Geometric Algebra Framework Applied to Circuits with Non-sinusoidal Voltages and Currents

We apply a well known technique of theoretical physics, known as Geometric Algebra or Clifford algebra, to linear electrical circuits with non-sinusoidal voltages and currents. We rederive from the first principles the Geometric Algebra approach to the apparent power decomposition. The important new point consists in a choice of a natural convenient basis in the Clifford vector space which simplifies considerably the presentation. Thus we are able to derive a number of general results which are missing in the former papers. In particular, a natural correspondence with the Current Physical Components approach is shown.

On solutions of the Yang-Mills equations in the algebra of h-forms

We study the Yang-Mills equations in the algebra of h-forms, which is developed in the works of N. G. Marchuk and the author. The algebra of h-forms is a special geometrization of the Clifford algebra and is a generalization of the Atiyah-K¨ahler algebra. We discuss an invariant subspace of the constant Yang-Mills operator in the algebra of h-forms and present particular classes of solutions of the Yang-Mills equations.

Theory of Spinors in Curved Space-Time

By means of Clifford Algebra, a unified language and tool to describe the rules of nature, this paper systematically discusses the dynamics and properties of spinor fields in curved space-time, such as the decomposition of the spinor connection, the classical approximation of the Dirac equation, the energy-momentum tensor of spinors and so on. To split the spinor connection into the Keller connection Υμ∈Λ1 and the pseudo-vector potential Ωμ∈Λ3 not only makes the calculation simpler, but also highlights their different physical meanings. The representation of the new spinor connection is dependent only on the metric, but not on the Dirac matrix. Only in the new form of connection can we clearly define the classical concepts for the spinor field and then derive its complete classical dynamics, that is, Newton’s second law of particles. To study the interaction between space-time and fermion, we need an explicit form of the energy-momentum tensor of spinor fields; however, the energy-momentum tensor is closely related to the tetrad, and the tetrad cannot be uniquely determined by the metric. This uncertainty increases the difficulty of deriving rigorous expression. In this paper, through a specific representation of tetrad, we derive the concrete energy-momentum tensor and its classical approximation. In the derivation of energy-momentum tensor, we obtain a spinor coefficient table Sabμν, which plays an important role in the interaction between spinor and gravity. From this paper we find that Clifford algebra has irreplaceable advantages in the study of geometry and physics.

Projective Metric Geometry and Clifford Algebras

Each vector space that is endowed with a quadratic form determines its Clifford algebra. This algebra, in turn, contains a distinguished group, known as the Lipschitz group. We show that only a quotient of this group remains meaningful in the context of projective metric geometry. This quotient of the Lipschitz group can be viewed as a point set in the projective space on the Clifford algebra and, under certain restrictions, leads to an algebraic description of so-called kinematic mappings.