RELATIONS AND EQUATIONS OF PHYSICS IN HYPERSPACE OF DOUBLE QUATERNIONS

Abstract

The use of algebraic methods for a priori modeling of natural phenomena and processes allows us to identify general patterns in the physical world. The construction of covering algebras leads to new numerical systems that can be used to obtain not only known relations and equations, but also to establish new dependencies. In addition, this approach opens up the possibility of discovering new algebras with unique properties that reflect unknown characteristics of material objects. In this work, we construct an algebra that is isomorphic to the algebra of complex Hamilton quaternions and the algebra of Pauli matrices, as well as an algebra whose product of numbers contains left and right trivial terms. Generalizations of Cauchy-Riemann conditions for double quaternions are obtained, and it is shown that the first condition of them corresponds to a large number of physical equations (local conservation laws of mass and charge, the Lorentz gauge condition, Fick's diffusion equation, and others). The second condition of hyperanalyticity of a double quaternion sets the evolution of the vector component of a hyperfunction. Quaternion expressions are established that represent the formulas of relativistic and classical mechanics under various parameterizations, and in particular, the formulas for the Lorentz transformation are obtained. The introduction of angular double quaternions allows us to obtain formulas for the Lorentz scalar and de Broglie relations for the energy and momentum of a wave. The search for eigenquaternions for the 4-gradient operator reduces to solving equations of damped or growing oscillations. The presence of special quaternions in hyperspace corresponds to the conservation of 4-potentials at the substantive level when they change in a local region. In this case, the components of the quaternion satisfy the equations of Maxwell's theory. An expression for the substantial operator has been found; the presence of substantial hyperanalyticity of the 4-potential in the absence of its hyperanalyticity at the local level has been demonstrated using the example of the Maxwell field. The form of the 4-force acting on a charged body has been established.