Abstract:
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 1972, Teukolsky wrote the Dirac equation in Kerr geometry. Chandrasekhar separated it into radial and angular parts in 1976. Later Chakrabarti solved the angular equation in 1984. In 1999 Mukhopadhyay and Chakrabarti have solved the radial Dirac equation in Kerr geometry in a spatially complete manner. In this review we will discuss these developments systematically and present some solutions.

Abstract:
In this paper, we will first clarify the physical meaning of having a minimum measurable time. Then we will combine the deformation of the Dirac equation due to the existence of minimum measurable length and time scales with its deformation due to the doubly special relativity. We will also analyse this deformed Dirac equation in curved spacetime, and observe that this deformation of the Dirac equation also leads to a non-trivial modification of general relativity. Finally, we will analyse the stochastic quantization of this deformed Dirac equation on curved spacetime.

Abstract:
In the first part of the paper we give a tensor version of the Dirac equation. In the second part we formulate and analyse a simple model equation which for weak external fields appears to have properties similar to those of the 2--dimensional Dirac equation.

Abstract:
In this paper, we propose finite temperature Dirac equation, which can describe the quantum systems in an arbitrary temperature for a relativistic particle of spin-1/2. When the temperature T=0, it become Dirac equation. With the equation, we can study the relativistic quantum systems in an arbitrary temperature.

Abstract:
We point out that the anticommutation properties of the Dirac matrices can be derived without squaring the Dirac hamiltonian, that is, without explicit reference to the Klein-Gordon equation. We only require the Dirac equation to admit two linearly independent plane wave solutions with positive energy for all momenta. The necessity of negative energies as well as the trace and determinant properties of the Dirac matrices are also a direct consequence of this simple and minimal requirement.

Abstract:
We present a general description of topological insulators from the point of view of Dirac equations. The Z_{2} index for the Dirac equation is always zero, and thus the Dirac equation is topologically trivial. After the quadratic B term in momentum is introduced to correct the mass term m or the band gap of the Dirac equation, the Z_{2} index is modified as 1 for mB>0 and 0 for mB<0. For a fixed B there exists a topological quantum phase transition from a topologically trivial system to a non-trivial one system when the sign of mass m changes. A series of solutions near the boundary in the modified Dirac equation are obtained, which is characteristic of topological insulator. From the solutions of the bound states and the Z_{2} index we establish a relation between the Dirac equation and topological insulators.

Abstract:
A symmetry reduction of the Dirac equation is shown to yield the system of ordinary differential equations whose integrability by quadratures is closely connected to the stationary mKdV hierarchy.

Abstract:
Using the China unitary principle to test the Dirac theoryfor the hydrogen atomic spectrum shows that the standard Dirac function withthe Dirac energy levels is only one the formal solutions of theDirac-Coulomb equation, which conceals some pivotal mathematicalcontradictions. The theorem of existence of solution of the Dirac equationrequires an important modification to the Dirac angular momentum constantthat was defined by Dirac's algebra. It derives the modified radial Diracequation which has the consistency solution involving the quantum neutronradius and the neutron binding energy. The inevitable solution for otheratomic energy states is only equivalent to the Bohr solution. It concludesthat the Dirac equation is more suitable to describe the structure ofneutron. How to treat the difference between the unitary energy levels andthe result of the experimental observation of the atomic spectrums for thehydrogen atom needs to be solved urgently.