Biquaternion generalizations of Maxwell's and Dirac's equations and properties of their solutions

Abstract

The differential algebra of biquaternions is used to construct generalized solutions of the biquaternion wave (biquaternion) equation, special cases of which are the systems of equations of Maxwell and Dirac. The method of generalized functions is used to construct solutions and study their properties. Fundamental and generalized solutions are constructed, an analogue of the Kirchhoff formula for the wave equation, which gives a solution to the Cauchy problem. A dynamic analog of the Gauss formula is obtained, which gives a solution to the biwave equation in a limited region under known boundary and initial conditions for the desired biquaternion. Using the constructed analogues of the Kirchhoff and Green's formula, one can obtain analogues of the Green's formula under nonzero initial conditions by decomposing the solution of Eq. (1) into two biquaternions, one of which satisfies the initial conditions. Conditions on the boundary for the second biquaternion will be obtained using the boundary values of the original biquaternion and the first constructed one. After that, formulas similar to (30) give an integral representation of the second biquaternion. Since the equations of Maxwell and Dirac are a special case of biwave equations, the constructed solutions can be used for problems of electrodynamics and field theory. They can be used in experiments, so the field characteristics of EM fields at the boundary can be measured experimentally without solving the SHIE. The constructed solutions can be used in field theory, elementary particles and electro-gravimagnetic interactions.