We are familiar with Dirac equation in flat space by which we can investigate the behaviour of half-integral spin particle. With the introduction of general relativistic effects the form of the Dirac equation will be modified. For the cases of different background geometry like Kerr, Schwarzschild etc. the corresponding form of the Dirac equation as well as the solution will be different. In 19...

In this article, we study an inverse boundary value problem for a Dirac-type equation. Let V be a bundle with a base manifold M which has a Dirac structure with a chirality operator F . We assume that we can observe on the lateral boundary ∂M ×R+ the boundary values of solutions of the Dirac-type equation in M × R+. When the coefficients of the Dirac-type equation are time-independent, i.e. we ...

Solving the Dirac equation is a formidable task due to the high frequency and the degrees of freedom involved. However, this high frequency allows one to obtain an approximation to the equation. Here, we directly solve the Dirac equation using an envelope method and derive analytical solutions of Dirac wave packets to first order for the small momentum spread. We apply the insight gained from t...

We construct exact solutions of the Einstein-Dirac equation, which couples the gravitational eld with an eigenspinor of the Dirac operator via the energy-momentum tensor. For this purpose we introduce a new eld equation generalizing the notion of Killing spinors. The solutions of this spinor eld equation are called weak Killing spinors (WK-spinors). They are special solutions of the Einstein-Di...

The spectrum of a Schrödinger operator with a perfect honeycomb lattice potential has special points, called Dirac points where the lowest two branches of the spectrum touch. Deformations can result in the merging and disappearance of the Dirac points and the originally intersecting dispersion relation branches separate. Corresponding to these deformations, nonlinear envelope equations are deri...

Tensor, matrix and quaternion formulations of Dirac-Kähler equation for massive and massless fields are considered. The equation matrices obtained are simple linear combinations of matrix elements in the 16-dimensional space. The projection matrix-dyads defining all the 16 independent equation solutions are found. A method of computing the traces of 16-dimensional Petiau-Duffin-Kemmer matrix pr...

The (3+1)-dimensional Dirac equation with position dependent mass in 4-vector electromagnetic fields is considered. Using two over-simplified examples (the Dirac-Coulomb and Dirac-oscillator fields), we report energylevels crossing as a spectral property or as an effect of the hidden supersymmetric quantum mechanical language and/or quasi-parity signatures. Under different settings of the relat...

Tensor and matrix formulations of Dirac-Kähler equation for massive and massless fields are considered. The equation matrices obtained are simple linear combinations of matrix elements in the 16-dimensional space. The projection matrix-dyads defining all the 16 independent equation solutions are found. A method of computing the traces of 16-dimensional Petiau-Duffin-Kemmer matrix product is con...

We consider the behavior of particles at ultra relativistic energies, for both the Klein-Gordon and Dirac equations. We observe that the usual description is valid for energies such that we are outside the particle’s Compton wavelength. For higher energies however, both the Klein-Gordon and Dirac equations get modified and this leads to some new effects for the particles, including the appearan...