Hawking Radiation of Dirac Particles in a Variable-Mass Kerr Space-Time

The Hawking effect of Dirac particles in a variable-mass Kerr space-time is investigated by using a method called as the generalized tortoise coordinate transformation. The location and the temperature of the event horizon of the non-stationary Kerr black hole are derived. It is shown that the temperature and the shape of the event horizon depend not only on the time but also on the angle. However, the Fermi–Dirac spectrum displays a residual term which is absent from that of Bose–Einstein distribution.     

Hawking radiation of scalar and Dirac quanta from a Kerr black hole

Unruh’s technique of replacing collapse by boundary conditions on the past horizon (theξ-quantisation scheme) for the derivation of the well-known Hawking radiation is extended to the Kerr black hole for the scalar and especially for the spin half field. The expectation value of the energy momentum tensor is evaluated asymptotically in theξ-vacuum state yielding explicitly the net Hawking flux of scalar and spin half quanta. The appropriate statistical distribution that emerges naturally for Dirac quanta validates the ξ-scheme for fermions and confirms the association of temperature with a Kerr black hole.     

The Dirac Equation in the Bertotti-Robinson Space-Time

The Dirac equation is considered in the uniform electromagnetic field space of Bertotti-Robinson with charge coupling. The methods of separation of variables and decoupling are easily achieved. The separated axial equation is reduced to a rare Riccati type of differential equation. The behaviour of potentials, their asymptotic solutions and the conserved currents of the Dirac equation are found.     

Optical analogue of relativistic Dirac solitons in binary waveguide arrays

We study analytically and numerically an optical analogue of Dirac solitons in binary waveguide arrays in the presence of Kerr nonlinearity. Pseudo-relativistic soliton solutions of the coupled-mode equations describing dynamics in the array are analytically derived. We demonstrate that with the found soliton solutions, the coupled mode equations can be converted into the nonlinear relativistic 1D Dirac equation. This paves the way for using binary waveguide arrays as a classical simulator of quantum nonlinear effects arising from the Dirac equation, something that is thought to be impossible to achieve in conventional (i.e. linear) quantum field theory.     

Integration method for the Dirac equation

An integration method for the Dirac equation is proposed. The method, based on diagonalization, reduces the problem to one of integration of independent second-order differential equations.     

Equivalence and hermiticity of Dirac Hamiltonians in the Kerr gravitational field

In the paper, for the Kerr field, we prove that Chandrasekhar's Dirac Hamiltonian and the self‐adjoint Hamiltonian with a flat scalar product of the wave functions are physically equivalent. Operators of transformation of Chandrasekhar's Hamiltonian and wave functions to the η representation with a flat scalar product are defined explicitly. If the domain of the wave functions of Dirac's equation in the Kerr field is bounded by two‐dimensional surfaces of revolution around the z axis, Chandrasekhar's Hamiltonian and the self‐adjoint Hamiltonian in the η representation are Hermitian with equality of the scalar products, .

     

Higher dimensional supersymmetric quantum mechanics and Dirac equation

We exhibit the supersymmetric quantum mechanical structure of the full 3+1 dimensional Dirac equation considering ‘mass’ as a function of coordinates. Its usefulness in solving potential problems is discussed with specific examples. We also discuss the ‘physical’ significance of the supersymmetric states in this formalism.     

On the Eigenvalues of the Fermionic Angular Eigenfunctions in the Kerr Metric

In view of a result recently published in the context of the deformation theory of linear Hamiltonian systems, we reconsider the eigenvalue problem associated with the angular equation arising after the separation of the Dirac equation in the Kerr metric, and we show how a quasilinear first order PDE for the angular eigenvalues can be derived efficiently. We also prove that it is not possible to obtain an ordinary differential equation for the eigenvalues when the role of the independent variable is played by the particle energy or the black hole mass. Finally, we construct new perturbative expansions for the eigenvalues in the Kerr case and obtain an asymptotic formula for the eigenvalues in the case of a Kerr naked singularity.     

Nodal theorems for the Dirac equation in d ≥ 1 dimensions

A single particle obeys the Dirac equation in spatial dimensions and is bound by an attractive central monotone potential that vanishes at infinity. In one dimension, the potential is even, and monotone for The asymptotic behavior of the wave functions near the origin and at infinity are discussed. Nodal theorems are proven for the cases and , which specify the relationship between the numbers of nodes n₁ and n₂ in the upper and lower components of the Dirac spinor. For , whereas for if and if where and This work generalizes the classic results of Rose and Newton in 1951 for the case Specific examples are presented with graphs, including Dirac spinor orbits      

Solution of the Dirac equation in plane wave metric backgrounds

We present a new solution of the Dirac equation in the background of a plane wave metric. We examine the relation between sections of the exterior and Clifford bundles of a (pseudo-)Riemannian manifold. A spinor calculus is established and used to investigate a new solution of the Dirac equation lying in a minimal left ideal characterized by a certain idempotent projector.     

在赝自旋对称性势场中运动的相对论粒子的束缚态

在赝自旋对称性条件下,分别求解了在Kratzer型、Hulthén型和Poschl-Teller型标量势与矢量势场中运动的相对论粒子的Klein-Gordon方程和Dirac方程,给出了它们的束缚态能谱和相对论性波函数.     

A remark on the Dirac equation

We give a simple deductive derivation of the Dirac equation for a free particle. Our construction provides a clear distinction between what are the physical contents and what are purely mathematical expedients in the formalism of quantum mechanics (QM).     

A simple efficient methodology for Dirac equation in minimal length quantum mechanics

We solve the modified Dirac equation by adding a harmonic oscillator potential and implementing the Nikiforov–Uvarov technique. The closed forms of solutions are reported in a quite simple and systematic manner.     

Spin symmetric solutions of Dirac equation with PSschl-Teller potential

The approximate analytical solutions of the Dirac equation with the Poeschl-Teller potential is presented for arbitrary spin-orbit quantum number κ within the framework of the spin symmetry concept. The energy eigenvalues and the corresponding two Dirac spinors are obtained approximately in closed forms. The limiting cases of the energy eigenvalues and the two Dirac spinors are briefly discussed.     

具有Kratzer型标量势与矢量势的Klein-Gordon方程和Dirac方程的束缚态

给出了具有Kratzer型标量势与矢量势的Klein-Gordon方程和Dirac方程的s波束缚态解.     

Quantum-classical correspondence of the Dirac equation with a scalar-like potential

Quantum matrix elements of the coordinate, momentum and the velocity operator for a spin-1/2 particle moving in a scalar-like potential are calculated. In the large quantum number limit, these matrix elements give classical quantities for a relativistic system with a position-dependent mass. Meanwhile, the Klein-Gordon equation for the spin-0 particle is discussed too. Though the Heisenberg equations for both the spin-0 and spin-1/2 particles are unlike the classical equations of motion, they go to the classical equations in the classical limit.      

Yangian symmetries and integrability of the Dirac equation with spin symmetry

We show that a Yangian symmetry, namely, Y(su(2))$Y\left(su\left(2\right)\right)$, exists in the Dirac equation with spin symmetry when the potential term takes a Coulomb form. We construct the generators of Y(su(2))$Y\left(su\left(2\right)\right)$ explicitly and get the energy spectrum of this model from the representation theory for Y(su(2))$Y\left(su\left(2\right)\right)$. We also show that this model is integrable, from RTT relations.     

Invariance properties of the Dirac equation with external electro-magnetic field

In this paper, we attempt to obtain the nature of the external field such that the Dirac equation with external electro-magnetic field is invariant. The Poincaré group, which is the maximal symmetry group for field free case, is constrained by the presence of the external field. Introducing infinitesimal transformation ofx and ψ, we apply Lie’s extended group method to obtain the class of external field which admit of the invariance of the equation. It is important to note that the constraints for the existence of invariance are explicity on the electric and magnetic field, though only potentials explicity appears in the equation. Presented at the Sixth Chittagong Conference on Mathematical Physics, January 2001.