On the weak‐field approximation of nonlinear vacuum electrodynamics of the Plebański class

Plebański's class of nonlinear vacuum electrodynamics is considered, which is for several reasons of interest at the present time. In particular, the question is answered under which circumstances Maxwell's original field equations are recovered approximately and which ‘post‐Maxwellian’ effects could arise. To this end, a weak field approximation method is developed, allowing to calculate ‘post‐Maxwellian’ corrections up to Nth order. In some respect, this is analogue of determining ‘post‐Newtonian’ corrections from relativistic mechanics by a low velocity approximation. As a result, we got a series of linear field equations that can be solved order by order. In this context, the solutions of the lower orders occur as source terms inside the higher order field equations and represent a ‘post‐Maxwellian’ self‐interaction of the electromagnetic field, which increases order by order. It becomes apparent that one has to distinguish between problems with and without external source terms because without sources also high frequency solutions can be approximately described by Maxwell's original equations. The higher order approximations, which describe ‘post‐Maxwellian’ effects, can give rise to experimental tests of Plebańksi's class. Finally, two boundary value problems are discussed to have examples at hand.     

‘Majorana Mass’ Fermions as Untrue Majorana Particles,Rather Endowed with Pseudoscalar-Type Charges than Genuinely Neutral

The idea of a ‘Majorana mass’ to make a chiral neutrino really neutral is here reconsidered. It is pointed out that such an approach, unlike Majorana’s (non-chiral) old one, does not strictly lead, in general, to a true self-conjugate particle. This can be seen on directly using the basic definition (or fundamental representation) of charge conjugation C in Quantum Field Theory, as an operation just acting on annihilation and creation operators and just expressing particle–antiparticle interchange. It is found, indeed, that the ‘active’ and ‘sterile’ whole fields which can be obtained from mixing the chiral components of two mutually charge-conjugate Dirac fields are themselves ‘charge conjugate’ to each other (rather than individually self-conjugate). These fields, taken as mass eigenfields (as in the ‘Majorana mass’ case), are shown to describe particles carrying pseudoscalar-type charges and being neutral relative to scalar-type charges only. For them, ‘CP symmetry’ would be nothing but pure mirror symmetry, and C violation (already implied in their respective ‘active’ and ‘sterile’ behaviors) should then involve time-reversal violation as well. The new (no longer strictly chargeless) ‘Majorana mass’ neutrino model still proves, however, neither to affect the usual expectation for a neutrinoless double β-decay, nor to prevent ‘active’ and ‘sterile’ neutrino varieties from generally taking different mass values. One has, on the other hand, that any fermion being just a genuine (i.e. really self-conjugate) Majorana particle cannot truly exist in two distinct—‘active’ and ‘sterile’—versions, and it can further bear only a unified mass kind which may at once be said to be either a ‘Majorana-like’ or a ‘Dirac-like’ mass kind.     

Nonexistence of quantum fields associated with two-dimensional spacelike planes

It is shown that in a relativistic quantum field theory satisfying Wightman's axioms, there are no nontrivial field-like operators, or even bilinear forms, associated to a two (or less)-dimensional spacelike plane in Minkowski space. This generalizes Wightman's result that fields can not be defined as operators at a point and stands in contrast to Borchers' result that field operators can be associated with one-dimensional timelike planes.     

Particle motion and electromagnetic fields of rotating compact gravitating objects with gravitomagnetic charge

The exact solution for the electromagnetic field occuring when the Kerr–Taub–NUT compact object is immersed (i) in an originally uniform magnetic field aligned along the axis of axial symmetry (ii) in dipolar magnetic field generated by current loop has been investigated. Effective potential of motion of charged test particle around Kerr–Taub–NUT gravitational source immersed in magnetic field with different values of external magnetic field and NUT parameter has been also investigated. In both cases presence of NUT parameter and magnetic field shifts stable circular orbits in the direction of the central gravitating object. Finally we find analytical solutions of Maxwell equations in the external background spacetime of a slowly rotating magnetized NUT star. The star is considered isolated and in vacuum, with monopolar configuration model for the stellar magnetic field.     

Existence of dual mass: Linear theory

The field equations of general relativity are symmetrized in a manner similar to Dirac's symmetrization of electromagnetism. This symmetrization allows us to predict the existence of a gravitational, magnetic-like mass which we call dual mass. Time-independent solutions for both rotating and nonrotating sources are constructed for these generalized equations. The gravitational field produced by the dual mass source is compared with the gravitational field that follows from the linearized NUT and Kerr-NUT metrics, and an identification of the NUT parameter with a dual mass monopole is made.     

Exact uncertainty approach in quantum mechanics and quantum gravity

The assumption that an ensemble of classical particles is subject to nonclassical momentum fluctuations, with the fluctuation uncertainty fully determined by the position uncertainty, has been shown to lead from the classical equations of motion to the Schrödinger equation. This ‘exact uncertainty’ approach may be generalised to ensembles of gravitational fields, where nonclassical fluctuations are added to the field momentum densities, of a magnitude determined by the uncertainty in the metric tensor components. In this way one obtains the Wheeler-DeWitt equation of quantum gravity, with the added bonus of a uniquely specified operator ordering. No a priori assumptions are required concerning the existence of wave functions, Hilbert spaces, Planck's constant, linear operators, etc. Thus this approach has greater transparency than the usual canonical approach, particularly in regard to the connections between quantum and classical ensembles. Conceptual foundations and advantages are emphasised.     

Symmetry and invariance

The concept of invariance relates to both the intrinsic symmetries of physical systems and the symmetry of the set of equivalent reference frames used to observe them. Standard algebraic expressions for electrostatic potentials and crystal-field effective operators display both types of invariance. The concept of a reference frame is generalized to that of an ‘observing system’, which can, for example, be the basis states of a quantum system. This idea is related to Racah’s mathematical machinery for evaluating the matrix elements of many-electron 4f open-shell states in lanthanide ions. It is argued, on the basis of computational flexibility and ease of interpretation, that all equations that represent physical processes be expressible in terms of invariants of the set of observing systems. This ‘Principle of Invariance’ is then applied to special relativity, leading to a simple geometrical interpretation of Maxwell’s electromagnetic field equations. The close relationship between Dirac’s relativistic wave equation and Maxwell’s equations is then exposed. This leads to the concept of an inner structure of space-time and the reinterpretation of particle spin. Finally, it is shown that the use of invariants in relativity theory identifies a set of observing systems with a higher symmetry than that of Minkowski space-time.     

The exact response of a two-dimensional flat-layered medium to a probing pulse: An application to a layer stripping reconstruction procedure

Abstract

We consider a simple model problem that can be found in many fields of application such as, for example, reflection seismology. That is we consider an initial boundary value problem on a half-plane for a class of two-dimensional wave equations with a piecewise-constant coefficient. This coefficient describes the flat layered medium under consideration. An initial pulse located on the boundary of the half-plane is used to probe the medium. An integral representation of the solution of this problem is obtained by studying the spectral measures of some differential operators in one variable. This integral representation is exploited to obtain an ‘explicit’ formula for the solution of the problem considered evaluated at the location of the probing pulse. This ‘explicit’ formula is exploited to reconstruct the structure of the medium from its response to a probing pulse via a layer stripping procedure. Some numerical results obtained with this procedure on test problems are shown. The ‘explicit’ formula obtained can be used in several other contexts such as, for example, the study of perturbed flat layered media or the study of random flat layered media.     

Robinson–Trautman solutions to Einstein’s equations

Solutions to Einstein’s equations in the form of a Robinson–Trautman metric are presented. In particular, we derive a pure radiation solution which is non-stationary and involves a mass m, The resulting spacetime is of Petrov Type II A special selection of parametric values throws up the feature of the particle ‘rocket’, a Type D metric. A suitable transformation of the complex coordinates allows the metrics to be expressed in real form. A modification, by setting m to zero, of the Type II metric thereby converting it to Type III, is then shown to admit a null Einstein–Maxwell electromagnetic field.     

The Sugawara model in general relativity. A. The case for three current vectors (SU (2))

A general method of solving the equations of Sugawara's field theory of currents has been developed, and illustrated by applying it to the set of three currents. These are inserted into Einstein's field equations which have been solved together with the co-variant ‘gauge’ conditions for a gravitational field involving cylindrical symmetry. A further transformation exhibits the triad formed by the current vectors and exhibits clearly the deviations of the line-element from Schwarzschild's exterior solution. In a subsequent paper the case for eight vector currents corresponding toSU (3) will be treated in similar fashion.     

Non-static electromagnetic fields in general relativity. IV

An investigation is made of Rainich's ‘already unified field theory’ from the standpoint of continuous groups of motions, and a solution is obtained for the case where the electromagnetic field is non-static and the space-time admits a three-parameter continuous group of motions whose minimum invariant varieties are three-dimensional spaces. There exists a divergence-free electromagnetic field for values of t at which the metric is not singular.     

On the physical meaning of the NUT parameter

Exact interpretation of the NUT parameter becomes possible when a static Schwarzchild mass is immersed in a stationary, source free electromagnetic universe. We explain this on the basis of our solution (Halilsoy and Al-Badawi in Class Quantum Grav 12:3013, 1995), which represents the non-linear superposition of the Schwarzschild solution and the generalized Bertotti–Robinson solution.     

Pathologies in the sticky limit of hard sphere Yukawa models for colloidal fluids: a possible correction

A known ‘sticky hard sphere’ model, starting from a hard sphere Yukawa potential and taking the limit of infinite amplitude and vanishing range with their product remaining constant, is shown to be ill-defined. This is because its Hamiltonian (which we call SHS2) leads to an exact second virial coefficient which diverges, unlike that of Baxter's original model (SHS1). This deficiency has never been observed so far, since the linearization implicit in the ‘mean spherical approximation’ (MSA), within which the model is analytically solvable, partly masks such a pathology. To overcome this drawback and retain some useful features of SHS2, we propose both a new model (SHS3) and a new closure (‘modified MSA’), whose combination yields an analytical solution formally identical with the SHS2-MSA solution. This mapping allows the recovery of many results derived from SHS2, after a re-interpretation within a correct framework. Possible developments are indicated.     

The tree graphs of quantum gravity and the Reissner-Nordström solution

It is shown that the manifestly covariant quantization of gravity correctly reproduces the classical Reissner-Nordström solution in the ? → 0 limit. This is explicitly verified by evaluating the lowest order tree graph contribution to the vacuum expectation value of the gravitational field produced by a spherically symmetric c-number charged source. The generalization from a point source to that of finite extension is unavoidable if the ‘trees’ are not to lead to divergent expressions. Moreover, the mass which appears in the R-N solution is seen to be positive definite. For convenience, the source is taken to be a sphere of uniform charge and matter densities. Owing to a mass renormalization relating the total mass of the sphere to its bare mass, charge and invariant extension, both exterior and interior solutions may then be generated. This mass renormalization formula is in complete agreement with that obtained by purely classical reasoning.     

On Special Requantization of a Black Hole

Quantized expressions for the gravitational energy and momentum are derived from a linearized theory of teleparallel gravity. The derivation relies on a second-quantization procedure that constructs annihilation and creation operators for the graviton. The resulting gravitational field is a collection of gravitons, each of which has precise energy and momentum. On the basis of the weak-field approximation of Schwarzschild’s solution, a new form for the quantization of the mass of a black hole is derived.     

Applications of Singh-Rajput Mes in Recall Operations of Quantum Associative Memory for a Two- Qubit System

Recall operations of quantum associative memory (QuAM) have been conducted separately through evolutionary as well as non-evolutionary processes in terms of unitary and non- unitary operators respectively by separately choosing our recently derived maximally entangled states (Singh-Rajput MES) and Bell’s MES as memory states for various queries and it has been shown that in each case the choices of Singh-Rajput MES as valid memory states are much more suitable than those of Bell’s MES. it has been demonstrated that in both the types of recall processes the first and the fourth states of Singh-Rajput MES are most suitable choices as memory states for the queries ‘11’ and ‘00’ respectively while none of the Bell’s MES is a suitable choice as valid memory state in these recall processes. It has been demonstrated that all the four states of Singh-Rajput MES are suitable choice as valid memory states for the queries ‘1?’, ‘?1’, ‘?0’ and ‘0?’ while none of the Bell’s MES is suitable choice as the valid memory state for these queries also.     

Geodesics for the NUT metric and gravitational monopoles

In order to provide insight about the physical interpretation of the NUT parameter, we solve the geodesic equations for the NUT metric. We show that the properties of NUT geodesics are similar to the properties of trajectories for charged particles orbiting about a magnetic monopole. In summary, we show that (1) the orbits lie on the surface of a cone, (2) the conserved total angular momentum is the sum of the orbital angular momentum plus the angular momentum due to the monopole field, (3) the monopole field angular momentum is independent of the separation between the source of the gravitational field and the test particle, and (4) the geodesics are almost spherically symmetric. The strong similarities between the NUT geodesics and the electromagnetic monopole suggest that the NUT metric is an exact solution for a gravitational magnetic monopole. However, the subtle difference of being only almost spherically symmetric implies that the analogy is not perfect. The almost spherically symmetric nature of the NUT geodesics suggest that the energy of the Dirac string makes a contribution to the solution. We also construct exact solutions for special orbits, discuss a twin paradox, and speculate about the Dirac quantization condition for a gravitational magnetic monopole.     

Algebraic approach to quantum field theory on a class of noncommutative curved spacetimes

In this article we study the quantization of a free real scalar field on a class of noncommutative manifolds, obtained via formal deformation quantization using triangular Drinfel’d twists. We construct deformed quadratic action functionals and compute the corresponding equation of motion operators. The Green’s operators and the fundamental solution of the deformed equation of motion are obtained in terms of formal power series. It is shown that, using the deformed fundamental solution, we can define deformed *-algebras of field observables, which in general depend on the spacetime deformation parameter. This dependence is absent in the special case of Killing deformations, which include in particular the Moyal-Weyl deformation of the Minkowski spacetime.