The relativistic generalization of the screened potential problem is given. The Klein–Gordon and Dirac equations in the presence of the Hulthén potential V(r)=−αδe−δr/(1−e−δr) are solved by using the usual approximation of the centrifugal potential. The approach proposed by Biedenharn for the Dirac–Coulomb problem is applied to the spin- case. The energy spectrum and the scattering function are obtained both for spin-0 and spin- particles. The nonrelativistic limit is discussed. When the screening parameter δ vanishes, it is shown that the obtained wavefunctions and energies become the same as that of the Coulomb potential.

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The supersymetric path integrals in solving the problem of relativistic spinning particle interacting with pseudoscalar potentials is examined. The relative propagator is presented by means of path integral, where the spin degrees of freedom are described by odd Grassmannian variables and the gauge invariant part of the effective action has a form similar to the standard pseudoclassical action given by Berezin and Marinov. After integrating over fermionic variables (Grassmannian variables), the problem is reduced to a nonrelativistic one with an effective supersymetric potential. Some explicit examples are considered, where we have extracted the energy spectrum of the electron and the wave functions.

In this paper we have studied the problem of scalar particles pair creation by an electric field in the presence of a minimal length. Two sets of exact solutions for the Klein Gordon equation are given in momentum space. Then the canonical method based on Bogoliubov transformation connecting the "in" with the "out" states is applied to calculate the probability to create a pair of particles and the mean number of created particles. The number of created particles per unit of time per unit of length, which is related directly to the experimental measurements, is calculated. It is shown that, with an electric field less than the critical value, the minimal length minimizes the particle creation. It is shown, also, that the limit of zero minimal length reproduces the known results corresponding to the ordinary quantum fields.

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The (3+1)-dimensional Dirac equation is solved in the presence of the radial pseudoscalar Hulthén potential by using the usual approximation of the centrifugal potential. The approach proposed by Biedenharn for the Dirac-Coulomb problem is applied. Analytic bounded solutions of the Dirac equation with the pseudoscalar Hulthén potential are obtained in contrast to the pseudoscalar Coulomb one where there is no bounded solutions.

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The problem of particle creation from vacuum in a flat Robertson-Walker spacetime is studied. Two sets of exact solutions for the Klein-Gordon equation are given when the scale factor is a 2 (η) = a + b tanh(λη) + c tanh 2 (λη). Then the canonical method based on Bogoliubov transformation is applied to calculate the pair creation probability and the density number of created particles. Some particular cosmological models such as radiation dominated universe and Milne universe are discussed. For both cases the vacuum to vacuum transition probability is calculated and the imaginary part of the effective action is extracted.