Related papers: Quaternionic Electron Theory: Geometry, Algebra an…
In this paper, we present some applications of quaternions and octonions. We present the real matrix representations for complex octonions and some of their properties which can be used in computations where these elements are involved.…
On base of differential biquaternions algebra and generalized functions theory the biquaternionic wave equation is considered under vector representation of its structural coefficient. Its generalized solutions are constructed, which…
We show that Dirac 4-spinors admit an entirely equivalent formulation in terms of 2-spinors defined over the split-quaternions. In this formalism, a Lorentz transformation is represented as a $2 \times 2$ unitary matrix over the…
Consider a formally self-adjoint first order linear differential operator acting on pairs (2-columns) of complex-valued scalar fields over a 4-manifold without boundary. We examine the geometric content of such an operator and show that it…
Using the language of the Geometric Algebra, we recast the massless Dirac bispinor as a set of Lorentz scalar, bivector, and pseudoscalar fields that obey a generalized form of Maxwell's equations of electromagnetism. The spinor's unusual…
The recent literature shows a renewed interest, with various independent approaches, in the classical theories for spin. Considering the possible interest of those results, at least for the electron case, we purpose in this paper to explore…
We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…
A new approach is proposed for an electromagnetic field geometrisation. We show that interacting Maxwell and Dirac fields can be considered as a single connected space-time 4-manifold. The Dirac spinors appear wihtin such approach as basic…
Complex geometry represents a fundamental ingredient in the formulation of the Dirac equation by the Clifford algebra. The choice of appropriate complex geometries is strictly related to the geometric interpretation of the complex imaginary…
Quaternions were appeared through Lagrangian formulation of mechanics in Symplectic vector space. Its general form was obtained from the Clifford algebra, and Frobenius' theorem, which says that "the only finite-dimensional real division…
Previously (A. Akhmeteli, J. Math. Phys., v. 52, p. 082303 (2011)), the Dirac equation in an arbitrary electromagnetic field was shown to be generally equivalent to a fourth-order equation for just one component of the four-component Dirac…
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…
The interior structure of arbitrary sets of quaternion units is analyzed using general methods of the theory of matrices. It is shown that the units are composed of quadratic combinations of fundamental objects having a dual mathematical…
Quaternion (Q-) mathematics formally contains many fragments of physical laws; in particular, the Hamiltonian for the Pauli equation automatically emerges in a space with Q-metric. The eigenfunction method shows that any Q-unit has an…
A representation of the Dirac algebra, derived from first principles, can be related to the combinations of unit charges which determine particle structures. The algebraic structure derives from a broken symmetry between 4-vectors and…
A new concept for the geometrisation of electromagnetic interaction is proposed. Instead of the concept "extended field--point sources", interacting Maxwell's and Dirac's fields are considered as a unified closed noneuclidean and…
The known equivalence of 8-dimensional chiral spinors and vectors, also referred to as triality, is discussed for (4+4)-space. Split octonionic representation of SO(4,4) and Spin(4,4) groups and the trilinear invariant form are explicitly…
We extract the square root of the Minkowski metric using Dirac/Clifford matrices. The resulting $4\times 4$ operator $d{\bf S}$ that represents the square root, can be used to transform four vectors between relatively moving observers. This…
The paper aims to adopt the complex quaternion and octonion to formulate the field equations for electromagnetic and gravitational fields. Applying the octonionic representation enables one single definition to combine some physics contents…
Starting with the quaternionic Minkowski space-time and its four-vector representation, a rotational analogue of the quaternionic Dirac equation in the electromagnetic field is developed, which includes not only the energy solutions but…