A Spinor Model for Cascading Two-port Scattering Matrices In Conformal Geometric Algebra

Building on the work in [1], this paper shows how Conformal Geometric Algebra (CGA) can be used to model an arbitrary two-port scattering matrix as a rotation in four dimensional Minkowski space, known as a spinor. This spinor model plays the role of the wave-cascading matrix in conventional microwave network theory. Techniques to translate two-port scattering matrix in and out of spinor form are given. Once the translation is laid out, geometric interpretations are given to the physical properties of reciprocity, loss, and symmetry and some mathe- matical groups are identified. Methods to decompose a network into various sub-networks, are given. An example application of interpolating a 2-port network is provided demonstrating an advantage of the spinor model. Since rotations in four dimensional Minkowski space are Lorentz transformations, this model opens up the field of network theory to physicists familiar with relativity, and vice versa.

A note on spinor form of Lovelock differential identity

The paper is devoted to 2-spinor calculus methods in general relativity. New spinor form of the Lovelock differential identity is suggested. This identity is second-order identity for the Riemann curvature tensor. We provide an example that our spinorial treatment of Lovelock identity is effective for the description of solutions of Einstein–Maxwell equations. It is shown that the covariant divergence of Lipkin’s zilch tensor for the free Maxwell field vanishes on the solutions of Einstein–Maxwell equations if and only if the energy–momentum tensor of the electromagnetic field is Weyl-compatible.

ON THE EQUIVALENCE OF THE MASSLESS DKP EQUATION AND THE MAXWELL EQUATIONS IN THE SHUWER

In this paper, a general relativistic wave equation is written to deal with electromagnetic waves in the background of the Shuwer. We obtain the exact form of this equation in a second-order form. On the other hand, by using spinor form of the Maxwell equations the propagation problem is reduced to the solution of the second-order differential equation of complex combination of the electric and magnetic fields. For these two different approaches, we obtain the spinors in terms of field strength tensor. We show that the Maxwell equations are equivalence to the mDKP equation in the Shuwer.

TWISTORS IN CONFORMALLY FLAT EINSTEIN FOUR-MANIFOLDS

This paper studies the two-component spinor form of massive spin-[Formula: see text] potentials in conformally flat Einstein four-manifolds. Following earlier work in the literature, a nonvanishing cosmological constant makes it necessary to introduce a supercovariant derivative operator. The analysis of supergauge transformations of primary and secondary potentials for spin-[Formula: see text] shows that the gauge freedom for massive spin-[Formula: see text] potentials is generated by solutions of the supertwistor equations. The supercovariant form of a partial connection on a nonlinear bundle is then obtained, and the basic equation of massive secondary potentials is shown to be the integrability condition on super β-surfaces of a differential operator on a vector bundle of rank 3. Moreover, in the presence of boundaries, a simple algebraic relation among some spinor fields is found to ensure the gauge invariance of locally supersymmetric boundary conditions relevant for quantum cosmology and supergravity.

SPIN-$\frac{3}{2}$ POTENTIALS IN BACKGROUNDS WITH BOUNDARY

This paper studies the two-spinor form of the Rarita-Schwinger potentials subject to local boundary conditions compatible with local supersymmetry. The massless Rarita-Schwinger field equations are studied in four-real-dimensional Riemannian backgrounds with boundary. Gauge transformations on the potentials are shown to be compatible with the field equations providing the background is Ricci-flat, in agreement with previous results in the literature. However, the preservation of boundary conditions under such gauge transformations leads to a restriction of the gauge freedom. The recent construction by Penrose of secondary potentials which supplement the Rarita-Schwinger potentials is then applied. The equations for the secondary potentials, jointly with the boundary conditions, imply that the background four-geometry is further restricted to be totally flat. The analysis of other gauge transformations confirms that, in the massless case, the only admissible class of Riemannian backgrounds with boundary is totally flat.

COMPLEXIFIED THEORY OF POSITIVE-FREQUENCY MAXWELL-DIRAC FIELDS

We present a method whereby the equations of motion yielding the explicit two-component spinor form of the complete Maxwell-Dirac theory in complex Minkowski space may be directly derived from two variational principles. One of these dynamical statements gives rise to the first half of the electromagnetic theory which particularly appears as the equations of motion involving a slightly modified version of the free part of the conventional Maxwell Lagrangian density. The other principle actually involves a holomorphic two-spinor expression for the full Maxwell-Dirac Lagrangian density which leads to the second half along with the Dirac equations that carry the relevant covariant derivative operator. It is shown explicitly how the Bianchi identities of the complete theory can be established in a manifestly covariant way. A system of essentially equivalent wave equations for positive-frequency photons and electrons is then exhibited. In particular, we make use of certain skew-symmetric operators to obtain an extended form of the Feynman-Gell-Mann equations.