scholarly journals Separation of variables and exact solution of the Klein–Gordon and Dirac equations in an open universe

2002 ◽  
Vol 43 (10) ◽  
pp. 4909 ◽  
Author(s):  
Vı́ctor M. Villalba ◽  
Esteban Isasi Catalá
Keyword(s):  
Exact Solution ◽  
Separation Of Variables ◽  
Dirac Equations ◽  
Open Universe ◽  
Klein Gordon

Vacuum effects in a spatially homogeneous and isotropic cosmological background

1992 ◽  
Vol 70 (2-3) ◽  
pp. 143-147
Author(s):  
Victor M. Villalba ◽  
Umberto Percoco
Keyword(s):  
Separation Of Variables ◽  
Particle Creation ◽  
Coupling Constant ◽  
Positive Cosmological Constant ◽  
Dirac Equations ◽  
Klein Gordon ◽  
Dimensionless Coupling ◽  
Spatially Homogeneous ◽  
Dimensionless Coupling Constant ◽  
Vacuum Effects

In this article we obtain, by separation of variables, an exact solution to the Klein–Gordon and Dirac equations in a cosmological, spatially-closed, Robertson–Walker space-time with a positive cosmological constant Λ. The model is associated with a universe filled with radiation. We analyze the phenomenon of particle creation for different values of the dimensionless coupling constant ξ.

Separation of variables for the Klein–Gordon and Dirac equations in two-dimensional space-times with a null coordinate

1990 ◽  
Vol 68 (11) ◽  
pp. 1243-1246
Author(s):  
Victor M. Villalba
Keyword(s):  
Dimensional Space ◽  
Separation Of Variables ◽  
Constant Electric Field ◽  
Coordinate Systems ◽  
Constant Acceleration ◽  
Quantum Distribution ◽  
Dirac Equations ◽  
Curvilinear Coordinate ◽  
Klein Gordon ◽  
Free Case

The separation of variables for the Klein–Gordon and Dirac equations, in the presence of electromagnetic fields, for a class of curvilinear coordinate systems with a null coordinate is presented. We show that these coordinates can be associated with a system with constant acceleration. Exact solutions for the free case and for a particle in a constant electric field are obtained. Finally, the quantum distribution of scalar massless particles in the accelerated frame of reference is computed.

scholarly journals QUANTUM SINGULARITIES IN STATIC SPACETIMES

2011 ◽  
Vol 20 (05) ◽  
pp. 729-743 ◽  
Author(s):  
JOÃO PAULO M. PITELLI ◽  
PATRICIO S. LETELIER
Keyword(s):  
Quantum Mechanics ◽  
Wave Operator ◽  
Point Of View ◽  
Quantum Mechanical ◽  
Mathematical Framework ◽  
Physical Content ◽  
Dirac Equations ◽  
Quantum Particles ◽  
Klein Gordon ◽  
Quantum Singularities

We review the mathematical framework necessary to understand the physical content of quantum singularities in static spacetimes. We present many examples of classical singular spacetimes and study their singularities by using wave packets satisfying Klein–Gordon and Dirac equations. We show that in many cases the classical singularities are excluded when tested by quantum particles but unfortunately there are other cases where the singularities remain from the quantum mechanical point of view. When it is possible we also find, for spacetimes where quantum mechanics does not exclude the singularities, the boundary conditions necessary to turn the spatial portion of the wave operator to be self-adjoint and emphasize their importance to the interpretation of quantum singularities.

Relativistic quantum bouncing particles in a homogeneous gravitational field

2021 ◽  
pp. 2150098
Author(s):  
Ar Rohim ◽  
Kazushige Ueda ◽  
Kazuhiro Yamamoto ◽  
Shih-Yuin Lin
Keyword(s):  
Gravitational Field ◽  
Relativistic Effect ◽  
Relativistic Quantum ◽  
Energy Levels ◽  
Nonrelativistic Limit ◽  
Wave Functions ◽  
Dirac Equations ◽  
Rindler Coordinates ◽  
Klein Gordon ◽  
Homogeneous Gravitational Field

In this paper, we study the relativistic effect on the wave functions for a bouncing particle in a gravitational field. Motivated by the equivalence principle, we investigate the Klein–Gordon and Dirac equations in Rindler coordinates with the boundary conditions mimicking a uniformly accelerated mirror in Minkowski space. In the nonrelativistic limit, all these models in the comoving frame reduce to the familiar eigenvalue problem for the Schrödinger equation with a fixed floor in a linear gravitational potential, as expected. We find that the transition frequency between two energy levels of a bouncing Dirac particle is greater than the counterpart of a Klein–Gordon particle, while both are greater than their nonrelativistic limit. The different corrections to eigen-energies of particles of different nature are associated with the different behaviors of their wave functions around the mirror boundary.

scholarly journals Hawking radiation from cubic and quartic black holes via tunneling of GUP corrected scalar and fermion particles

2019 ◽  
Vol 34 (09) ◽  
pp. 1950057 ◽  
Author(s):  
Wajiha Javed ◽  
Rimsha Babar ◽  
Ali Övgün
Keyword(s):  
Black Holes ◽  
Hawking Radiation ◽  
Generalized Uncertainty Principle ◽  
Hawking Temperature ◽  
Jacobi Method ◽  
Dirac Equations ◽  
Tensor Theory ◽  
Klein Gordon ◽  
The Given ◽  
Do So

We analyze the effect of the generalized uncertainty principle (GUP) on the Hawking radiation from the hairy black hole in U(1) gauge-invariant scalar–vector–tensor theory by utilizing the semiclassical Hamilton–Jacobi method. To do so, we evaluate the tunneling probabilities and Hawking temperature for scalar and fermion particles for the given spacetime of the black holes with cubic and quartic interactions. For this purpose, we utilize the modified Klein–Gordon equation for the Boson particles and then Dirac equations for the fermion particles, respectively. Next, we examine that the Hawking temperature of the black holes do not depend on the properties of tunneling particles. Moreover, we present the corrected Hawking temperature of scalar and fermion particles which look similar in both interactions, but there are different mass and momentum relationships for scalar and fermion particles in cubic and quartic interactions.

scholarly journals Comparison of the Effectiveness of Different Methods for Constructing Exact Solutions to Nonlinear PDEs. Generalizations and New Solutions

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 386 ◽  
Author(s):  
Andrei D. Polyanin
Keyword(s):  
Exact Solutions ◽  
Direct Method ◽  
Separation Of Variables ◽  
Surface Condition ◽  
Reaction Diffusion ◽  
Variable Coefficients ◽  
Nonlinear Pdes ◽  
Boundary Layer Equations ◽  
Compatibility Analysis ◽  
Klein Gordon

The paper shows that, in looking for exact solutions to nonlinear PDEs, the direct method of functional separation of variables can, in certain cases, be more effective than the method of differential constraints based on the compatibility analysis of PDEs with a single constraint (or the nonclassical method of symmetry reductions based on an invariant surface condition). This fact is illustrated by examples of nonlinear reaction-diffusion and convection-diffusion equations with variable coefficients, and nonlinear Klein–Gordon-type equations. Hydrodynamic boundary layer equations, nonlinear Schrödinger type equations, and a few third-order PDEs are also investigated. Several new exact functional separable solutions are given. A possibility of increasing the efficiency of the Clarkson–Kruskal direct method is discussed. A generalization of the direct method of the functional separation of variables is also described. Note that all nonlinear PDEs considered in the paper include one or several arbitrary functions.

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