative Clifford algebras to establish an eight-dimensional sub-algebra. This paper is a continuation of the exploration into normed division algebras beyond the traditional dimensions, associating them in a manner that sustains linear algebraic properties without resorting to non-associativity as in classical octonions.

Overview

The heart of the paper lies in demonstrating that an eight-dimensional subset of the 16-dimensional associative Clifford algebra, denoted as $Cl_{4,0}$, can support an octonion-like structure that remains associative. Traditional octonions, an eight-dimensional normed division algebra, are non-associative, leading to intriguing but complex algebraic properties. By contrast, the author's approach retains associativity, preserving simpler computational manipulations while still supporting division algebra functionality.

The paper utilizes the language of Geometric Algebra and introduces an algebraic structure, ${\cal K}^{\lambda}$, comprising linearly independent elements that obey a norm relation similar to octonions: $||{\bf X}{\bf Y}|| = ||{\bf X}||\,||{\bf Y}||$. This norm is derived using the Geometric Product instead of the usual scalar product, embedding complex algebraic behavior akin to split-complex numbers, rather than purely real coefficients.

Strong Numerical Results and Claims

Christian provides rigorous mathematical forms showing that the eight-dimensional algebra not only supports closure under multiplication but also satisfies a unique norm property. Specifically, for two multivectors ${\bf X}$ and ${\bf Y}$ in ${\cal K}^{\lambda}$, their norm follows the property $||{\bf X}{\bf Y}|| = ||{\bf X}||\,||{\bf Y}||$. This relationship is foundational for supporting the claim of ${\cal K}^{\lambda}$ as a normed division algebra. The paper also verifies that this algebra maintains its closed nature under multiplication, maintaining the topological and algebraic integrity of the subset as a 7-sphere in specific normalization conditions.

Implications and Future Developments

The paper has substantial implications for both theoretical and practical applications, particularly in areas requiring complex algebraic manipulations such as quantum mechanics and computer graphics. The introduction of a new associative division algebra in eight dimensions opens avenues for reinterpreting mathematical structures underlying physical systems without relying on non-associative structures, which often complicates theoretical physics formulations.

In terms of future developments, additional research might focus on extending these algebraic formulations into other dimensions or exploring broader implications for conformal geometric algebra in engineering applications. For instance, developments could explore whether such algebraic structures could offer new insights or methods in tackling singularities in conformal mappings or quantum field theories.

Christian's framework provides a basis that may enable extensions of Jordan's proposals concerning generalized quantum mechanics based on non-associative systems. By employing an associative form, it might resolve some challenges identified by previous formulations. As computational methodologies continue to evolve, incorporating such associative algebraic systems could streamline algorithms and enhance computational stability in large-scale simulations.

Overall, this paper not only contributes to the corpus of mathematical algebraic structures but also provides potential applications that could significantly impact computational and theoretical physics. The intriguing aspect of maintaining association while extending into higher dimensions underscores a nuanced understanding of algebraic properties, bringing forth new challenges and opportunities for scientific exploration.