Gauge Condition Research Articles

ELECTRIC AND MAGNETIC FIELDS DUE TO MASSIVE PHOTONS AND THEIR CONSEQUENCES

Allowing photons to bear mass, the electric and magnetic flelds of a steadily moving charge are no longer perpendicular to each other, as anticipated from Biot-Savart law. The electric and magnetic flelds of such a particle depend on the gauge potentials, ' and ~ A. The orthogonality relations of the particle flelds and the direction of motion depend on the mass of the photon. The non-relativistic correction to the particle flelds was found to be related to the Lorenz gauge condition. It is shown that the existence of magnetic monopoles inside matter is inevitable when magnetic fleld is applied in a conductor. Their existence is a manifestation of the massive nature of the photon inside matter. Neither electric nor magnetic current is separately conserved for photons, but their sum is. Massive photons are found to produce electric and magnetic flelds. A force proportional to the square of the current is found to act along the wire, F = 1 0I 2 , where 0 is vacuum permeability.

THE EXTENDED GAUGE TRANSFORMATIONS

In this work, new \extended gauge transformations involving current and flelds are presented. The transformation of Maxwell's equations under these gauges leads to a massive boson fleld (photon) that is equivalent to Proca fleld. The charge conservation equation and Proca equations are invariant under the new extended gauge transformations. Maxwell's equations formulated with Lorenz gauge condition violated give rise to massive vector boson. The inclusion of London supercurrent in Maxwell's equations yields a massive scalar boson satisfying Klein-Gordon equation. It is found that in superconductivity Lorenz gauge condition is violated, and consequently massive spin-0 bosons are created. However, the charge conservation is restored when the total current and charge densities are considered.

Two-point function for the Maxwell field in flat Robertson-Walker spacetimes

We obtain an explicit two-point function for the Maxwell field in flat Roberson-Walker spaces, thanks to a new gauge condition which takes the scale factor into account and assume a simple form. The two-point function is found to have the short distance Hadamard behavior.

Comparing gravitational waveform extrapolation to Cauchy-characteristic extraction in binary black hole simulations

We extract gravitational waveforms from numerical simulations of black hole binaries computed using the Spectral Einstein Code. We compare two extraction methods: direct construction of the Newman-Penrose (NP) scalar ${\ensuremath{\Psi}}_{4}$ at a finite distance from the source and Cauchy-characteristic extraction (CCE). The direct NP approach is simpler than CCE, but NP waveforms can be contaminated by near-zone effects---unless the waves are extracted at several distances from the source and extrapolated to infinity. Even then, the resulting waveforms can in principle be contaminated by gauge effects. In contrast, CCE directly provides, by construction, gauge-invariant waveforms at future null infinity. We verify the gauge invariance of CCE by running the same physical simulation using two different gauge conditions. We find that these two gauge conditions produce the same CCE waveforms but show differences in extrapolated-${\ensuremath{\Psi}}_{4}$ waveforms. We examine data from several different binary configurations and measure the dominant sources of error in the extrapolated-${\ensuremath{\Psi}}_{4}$ and CCE waveforms. In some cases, we find that NP waveforms extrapolated to infinity agree with the corresponding CCE waveforms to within the estimated error bars. However, we find that in other cases extrapolated and CCE waveforms disagree, most notably for $m=0$ ``memory'' modes.

Cylindrical quasi-Gaussian beams

Making use of the complex-source-point method in cylindrical coordinates, an exact solution representing a cylindrical quasi-Gaussian beam of arbitrary waist w(0) satisfying both the Helmholtz and Maxwell's equations is introduced. The Cartesian components of the electromagnetic field are derived stemming from different polarizations of the magnetic and electric vector potentials based on Maxwell's vectorial equations and Lorenz's gauge condition, without any approximations. Computations illustrate the theory for tightly focused and quasi-collimated cylindrical beams. The results are particularly useful in beam-forming design using high-aperture or collimated cylindrical laser beams in imaging microscopy, particle manipulation, optical tweezers, and the study of scattering, radiation forces, and torque on cylindrical structures.

Generalized Bondi-Sachs equations for characteristic formalism of numerical relativity

The Cauchy formalism of numerical relativity has been successfully applied to simulate various dynamical spacetimes without any symmetry assumption. But discovering how to set a mathematically consistent and physically realistic boundary condition is still an open problem for Cauchy formalism. In addition, the numerical truncation error and finite region ambiguity affect the accuracy of gravitational wave form calculation. As to the finite region ambiguity issue, the characteristic extraction method helps much. But it does not solve all of the above issues. Besides the above problems for Cauchy formalism, the computational efficiency is another problem. Although characteristic formalism of numerical relativity suffers the difficulty from caustics in the inner near zone, it has advantages in relation to all of the issues listed above. Cauchy-characteristic matching (CCM) is a possible way to take advantage of characteristic formalism regarding these issues and treat the inner caustics at the same time. CCM has difficulty treating the gauge difference between the Cauchy part and the characteristic part. We propose generalized Bondi-Sachs equations for characteristic formalism for the Cauchy-characteristic matching end. Our proposal gives out a possible same numerical evolution scheme for both the Cauchy part and the characteristic part. And our generalized Bondi-Sachs equations have one adjustable gauge freedom which can be used to relate the gauge used in the Cauchy part. Then these equations can make the Cauchy part and the characteristic part share a consistent gauge condition. So our proposal gives a possible new starting point for Cauchy-characteristic matching.

Canonical quantization theory of general singular QED system of Fermi field interaction with generally decomposed gauge potential

Quantization theory gives rise to transverse phonons for the traditional Coulomb gauge condition and to scalar and longitudinal photons for the Lorentz gauge condition. We describe a new approach to quantize the general singular QED system by decomposing a general gauge potential into two orthogonal components in general field theory, which preserves scalar and longitudinal photons. Using these two orthogonal components, we obtain an expansion of the gauge-invariant Lagrangian density, from which we deduce the two orthogonal canonical momenta conjugate to the two components of the gauge potential. We then obtain the canonical Hamiltonian in the phase space and deduce the inherent constraints. In terms of the naturally deduced gauge condition, the quantization results are exactly consistent with those in the traditional Coulomb gauge condition and superior to those in the Lorentz gauge condition. Moreover, we find that all the nonvanishing quantum commutators are permanently gauge-invariant. A system can only be measured in physical experiments when it is gauge-invariant. The vanishing longitudinal vector potential means that the gauge invariance of the general QED system cannot be retained. This is similar to the nucleon spin crisis dilemma, which is an example of a physical quantity that cannot be exactly measured experimentally. However, the theory here solves this dilemma by keeping the gauge invariance of the general QED system.

BRST symmetry andW-algebra in higher derivative models

In this paper we discuss the (anti)BRST symmetries and $W$-algebra of higher derivative theories of relativistic particles satisfying general gauge conditions. Using this formalism, the connection between the (anti)BRST symmetries and $W$-algebra for the massless particle with rigidity is established. Incidentally, the full $W$-algebra emerges only when the anti-BRST transformations are considered in tandem with the BRST ones. Further, the BRST symmetry is made finite and coordinate dependent. We show that such finite coordinate-dependent BRST symmetry changes the BRST invariant gauge-fixing fermion within the functional integration. This is exploited to connect two different arbitrary gauge conditions.

On the total mass of closed universes

The total mass, the Witten type gauge conditions and the spectral properties of the Sen-Witten and the 3-surface twistor operators in closed universes are investigated. It has been proven that a recently suggested expression ${\tt M}$ for the total mass density of closed universes is vanishing if and only if the spacetime is flat with toroidal spatial topology; it coincides with the first eigenvalue of the Sen-Witten operator; and it is vanishing if and only if Witten's gauge condition admits a non-trivial solution. Here we generalize slightly the result above on the zero-mass configurations: ${\tt M}=0$ if and only if the spacetime is holonomically trivial with toroidal spatial topology. Also, we show that the multiplicity of the eigenvalues of the (square of the) Sen-Witten operator is at least two, and a potentially viable gauge condition is suggested. The monotonicity properties of ${\tt M}$ through the examples of closed Bianchi I and IX cosmological spacetimes are also discussed. A potential spectral characterization of these cosmological spacetimes, in terms of the spectrum of the Riemannian Dirac operator and the Sen-Witten and the 3-surface twistor operators, is also indicated.

Cosmological post-Newtonian equations from nonlinear perturbation theory

We derive the basic equations of the cosmological first-orderpost-Newtonian approximation from the recently formulatedfully nonlinear and exact cosmological perturbation theory in Einstein'sgravity. Apparently the latter, being exact, should include theformer, and here we use this fact as a new derivation of the former.The complete sets of equations in both approaches are presented withoutfixing the temporal gauge conditions so that we can use the gaugechoice as an advantage. Comparisons between the two approaches are made. Both are potentially important in handling relativistic aspects of nonlinearprocesses occurring in cosmological structure formation. We consider an ideal fluid and include the cosmologicalconstant.

High-order Bessel non-vortex beam of fractional type α: II. Vector wave analysis for standing and quasi-standing laser wave tweezers

Based on the vector Maxwell’s equations and Lorenz’ gauge condition, full vector-wave derivations for the electric and magnetic fields components of a high-order Bessel non-vortex beam of fractional type α (HOBNVB-Fα) are presented. The field corresponds to the most generalized case of quasi-standing waves that reduce to perfect (i.e. equi-amplitude) standing waves or progressive waves with appropriate choice of the quasi-standing wave coefficient Υ. Particular emphasis is given on the polarization states of the vector potentials used to derive the field’s components and the transition from the progressive to perfect standing wave behavior. The results are of particular importance in the study of the optical/electromagnetic wave scattering, radiation force and torque in dual-beam optical laser-wave tweezers operating with this fractional type of non-diffracting non-vortex beams.

Fully non-linear and exact perturbations of the Friedmann world model

In 1988 Bardeen has suggested a pragmatic formulation of cosmological perturbation theory which is powerful in practice to employ various fundamental gauge conditions easily depending on the character of the problem. The perturbation equations are presented without fixing the temporal gauge condition and are arranged so that one can easily impose fundamental gauge conditions by simply setting one of the perturbation variables in the equations equal to zero. In this way one can use the gauge degrees of freedom as an advantage in handling problems. Except for the synchronous gauge condition, all the other fundamental gauge conditions completely fix the gauge mode, and consequently, each variable in such a gauge has a unique gauge invariant counterpart, so that we can identify the variable as the gauge-invariant one. Here, we extend Bardeen's linear formulation to fully nonlinear order in perturbations, with the gauge advantage kept intact. Derived equations are exact, and from these we can easily expand to higher order perturbations in a gauge-ready form. We consider scalar- and vector-type perturbations of an ideal fluid in a flat background; we also present the multiple components of ideal fluid case. As applications we present fully nonlinear density and velocity perturbation equations in Einstein's gravity in the zero-pressure medium, vorticity generation from pure scalar-type perturbation, and fluid formulation of a minimally coupled scalar field, all in the comoving gauge. We also present the equation of gravitational waves generated from pure scalar- and vector-type perturbations.

Reconstruction of the solar coronal magnetic field in spherical geometry

Context. High-resolution vector magnetographs either onboard spacecrafts or satellites (HMI/SDO, etc.) or ground based (SOLIS, etc.) now gives access to vector synoptic maps, composite magnetograms made of multiple interactive active regions, and full disk magnetograms. It thus become possible to reconstruct the coronal magnetic field on the full Sun scale.Aims. We present a method for reconstructing the global solar coronal magnetic field as a nonlinear force-free field. It is based on a well-posed Grad-Rubin iterative scheme adapted to spherical coordinates Methods. This method is a natural extension to spherical geometry of the one we previously developed in Cartesian geometry. It is implemented in the code XTRAPOLS, which is a massively parallel code. It allows dealing with the strong constraints put on the computational methods by having to handle the very large amounts of data contained in high-resolution large-scale magnetograms. The method exploits the mixed elliptic-hyperbolic nature of the Grad-Rubin boundary value problem. It uses a finite-difference method for the elliptic part and a method of characteristics for the hyperbolic part. The computed field guarantees to be divergence free up to round-off errors, by introducing a representation in terms of a vector potential satisfying specific gauge conditions. The construction of the latter – called here the restricted DeVore gauge – is described in detail in an appendix. Results. We show that XTRAPOLS performs well by applying it to the reconstruction of a particular semi-analytic force-free field that has already been considered by various authors.

Consistent quantization of massless fields of any spin

Consistent quantization of massless fields of any spin is developed from first principles. Probability currents and Lagrangians are derived allowing a first quantized formalism. A simple procedure was derived of connecting the wavefunctions with potentials and gauge conditions. We emphasize the spin one and spin two cases, relating them to the quantum photon (via modified Maxwell's equations) and to the quantum graviton respectively.

Dirac prescription from BRST symmetry in FRW space-time

A procedure to define the Becchi-Rouet-Stora-Tyutin (BRST) charge from the Noether one in extended phase space is given. It is outlined how this prescription can be applied to a Friedmann-Robertson-Walker space-time with a differential gauge condition and it allows us to reproduce the results of Ref. [21]. Then, we discuss the cohomological classes associated with functions in extended phase space having ghost number one and we recover the frozen formalism for classical observables. Finally, we consider the quantization of Becchi-Rouet-Stora-Tyutin--closed states and we define a scalar product which implements the super-Hamiltonian constraint.

Newtonian limit of fully nonlinear cosmological perturbations in Einstein's gravity

We prove that in the infinite speed-of-light limit(i.e., non-relativistic and subhorizon limits),the relativistic fully nonlinear cosmological perturbationequations in two gauge conditions, the zero-shear gauge andthe uniform-expansion gauge, exactly reproduce the Newtonianhydrodynamic perturbation equations in the cosmological background;as a consequence, in the same two gauge conditions, the Newtonianhydrodynamic equations are exactly recovered in the Minkowsky background.

Loop quantum gravity without the Hamiltonian constraint

We show that under certain technical assumptions, including the existence of a constant mean curvature (CMC) slice and strict positivity of the scalar field, general relativity conformally coupled to a scalar field can be quantized on a partially reduced phase space, meaning reduced only with respect to the Hamiltonian constraint and a proper gauge fixing. More precisely, we introduce, in close analogy to shape dynamics, the generator of a local conformal transformation acting on both, the metric and the scalar field, which coincides with the CMC gauge condition. A new metric, which is invariant under this transformation, is constructed and used to define connection variables which can be quantized by standard loop quantum gravity methods. Since this connection is invariant under the local conformal transformation, the generator of which is shown to be a good gauge fixing for the Hamiltonian constraint, the Dirac bracket associated with implementing these constraints coincides with the Poisson bracket for the connection. Thus, the well developed kinematical quantization techniques for loop quantum gravity are available, while the Hamiltonian constraint has been solved (more precisely, gauge fixed) classically. The physical interpretation of this system is that of general relativity on a fixed spatial CMC slice, the associated ‘time’ of which is given by the CMC. While it is hard to address dynamical problems in this framework (due to the complicated ‘time’ function), it seems, due to good accessibility properties of the CMC gauge, to be well suited for problems such as the computation of black hole entropy, where actual physical states can be counted and the dynamics is only of indirect importance. The corresponding calculation yields the surprising result that the usual prescription of fixing the Barbero–Immirzi parameter β to a constant value in order to obtain the well-known formula S = a(Φ)A/(4G) does not work for the black holes under consideration, while a recently proposed prescription involving an analytic continuation of β to the case of a self-dual space-time connection yields the correct result. Also, the interpretation of the geometric operators gets an interesting twist, which exemplifies the deep relationship between observables and the choice of a time function and has consequences for loop quantum cosmology.

Quasi-Gaussian electromagnetic beams

A class of Maxwellian beams, which is an exact solution of the vector wave equation (Helmholtz equation) and Maxwell's equations, is introduced. The solution, termed a quasi-Gaussian electromagnetic (EM) beam, is formed from a superposition of sources and sinks with complex coordinates, and is characterized by an arbitrary waist ${w}_{0}$ and a diffraction convergence length known as the Rayleigh range ${z}_{R}$. An attractive feature of this beam is the description of strongly focused (or strongly divergent) EM-optical wave fields for $k{w}_{0}\ensuremath{\le}1$, where $k$ is the wave number. A vector wave analysis is developed to determine and compute the spatial Cartesian components of the electric and magnetic fields (valid in the near field and the far field) stemming from Maxwell's vector equations and the Lorenz gauge condition, with particular emphasis on the parameter $k{w}_{0}$ and the polarization states of the vector potentials used to derive the EM field's components. The results are potentially useful in the study of the axial and/or arbitrary wave scattering, radiation force, and torque in lasers operating with strongly focused (or strongly divergent) beams for particle manipulation in optical tweezers and imaging applications.

Dynamic Similarity in Titanosaur Sauropods: Ichnological Evidence from the Fumanya Dinosaur Tracksite (Southern Pyrenees)

The study of a small sauropod trackway from the Late Cretaceous Fumanya tracksite (southern Pyrenees, Catalonia) and further comparisons with larger trackways from the same locality suggest a causative relationship between gait, gauge, and body proportions of the respective titanosaur trackmakers. This analysis, conducted in the context of scaling predictions and using geometric similarity and dynamic similarity hypotheses, reveals similar Froude numbers and relative stride lengths for both small and large trackmakers from Fumanya. Evidence for geometric similarity in these trackways suggests that titanosaurs of different sizes moved in a dynamically similar way, probably using an amble gait. The wide gauge condition reported in trackways of small and large titanosaurs implies that they possessed similar body (trunk and limbs) proportions despite large differences in body size. These results strengthen the hypothesis that titanosaurs possessed a distinctive suite of anatomical characteristics that are well reflected in their tracks and trackways.

Conformally invariant formalism for the electromagnetic field with currents in Robertson-Walker spaces

We show that the Laplace-Beltrami equation □6a = j in $(\mathbb {R}^6,\eta )$(R6,η), η ≔ diag(+ − − − − +), leads under very moderate assumptions to both the Maxwell equations and the conformal Eastwood-Singer gauge condition on conformally flat spaces including the spaces with a Robertson-Walker metric. This result is obtained through a geometric formalism which gives, as byproduct, simplified calculations. In particular, we build an atlas for all the conformally flat spaces considered which allows us to fully exploit the Weyl rescalling to Minkowski space.