Narrow Results

A canonical relativistic quantization of the electromagnetic field is introduced in the presence of an anisotropic conductor magneto-dielectric medium in a standard way in the Gupta–Bleuler framework. The medium is modeled by a continuum collection of the vector fields and a continuum collection of the antisymmetric tensor fields of the second rank in Minkowski space–time. The collection of vector fields describes the conductivity property of the medium and the collection of antisymmetric tensor fields describes the polarization and the magnetization properties of the medium. The conservation law of the total electric charges, induced in the anisotropic conductor magneto-dielectric medium, is deduced using the antisymmetry conditions imposed on the coupling tensors that couple the electromagnetic field to the medium. Two relativistic covariant constitutive relations for the anisotropic conductor magneto-dielectric medium are obtained. The constitutive relations relate the antisymmetric electric–magnetic polarization tensor field of the medium and the free electric current density four-vector, induced in the medium, to the strength tensor of the electromagnetic field, separately. It is shown that for a homogeneous anisotropic medium the susceptibility tensor of the medium satisfies the Kramers–Kronig relations. Also it is shown that for a homogeneous anisotropic medium the real and imaginary parts of the conductivity tensor of the medium satisfy the Kramers–Kronig relations and a relation other than the Kramers–Kronig relations.

In recent papers,^1,2 it has been shown that the presence of negative norm states or negative frequency solutions are indispensable for a fully covariant quantization of the minimally coupled scalar field in de Sitter space. Their presence, while leaving unchanged the physical content of the theory, offers the advantage of eliminating any ultraviolet divergence in the vacuum energy² and infrared divergence in the two point function.³ We attempt here to extend this method to the interacting quantum field in Minkowski space-time. As an illustration of the procedure, we consider the λϕ⁴ theory in Minkowski space-time. The mathematical consequences of this method is the disappearance of the ultraviolet divergence to the one-loop approximation. This means, the effect of these auxiliary negative norm states is to allow an automatic renormalization of the theory in this approximation.

In this paper, Cayley formula is derived for 4 × 4 semi-skew-symmetric real matrices in . For this purpose, we use the decomposition of a semi-skew-symmetric matrix A = θ₁A₁ + θ₂A₂ by two unique semi-skew-symmetric matrices A₁ and A₂ satisfying the properties and Then, we find Lorentzian rotation matrices with semi-skew-symmetric matrices by Cayley formula. Furthermore, we give a way to find the semi-skew-symmetric matrix A for a given Lorentzian rotation matrix R such that R = Cay(A).

The main purpose of this paper is to investigate parallel frames of non-lightlike curves in Minkowski space-time. For this purpose, the relations between Frenet–Serret frame and parallel frame of a given non-lightlike curve are discussed in four different cases. Then, the curvature, first and second torsion functions κ(s), τ(s), σ(s) and principal curvature functions k₁(s), k₂(s), k₃(s) are expressed in terms of the angles α, β and θ.

In this paper, we define the Bishop frame of a null Cartan curve in Minkowski space-time ${𝔼}_1^4$. We obtain the Bishop’s frame equations of a null Cartan curve which lies in the timelike hyperplane of ${𝔼}_1^4$. We show that a null Cartan cubic lying in the timelike hyperplane of ${𝔼}_1^4$ has two Bishop frames, one of which coincides with its Cartan frame. We also derive the Bishop’s frame equation of the null Cartan curve which has the third Cartan curvature ${\kappa }_3(s)\ne 0$. As an application, we find a solution of the null Betchov-Da Rios vortex filament equation in terms of a null Cartan curve and its Bishop frame, which generates a timelike Hasimoto surface.

Starting with the quaternionic Minkowski space–time and its four-vector representation, a rotational analogue of the quaternionic Dirac equation in the electromagnetic field is developed, which includes not only the energy solutions but also the angular momentum solutions for rotating Dirac fermions. The striking feature of the quaternionic approach is that it depicts a unified representation of energy–momentum in a single framework as the space–time. Furthermore, we establish the generalized Schrodinger–Pauli-like equation associated with generalized electric and magnetic dipole energies that corresponds to their dipole moments. As such, the present analysis deals with the invariant behavior of the extended quaternionic rotational Dirac equation under Lorentz–Poincaré, Gauge, duality, and CPT invariance.

We investigate the parametrizations of loxodromes on the time-like rotational surfaces by using suitable Lorentzian angles in Minkowski space-time. Also, some examples and corresponding graphs are given by using Mathematica.