Triality, Biquaternion and Vector Representation of the Dirac Equation

Triality and Dual Equivalence Between Dirac Field and Topologically Massive Gauge Field

It is proved that there exist a vector representation of Dirac's spinor field and in one sense it is equivalent to biquaternion (i.e. complexified quaternion) representation. This can be considered as a generalization of Cartan's idea of triality to Dirac's spinors. In the vector representation the first order Dirac Lagrangian is dual equivalent to the two order Lagrangian of topologically massive gauge field. The potential field which corresponds to the Dirac field is obtained by using master (or parent) action approach. The novel gauge field is self-dual and contains both anti-symmetric Lie …

Comment on formulating and generalizing Dirac's, Proca's, and Maxwell's equations with biquaternions or Clifford numbers

Many difficulties of interpretation met by contemporary researchers attempting to recast or generalize Dirac's, Proca's, or Maxwell's theories using biquaternions or Clifford numbers have been encountered long ago by a number of physicists including Lanczos, Proca, and Einstein. In the modern approach initiated by Gursey, these difficulties are solved by recognizing that most generalizations lead to theories describing superpositions of particles of different intrinsic spin and isospin, so that the correct interpretation emerges from the requirement of full Poincare covariance, including space…

Triality and Quantization of Singularities in Massive Fermion

It is proved that fermions can acquire the mass through the additional non-integrable exponential factor. For this propose the special vector potential associated with the spinor field was introduced. Such a vector potential has close relation with the triality property in Dirac spinors and plays crucial role in the construction of massive term. It is shown that the change in phase of a wavefunction round any closed curve with the possibility of there being singularities in our vector potential will lead to the law of quantization of physical constants including the mass. The triality properti…

Biquaternions algebra and its applications by solving of some theoretical physics equations

The functional space of biquaternions is considered on Minkovskiy space. The scalar-vector biquaternions representation is used which was offered by W. Hamilton for quaternions. With introduction of differential operator - a mutual complex gradient (bigradients), which generalize the notion of a gradient on biquaternions space, biquaternionic wave (biwave) equations are considered, their invariance for group of the Lorentz-Puancare transformations is shown and their generalized solutions are obtained. Biquaternionic form of generalized Maxwell-Dirac equation is constructed and its solutions ar…

Quaternionic factorization of the Schroedinger operator and its applications to some first order systems of mathematical physics

We consider the following first order systems of mathematical physics. 1.The Dirac equation with scalar potential. 2.The Dirac equation with electric potential. 3.The Dirac equation with pseudoscalar potential. 4.The system describing non-linear force free magnetic fields or Beltrami fields with nonconstant proportionality factor. 5.The Maxwell equations for slowly changing media. 6.The static Maxwell system. We show that all this variety of first order systems reduces to a single quaternionic equation the analysis of which in its turn reduces to the solution of a Schroedinger equation with bi…

分裂八元数中的 Dirac 与 Maxwell 系统

Dirac and Maxwell systems in split octonions

本文利用分裂八元数表示 SO(4,4)/Spin(4,4) 及三线性不变量，构造推广 Dirac 与 Maxwell 系统的 Lagrangian，并证明其方程对应分裂八元数解析性条件。