八元数描述的电磁场方程组

Electromagnetic Field Equations Described with the Octonions

麦克斯韦首先同时采用矢量和四元数两种方法描述电磁场性质。这启发本文总结出三条基本假设,试图解决将八元数引进场论的问题。应用八元数可以描述电磁场的麦克斯韦方程组,从而部分程度实现采用八元数代数描述电磁场理论的努力目标。如果将八元数分解成为四元数和S-四元数两个部分,可以发现S-四元数适合于描述电磁场性质。这种方法在电磁场理论的以往研究中是不曾遇到过的。由八元数描述的电磁场的场源定义式可以分解出麦克斯韦方程组。同时电磁场势、场强定义式和麦克斯韦方程组均分别等同于采用矢量方法得到的结果,但位移电流方向和规范条件方程有所不同。研究结果揭示,应用八元数描述的电磁场理论可以涵盖经典电磁场理论中的大部分已有结论。采用S-四元数可以描述电磁场中的麦克斯韦方程组。

Inspired by J. C. Maxwell firstly using both the vector terminology and the algebra of quaternions to describe the property of electromagnetic fields, the paper summarizes 3 basic postulations to import the algebra of octonions into the field theory. In this case, the algebra of octonions can partly describe the electromagnetic theory by using the algebra of octonions to describe the Maxwell equations of electromagnetic field. The viewpoint of R. Descartes, M. Faraday, and A. Einstein etc claims that, the field is an irreducible element of physical description, while the space-time is only the extension of the field and does not claim existence on its own. According to the three basic postulations, the space-time extended from the electromagnetic field is different from the one extended from the gravitational field. They are quite similar but independent to each other. These two space-times, extended respectively from the electromagnetic field and the gravitational field, both can be considered as the quaternion space, and are perpendicular to each other so that they can combine together to become an octonion space. Therefore the octonion space can be used to describe the physical property of the electromagnetic field and the gravitational field. In order to maintain the dimensional homogeneity, the physical quantity of the electromagnetic field should be multiplied by the undetermined coefficient, when the two kinds of physical quantities exist in the same octonion formula. The undetermined coefficient can be determined by comparing with the classical theories of the electromagnetic field and the gravitational field. In math, the complex number can be divided into the real number and the imaginary number, while the imaginary number is the product of another real number and the imaginary unit. Just like the complex number, the octonion also can be divided into two components, the quaternion and the S-quaternion, while the S-quaternion is the product of another quaternion and the ‘fourth imaginary unit’. The ‘fourth imaginary unit’ is independent to the ‘three imaginary unit’ in the quaternion. After several years studies the author finds out that the quaternion is suitable for describing the property of the gravitational field, while the S-quaternion is suitable for describing the property of electromagnetic field. Comparing the physi-cal quantities describing by the algebra of octonions and the vector terminology can also prove this conclu-sion. This method is never been used before in all the theoretical studies of electromagnetic fields. And may be that can partly explain why it is not completely successful before to describe the electromagnetic field theory directly by the quaternion (rather than the S-quaternion). In the paper, the definition of field source described by the octonions can deduce Maxwell equation in the electromagnetic field. The Maxwell equa-tions, the electromagnetic potential, and the field strength definition, deduced by the field source definition of electromagnetic describe by octonions, are respectively identical with that deduced by the vector terminology in classical electromagnetic theory, but the direction of displacement current and the gauge equation are not. The study claims that the electromagnetic field theory described by the algebra of octonions can cover most of the existing conclusions of classical electromagnetic field theory. And the Maxwell equations in the elec-tromagnetic field can be described by S-quaternion.