The effect of the built-in supersymmetric quantum mechanical language on the spectrum of the (1+1)-Dirac equation, with position-dependent mass (PDM) and complexified Lorentz scalar interactions, is re-emphasized. The signature of the "quasi-parity" on the Dirac particles' spectra is also studied. A Dirac particle with PDM and complexified scalar interactions of the form S(z)=S(x-ib) (an inversely linear plus linear, leading to a PT-symmetric oscillator model), and S(x)=S_{r}(x)+iS_{i}(x) (a PT-symmetric Scarf II model) are considered. Moreover, a first-order intertwining differential operator and an $η$-weak-pseudo-Hermiticity generator are presented and a complexified PT-symmetric periodic-type model is used as an illustrative example.
We present a stochastic theory of charges moving in an electromagnetic field using nonequilibrium quantum field theory. We give a first principles' derivation of the Abraham-Lorentz-Dirac-Langevin equation which depicts the quantum expectation value for a particle's trajectory and its stochastic fluctuations by combining the worldline path integral quantization with the Feynman-Vernon influence functional or closed-time-path effective action methods. At lowest order, the equations of motion are approximated by a stochastic Lorentz-Dirac equation.
We have studied the relativistic effects on the mean spin entanglement of two massive Dirac particles using the simultaneous eigen-spinors of the Foldy-Woutheysen mean spin operator and the Dirac Hamiltonian. We have obtained the transformation matrix from the spinor with specific momentum to the spinor with a transformed momentum under an arbitrary Lorentz transformation. Using the transformation matrix we have shown the consistent monotonic behavior between the concurrence and the maximum value of Bell parameter in Bell inequality of transformed spin states.
We analyze recent results on a Dirac oscillator. We show that the truncation of the Frobenius series does not yield all the eigenvalues and eigenfunctions of the radial equation. For this reason the eigenvalues reported by the authors are useless and the prediction of allowed oscillator frequencies meaningless.
The lifelong efforts of Paul A. M. Dirac were to construct localized quantum systems in the Lorentz covariant world. In 1927, he noted that the time-energy uncertainty should be included in the Lorentz-covariant picture. In 1945, he attempted to construct a representation of the Lorentz group using a normalizable Gaussian function localized both in the space and time variables. In 1949, he introduced his instant form to exclude time-like oscillations. He also introduced the light-cone coordinate system for Lorentz boosts. Also in 1949, he stated the Lie algebra of the inhomogeneous Lorentz gro…
Jia and Dutra (J. Phys. A: Math. Gen. 39 (2006) 11877) have considered the one-dimensional non-Hermitian complexified potentials with real spectra in the context of position-dependent mass in Dirac equation. In their second example, a smooth step shape mass distribution is considered and a non-Hermitian non - PT- symmetric Lorentz vector potential is obtained. They have mapped this problem into an exactly solvable Rosen-Morse Schrodinger model and claimed that the energy spectrum is real. The energy spectrum they have reported is pure imaginary or at best forms an empty set. Their claim on the…
We consider two spacelike separated Dirac particles and construct five invariants under the spinor representations of the local proper orthochronous Lorentz groups. All of the constructed Lorentz invariants are identically zero for product states. The behaviour of the Lorentz invariants under local unitary evolutions that act unitarily on any subspace with fixed particle momenta is studied. All of the Lorentz invariants have invariant absolute values on such subspaces if the evolutions are generated by local zero-mass Dirac Hamiltonians. Some of them also for the case of nonzero-mass. Therefor…
We study the thermodynamic quantities such as the Helmholtz free energy, the mean energy and the specific heat for both the Klein-Gordon, and Dirac equations. Our analyze includes two main subsections: ($i$) statistical functions for the Klein-Gordon equation with a linear potential having Lorentz vector, and Lorentz scalar parts ($ii$) thermodynamic functions for the Dirac equation with a Lorentz scalar, inverse-linear potential by assuming that the scalar potential field is strong ($A \gg 1$). We restrict ourselves to the case where only the positive part of the spectrum gives a contribution…
We define a class of Lorentz invariant Bohmian quantum models for N entangled but noninteracting Dirac particles. Lorentz invariance is achieved for these models through the incorporation of an additional dynamical space-time structure provided by a foliation of space-time. These models can be regarded as the extension of Bohm's model for N Dirac particles, corresponding to the foliation into the equal-time hyperplanes for a distinguished Lorentz frame, to more general foliations. As with Bohm's model, there exists for these models an equivariant measure on the leaves of the foliation. This ma…
We propose a mathematical model of \textit{quantum} transistor in which bandgap engineering corresponds to the tuning of Dirac potential in the complex four-vector form. The transistor consists of $n$-relativistic spin qubits moving in \textit{classical} external electromagnetic fields. It is shown that the tuning of the direction of the external electromagnetic fields generates perturbation on the potential temporally and spatially, determining the type of quantum logic gates. The theory underlying of this scheme is on the proposal of the intertwining operator for Darboux transfomations on on…
We consider exact/quasi-exact solvability of Dirac equation with a Lorentz scalar potential based on factorizability of the equation. Exactly solvable and $sl(2)$-based quasi-exactly solvable potentials are discussed separately in Cartesian coordinates for a pure Lorentz potential depending only on one spatial dimension, and in spherical coordinates in the presence of a Dirac monopole.
We investigate the motion of a wave packet of a charged scalar particle linearly accelerated by a static potential in quantum electrodynamics. We calculate the expectation value of the position of the charged particle after the acceleration to first order in the fine structure constant in the hbar -> 0 limit. We find that the change in the expectation value of the position (the position shift) due to radiation reaction agrees exactly with the result obtained using the Lorentz-Dirac force in classical electrodynamics. We also point out that the one-loop correction to the potential may contribut…
We have shown the covariant relativistic spin operator is equivalent to the spin operator commuting with the free Dirac Hamiltonian. This implies that the covariant relativistic spin operator is a good quantum observable. The covariant relativistic spin operator has the pure quantum contribution which does not exist in the classical covariant spin operator. Based on this equivalence reduced spin states can be claerly defined. We have shown the change in the entropy of a reduced spin density matrix sweeps through the whole range according to the relative motion of an observer.
In the textbook proofs of Lorentz covariance of the Dirac equation, one treats the wave function as a spinor and gamma matrices as scalars, leading to a quite complicated formalism with several pedagogic drawbacks. As an alternative, I propose to teach Dirac equation and its Lorentz covariance by using a much simpler, but physically equivalent formalism, in which these drawbacks do not appear. In this alternative formalism, the wave function transforms as a scalar and gamma matrices as components of a vector, such that the standard physically relevant bilinear combinations do not change their …
The effects of Lorentz boosts on the quantum entanglement encoded by a pair of massive spin one-half particles are described according to the Lorentz covariant structure described by Dirac bispinors. The quantum system considered incorporates four degrees of freedom -- two of them related to the bispinor intrinsic parity and other two related to the bispinor spin projection, i.e. the Dirac particle helicity. Because of the natural multipartite structure involved, the Meyer-Wallach global measure of entanglement is preliminarily used for computing global quantum correlations, while the entangle…
We study the local group velocity defined as the weak value of the velocity operator in the (1+1) dimensional Klein-Gordon as well as Dirac theory. It was shown by Berry [ J. Phys. A 45, 185308 (2012)] that when the pre- and post-selected states for evaluating the weak value are chosen at random from an ensemble of available states, the local group velocity has a universal probability distribution which can have both subluminal and superluminal components. In this work, we numerically explore the role of Lorentz boost and its impact on the superluminal fraction of the total probability distrib…
Exact analytic solutions are found to the Dirac equation for a combination of Lorentz scalar and vector Coulombic potentials with additional non-Coulombic parts. An appropriate linear combination of Lorentz scalar and vector non-Coulombic potentials, with the scalar part dominating, can be chosen to give exact analytic Dirac wave functions.
By starting from the modified Maxwell theory coupled to gravity, the arising of geometric quantum phases in the relativistic and nonrelativistic quantum dynamics of a Dirac neutral particle from the effects of the violation of the Lorentz symmetry in the cosmic string spacetime is investigated. It is shown that the Dirac equation can be written in terms of an effective metric and a relativistic geometric phase stems from the topology of the cosmic string spacetime and the Lorentz symmetry breaking effects. Further, the nonrelativistic limit of the Dirac equation is discussed and it is shown th…
The $D$-dimensional $(β, β')$-two-parameter deformed algebra introduced by Kempf is generalized to a Lorentz-covariant algebra describing a ($D+1$)-dimensional quantized space-time. In the D=3 and $β=0$ case, the latter reproduces Snyder algebra. The deformed Poincaré transformations leaving the algebra invariant are identified. It is shown that there exists a nonzero minimal uncertainty in position (minimal length). The Dirac oscillator in a 1+1-dimensional space-time described by such an algebra is studied in the case where $β'=0$. Extending supersymmetric quantum mechanical and shape-invari…
Exact solutions are found to the Dirac equation for a combination of Lorentz scalar and vector Coulombic potentials with additional non-Coulombic parts. An appropriate linear combination of Lorentz scalar and vector non-Coulombic potentials, with the scalar part dominating, can be chosen to give exact analytic Dirac wave functions. The method works for the ground state or for the lowest orbital state with l=j-{1/2}, for any j.
In this paper we describe the transformation properties of quantum entanglement encoded in a pair of spin 1/2 particles described via Dirac bispinors. Due to the intrinsic parity-spin internal structure of the bispinors, the joint state is a four-qubit state exhibiting multipartite entanglement, and to compute global correlation properties we consider the averaged negativities over each possible bi-partition. We also consider specific bipartitions, such as the spin-spin and the particle-particle bipartitions. The particle-particle entanglement, between all degrees of freedom of one particle an…
The low energy Dirac and Weyl spectra are allowed to violate the Lorentz symmetry and thereby have a tilted energy dispersion. The tilt in the energy dispersion induces a Hall voltage in the plane spanned by the electric field and the tilt velocity. In the presence of a magnetic field the planar Hall conductivity and resistivity show Shubnikov de Haas oscillations. The oscillations in the planar Hall effect can become a fingerprint to spot the anomalous transport in Dirac and Weyl semimetals.
When Einstein formulated his special relativity in 1905, he established the law of Lorentz transformations for point particles. It is now known that particles have internal space-time structures. Particles, such as photons and electrons, have spin variables. Protons and other hadrons are regarded as bound states of more fundamental particles called quarks which have their internal variables. It is still one of the most outstanding problems whether these internal space-time variables are transformed according to Einstein's law of Lorentz transformations. It is noted that Wigner, Dirac, and Feyn…
We apply the open systems concept and the influence functional formalism introduced in Paper I to establish a stochastic theory of relativistic moving spinless particles in a quantum scalar field. The stochastic regime resting between the quantum and semi-classical captures the statistical mechanical attributes of the full theory. Applying the particle-centric world-line quantization formulation to the quantum field theory of scalar QED we derive a time-dependent (scalar) Abraham-Lorentz-Dirac (ALD) equation and show that it is the correct semiclassical limit for nonlinear particle-field syste…
We consider the consistent deformation of the relativistic quantum mechanics introducing the noncommutativity of the space-time and preserving the Lorentz symmetry. The relativistic wave equation describing the spinning particle on coordinate dependent noncommutative space-time (noncommutative Dirac equation) is proposed. The fundamental properties of this equation, like the Lorentz covariance and the continuity equation for the probability density are verified. To this end using the properties of the star product we derive the corresponding probability current density and prove its conservati…
The full spectrum of two-dimensional fermion states in a scalar soliton trap with a Lorentz breaking background is investigated in the context of the novel 2D materials, where the Lorentz symmetry should not be strictly valid. The field theoretical model with Lorentz breaking terms represents Dirac electrons in one valley and in a scalar field background. The Lorentz violation comes from the difference between the Dirac electron and scalar mode velocities, which should be expected when modelling the electronic and lattice excitations in 2D materials. We extend the analytical methods developed …
