In this paper the Proca field equations for a massive gauge particle are obtained in the presence of a natural momentum cutoff “ p_max ” based on a covariant generalization of a one-parameter extension of the Heisenberg algebra. The Yukawa potential for a static point source in the presence of p_max (generalized Yukawa potential) is obtained analytically and it is shown that in contrast with the Yukawa potential for a static point source in Proca electrodynamics, the generalized Yukawa potential has a finite value at the location of the static point source. Our calculations demonstrate that the Coulomb potential, the Yukawa potential, and the Coulomb potential in the presence of p_max can be derived from the generalized Yukawa poitential. We show that the free space solutions of Proca electrodynamics in the presence of p_max describe a massive gauge particle with the effective mass m_eff = m/√(1 - ( mc/p_max)^2) , where m is the rest mass of the ordinary Proca particle. Numerical estimations in Sect. 5, show that the lower bound for p_max must take the value ( p_max)_min = (91.187 ± 0.007) GeV/c in order to avoid complex values for the effective mass m_eff . This lower bound for p_max is near to the momentum scale of the electroweak interactions. It should be mentioned that for the very large values of p_max the results of this work reduce to the well-known results of standard Proca electrodynamics.

In this paper, a higher-derivative model for electrodynamics is presented in a D+1 dimensional Minkowski space-time by introducing a form factor into the kinetic term of Maxwell theory as -1/4µ0 FµνFµν→ -1/4µ0 FµνFHD2(ℓ2□)Fµν , where is a characteristic length scale. Our calculations show that for DÊÎ{3, 4, 5} the electrostatic potential of a point charge is finite at the position of the point charge in this higher-derivative modification of Maxwell's theory. For D=3 the explicit form of the potential and the electric field of a point charge are obtained analytically in this higher-derivative electrodynamics. According to numerical estimations, the upper bound for the characteristic length scale ℓ is ℓmax ~1/100ℓelectroweak , where ℓelectroweak= 10-18m is the electroweak length scale. Finally, it should be emphasized that for ℓ<<1 the results of this paper are compatible with the results of ordinary Maxwell theory.

In Maxwell electrostatics the electrostatic potential of a point charge is singular at the position of the point charge. In this paper, a nonsingular model for a point charge potential is presented in the presence of a momentum cutoff pmax based on a one-parameter deformation of the Heisenberg algebra in a 2D-dimensional phase space. For pmax→∞, the results of this paper reduce to the results of ordinary Maxwell electrostatics for a point charge.

Recently, a one-parameter extension of the covariant Heisenberg algebra with the extension parameter [Formula: see text] [Formula: see text] is a non-negative constant parameter which has a dimension of [Formula: see text] in a [Formula: see text]-dimensional globally flat spacetime has been presented which is a covariant generalization of the Kempf–Mangano algebra [see G. P. de Brito, P. I. C. Caneda, Y. M. P. Gomes, J. T. Guaitolini Junior and V. Nikoofard, Adv. High Energy Phys. 2017, 4768341 (2017) and A. Kempf and G. Mangano, Phys. Rev. D 55, 7909 (1997)]. The Abelian Proca model is reformulated from the viewpoint of the above one-parameter extension of the covariant Heisenberg algebra. It is shown that the free space solutions of the above modified Proca model describe two massive vector particles with different effective masses [Formula: see text] where [Formula: see text] is the characteristic length scale in our model. In addition, the Feynman propagator in momentum space for the modified Abelian Proca model is calculated analytically. Our numerical estimations show that the maximum value of [Formula: see text] in a four-dimensional spacetime is near the electroweak length scale, i.e. [Formula: see text]. We show that in the infrared/large-distance domain, the modified Proca model behaves like an Abelian massive Lee–Wick model which has been presented by Accioly and his co-workers in A. Accioly, J. Helayel-Neto, G. Correia, G. Brito, J. de Almeida and W. Herdy, Phys. Rev. D 93, 105042 (2016). The short-distance behavior of the modified Proca model is studied in the massless limit and the explicit forms of the inhomogeneous infinite derivative Maxwell equation and the infinite derivative Poisson equation are obtained. Finally, note that in the low-energy limit [Formula: see text], the results of this paper are compatible with the results of the usual Proca model.

Recently a one-parameter extension of the covariant Heisenberg algebra with the extension parameter l (l is a non-negative constant parameter which has a dimension of [momentum]^-1) in a (D+1)-dimensional Minkowski space-time has been presented [G. P. de Brito, P. I. C. Caneda, Y. M. P. Gomes, J. T. Guaitolini Junior and V. Nikoofard, Effective models of quantum gravity induced by Planck scale modifications in the covariant quantum algebra, Adv. High Energy Phys. 2017 (2017) 4768341]. The Abelian Proca model is reformulated from the viewpoint of the above one-parameter extension of the covariant Heisenberg algebra. It is shown that the free space solutions of the above modified Proca model satisfy the modified dispersion relation p^2/(1+Λ^2/2ħ^2p^2)^2=m^2c^2 where Λ=ħ l is the characteristic length scale in our model. This modified dispersion relation describes two massive vector particles with the effective masses M__±(Λ)=2m/1∓√(1-2(mcΛ/ħ)^2). Numerical estimations show that the maximum value of Λ in a four-dimensional space-time is near to the electroweak length scale, i.e., Λ__max∼ l__electroweak∼10^-18 m. We show that in the infrared/large-distance domain the modified Proca model behaves like an Abelian massive Lee-Wick model which has been presented by Accioly and his co-workers [A. Accioly, J. Helayel-Neto, G. Correia, G. Brito, J. de Almeida and W. Herdy, Interparticle potential energy for D-dimensional electromagnetic models from the corresponding scalar ones, Phys. Rev. D 93 (2016) 105042].

In 2017, G. P. de Brito and co-workers suggested a covariant generalization of the Kempf-Mangano algebra in a $(D+1)$-dimensional Minkowski space-time [A. Kempf and G. Mangano, Phys. Rev. D \textbf{55}, 7909 (1997); G. P. de Brito, P. I. C. Caneda, Y. M. P. Gomes, J. T. Guaitolini Junior, and V. Nikoofard, Adv. High Energy Phys. \textbf{2017}, 4768341 (2017)]. It is shown that reformulation of a real scalar field theory from the viewpoint of the covariant Kempf-Mangano algebra leads to an infinite derivative Klein-Gordon wave equation which describes two bosonic particles in the free space (a usual particle and a ghostlike particle). We show that in the low-energy (large-distance) limit our infinite derivative scalar field theory behaves like a Pais-Uhlenbeck oscillator for a spatially homogeneous field configuration $\phi(t,\vec{\textbf{x}})=\phi(t)$. Our calculations show that there is a characteristic length scale $\delta$ in our model whose upper limit in a four-dimensional Minkowski space-time is close to the nuclear scalar, i.e., $\delta_{max}\sim \delta_{nuclear\ scale}\sim 10^{-15}\, m$. Finally, we show that there is an equivalence between a non-local real scalar field theory with a non-local form factor ${\cal K}(x-y)= -\frac{\square_x}{(1-\frac{\delta^2}{2}\square_x)^2} \ \delta^{(D+1)}(x-y)$ and an infinite derivative real scalar field theory from the viewpoint of the covariant Kempf-Mangano algebra.

In the recent decade, many investigations have been done in the framework of generalized uncertainty principle (GUP), but the phenomenology of models in this framework are less studied. In this work, the applications of Biot–Savart law in the presence of a minimal length scale are investigated. We obtain the modified magnetostatic field from an infinitely long, straight wire carrying current [Formula: see text]. Also, the modified magnetostatic field from a circular loop carrying current [Formula: see text] and the modified magnetostatic field of an ideal solenoid are found. It is interesting to note that in the limit [Formula: see text], all of the modified magnetostatic fields become their usual forms.

Iacopini and Zavattini [Vacuum polarization effects in the [Formula: see text] atom and the Born–Infeld electromagnetic theory, Nuovo Cimento B 78 (1983) 38–52] proposed a [Formula: see text]-two-parameter modification of Born–Infeld electrodynamics, in which the classical self-energy for an electron takes a finite value for [Formula: see text]. In this paper, we want to study a cylindrical capacitor from the viewpoint of Iacopini–Zavattini nonlinear electrodynamics analytically. The capacitance, the electrostatic potential energy, and the potential difference between the plates of a cylindrical capacitor are calculated in the framework of Iacopini–Zavattini electrodynamics for two specific values of [Formula: see text] and [Formula: see text]. The study of the behavior of a nonlinear cylindrical capacitor in the weak electric fields shows that our results are compatible with the correspondence principle, i.e. we recover the results of Maxwell electrodynamics in the weak field regime. Finally, the invariance of Iacopini–Zavattini nonlinear electrodynamics under the duality transformation is investigated.

The solution of integro-differential Schrodinger equation (IDSE) which was introduced by physicists has a great role in the fields of science. The purpose of this paper comes in two parts. First, studying the relationship between integro-differential Schrodinger equation with a symmetric non-local potential and one-dimensional Schrodinger equation with a position-dependent effective mass. Second, we show that the quantum Hamiltonian for a particle with position-dependent mass after applying Liouville–Green transformations will be converted to a quantum Hamiltonian for a particle with constant mass.

The nonlinear capacitors are important devices in modern technologies and applied physics. The aim of this paper is to calculate exactly the capacitance and the electrostatic potential energy of a nonlinear parallel-plate capacitor by using a two-parameter modification of Born-Infeld electrodynamics. Our calculations show that the capacitance and the electrostatic potential energy of a nonlinear parallel-plate capacitor in modified Born-Infeld theory have the weak field expansions C=ϵ_0 A/d+ O(q^2) and U=q^2/2(ϵ_0 A/d)+ O(q^4), where q is the amount of electric charge on each plate of the capacitor. It is demonstrated that the results of this paper are in agreement with the results of Maxwell electrodynamics for weak electric fields. Numerical evaluations show that the nonlinear electrodynamical effects in modified Born-Infeld theory are negligible in the weak field regime.

. In 1934, Max Born and Leopold Infeld suggested and developed a nonlinear modification of Maxwell electrodynamics, in which the electrostatic self-energy of an electron was a finite value. In this paper, after a brief introduction to Lagrangian formulation of Born-Infeld electrodynamics with an external source, the explicit forms of Gauss’s law and the electrostatic energy density in Born-Infeld theory are obtained. The capacitance and the stored electrostatic energy for a parallel-plate and spherical capacitors are computed in the framework of Born-Infeld electrostatics. We show that the usual relations U=1/2C_Maxwell(ϕ)^2 and U=q^2/2C_ Maxwell are not valid for a capacitor in Born-Infeld electrostatics. Numerical estimations in this research show that the nonlinear corrections to the capacitance and the stored electrostatic energy for a capacitor in Born-Infeld electrostatics are considerable when the potential difference between the plates of a capacitor is very large.

More than 80 years ago, Born-Infeld electrodynamics was proposed in order to remove the point charge singularity in Maxwell electrodynamics. In this work, after a brief introduction to Lagrangian formulation of Abelian Born-Infeld model in the presence of an external source, we obtain the explicit forms of Gauss’s law and the energy density of an electrostatic field for Born-Infeld electrostatics. The electric field and the stored electrostatic energy per unit length for an infinite charged line and an infinitely long cylinder in Born-Infeld electrostatics are calculated. Numerical estimations in this paper show that the nonlinear corrections to Maxwell electrodynamics are considerable only for strong electric fields. We present an action functional for Abelian Born-Infeld model with an auxiliary scalar field in the presence of an external source. This action functional is a generalization of the action functional which was presented by Tseytlin in his studies on low energy dynamics of D -branes (Nucl. Phys. B469, 51 (1996); Int. J. Mod. Phys. A 19, 3427 (2004)). Finally, we derive the symmetric energy-momentum tensor for Abelian Born-Infeld model with an auxiliary scalar field.

In this note we consider the logarithmic curvature correction to Lifshitz and hyper scaling violation geometries. We investigate the effect of this correction to the gauge kinetic function f and the effective potential V . For the case of hyper scaling violating we show that the coupling between dilaton and the correction terms exhibit the logarithmic behavior for dilaton like for the case of squared curvature correction. We find the unexpected form of gauge kinetic function and potential in the presence of logarithmic terms. We try to resolve the IR singularity of Lifshitz and hyperscaling violation geometries by adding the logarithmic term.

In the 1990s, Kempf and his collaborators Mangano and Mann introduced a D-dimensional (beta, beta')-two-parameter deformed Heisenberg algebra which leads to an isotropic minimal length (Delta X-i)(min) = (h) over bar root D beta + beta', for all i is an element of {1, 2, ..., D}. In this work, the Lagrangian formulation of a magnetostatic field in three spatial dimensions (D = 3) described by Kempf algebra is presented in the special case of beta' = 2 beta up to the first-order over beta. We show that at the classical level there is a similarity between magnetostatics in the presence of a minimal length scale (modified magnetostatics) and the magnetostatic sector of the Abelian Lee-Wick model in three spatial dimensions. The integral form of Ampere's law and the energy density of a magnetostatic field in the modified magnetostatics are obtained. Also, the Biot-Savart law in the modified magnetostatics is found. By studying the effect of minimal length corrections to the gyromagnetic moment of the muon, we conclude that the upper bound on the isotropic minimal length scale in three spatial dimensions is 4.42 x 10(-19) m. The relationship between magnetostatics with a minimal length and the Gaete-Spallucci nonlocal magnetostatics [J. Phys. A: Math. Theor. 45, 065401 (2012)] is investigated.

In a series of papers, Quesne and Tkachuk (2006) presented a D + 1-dimensional (beta, beta')-two-parameter Lorentz-covariant deformed algebra which leads to a nonzero minimal measurable length. In this paper, the Lagrangian formulation of electrodynamics in a 3 + 1-dimensional spacetime described by Quesne-Tkachuk algebra is studied in the special case of beta' = 2 beta up to the first order over the deformation parameter beta. It is demonstrated that at the classical level there is a similarity between electrodynamics in the presence of a minimal measurable length (generalized electrodynamics) and Lee-Wick electrodynamics. We obtain the free space solutions of the inhomogeneous Maxwell's equations in the presence of a minimal length. These solutions describe two vector particles (a massless vector particle and a massive vector particle). We estimate two different upper bounds on the isotropic minimal length. The first upper bound is near to the electroweak length scale (l(electroweak) similar to 10(-18) m), while the second one is near to the length scale for the strong interactions (l(strong) similar to 10(-15) m). The relationship between the Gaete-Spallucci nonlocal electrodynamics (2012) and electrodynamics with a minimal length is investigated.

In this paper we study the interacting dark energy model in the framework of Hořava-Lifshitz cosmology. Using an additional canonical scalar field, we formulate Hořava-Lifshitz cosmology with an effective interacting dark energy sector. We show that the interacting dark energy model in the framework of Hořava gravity exhibiting phantom behavior.

In a series of papers, Kempf and co-workers (J. Phys. A: Math. Gen., 30 (1997) 2093; Phys. Rev. D, 52 (1995) 1108; Phys. Rev. D, 55 (1997) 7909) introduced a D-dimensional (beta, beta')-two-parameter deformed Heisenberg algebra which leads to a nonzero minimal observable length. In this work, the Lagrangian formulation of an electrostatic field in three spatial dimensions described by Kempf algebra is studied in the case in which beta' = 2 beta up to first order over the deformation parameter beta. It is shown that there is a similarity between electrostatics in the presence of a minimal length (modified electrostatics) and higher-derivative Podolsky's electrostatics. The important property of this modified electrostatics is that the classical self-energy of a point charge becomes a finite value. Two different upper bounds on the isotropic minimal length of this modified electrostatics are estimated. The first upper bound will be found by treating the modified electrostatics as a classical electromagnetic system, while the second one will be estimated by considering the modified electrostatics as a quantum field-theoretic model. It should be noted that the quantum upper bound on the isotropic minimal length in this paper is near to the electroweak length scale (L-electroweak similar to 10(-18) m). Copyright (C) EPLA, 2012

In 2006 Quesne and Tkachuk (J. Phys. A: Math. Gen.39, 10909, (2006)) introduced a (D+1)-dimensional (β, β′)-two-parameter Lorentz-covariant deformed algebra which leads to a nonzero minimal length. In this work, the Lagrangian formulation of the spinor field in a (3+1)-dimensional space–time described by Quesne–Tkachuk Lorentz-covariant deformed algebra is studied in the case where β′ = 2β up to first order over deformation parameter β. It is shown that the modified Dirac equation which contains higher order derivative of the wave function describes two massive particles with different masses. We show that physically acceptable mass states can only exist for [Formula: see text]. Applying the condition [Formula: see text] to an electron, the upper bound for the isotropic minimal length becomes about 3 ×10-13m. This value is near to the reduced Compton wavelength of the electron [Formula: see text] and is not incompatible with the results obtained for the minimal length in previous investigations.

The (D+1)-dimensional (β,β′)-two-parameter Lorentz-covariant deformed algebra introduced by Quesne and Tkachuk (J. Phys., A Math. Gen. 39, 10909, 2006), leads to a nonzero minimal uncertainty in position (minimal length). The Klein-Gordon equation in a (3+1)-dimensional space-time described by Quesne-Tkachuk Lorentz-covariant deformed algebra is studied in the case where β′=2β up to first order over deformation parameter β. It is shown that the modified Klein-Gordon equation which contains fourth-order derivative of the wave function describes two massive particles with different masses. We have shown that physically acceptable mass states can only exist for \(\beta<\frac{1}{8m^{2}c^{2}}\) which leads to an isotropic minimal length in the interval 10−17 m<(ΔX i )0<10−15 m. Finally, we have shown that the above estimation of minimal length is in good agreement with the results obtained in previous investigations.

We use a generalized scheme of supersymmetric quantum mechanics to obtain the energy spectrum and wave function for Dirac equation in (1+1)-dimensional spacetime coupled to a static scalar field.