Vacuum Polarization Charge Research Articles

Vacuum polarization near boundaries

We study the effect of boundary conditions on vacuum polarization for charged scalar fields in two space-time dimensions. We find that both Dirichlet and Neumann boundary conditions lead to screening. In the Dirichlet case, the vacuum polarization charge density vanishes at the boundary, whereas it attains its maximum there for Neumann boundary conditions. At a critical field strength, the vacuum polarization diverges for Neumann boundary conditions, an effect due to the instability of the lowest energy mode in the presence of the external field.

Planar density of vacuum charge induced by a supercritical Coulomb potential

Analytical expressions for the planar density of an induced vacuum charge are obtained in a strong Coulomb potential in coordinate space. Treatment is based on a self-adjoint extension approach for constructing of the Green's function of a charged fermion in an external electromagnetic field. Induced vacuum charge density is calculated and analyzed in subcritical and supercritical Coulomb potentials for massless and massive fermions. We argue that the virtual and so-called real vacuum polarizations contribute in an induced vacuum charge in a supercritical Coulomb potential. The behavior of the polarization vacuum charge density is investigated at long and short distances from the Coulomb center. The induced vacuum charge has a screening sign. Screening of a Coulomb impurity in graphene is briefly discussed. The real vacuum polarization charge density that acquires the quantum electrodynamics vacuum in a supercritical Coulomb potential due to the real vacuum polarization is calculated. It is shown that the vacuum charge densities essentially differ in massive and massless cases. We expect that our results can, as a matter of principle, be tested in graphene with a supercritical Coulomb impurity.

Vacuum polarization of planar charged fermions with Coulomb and Aharonov–Bohm potentials

Vacuum polarization of charged massless fermions is investigated in the superposition of Coulomb and Aharonov–Bohm (AB) potentials in 2 + 1 dimensions. For this purpose, we construct the Green function of the two-dimensional Dirac equation with Coulomb and AB potentials (via the regular and irregular solutions of the radial Dirac equation) and then calculate the vacuum polarization charge density in the so-called subcritical and supercritical regimes. In the supercritical regime, the Green function has a discontinuity in the complex plane of “energy” due to the singularities on the negative energy axis; these singularities are situated on the unphysical sheet and related to the creation of infinitely many quasistationary fermionic states with negative energies. We expect that our results will be helpful in gaining deeper understanding of the fundamental problem of quantum electrodynamics which can be applied to the problems of charged impurity screening in graphene taking into consideration the electron spin.

Space-time dynamics of the vacuum's polarization charge density

Using numerical solutions to the quantum-field theoretical Dirac equation, we study the space-time evolution of the vacuum's polarization charge and current densities induced by an external electric force in one spatial dimension. We discuss a simple analytical model that can predict the dynamics of the polarization dynamics for arbitrary external force configurations. We then study the corrections to these predictions when the external force becomes supercritical and real electron-positron pairs can be created. By coupling the Dirac equation to the Maxwell equations, we examine how the dynamics of the polarization density is affected, if we allow the corresponding induced charges to interact with each other.

Polarization charge distribution in gapped graphene: Perturbation theory and exact diagonalization analysis

We study the distribution of vacuum polarization charge induced by a Coulomb impurity in massive graphene. By analytically computing the polarization function, we show that the charge density is distributed in space in a non-trivial fashion, and on a characteristic length-scale set by the effective Compton wavelength. The density crosses over from a logarithmic behavior below this scale, to a power law variation above it. Our results in the continuum limit are confirmed by explicit diagonalization of the corresponding tight-binding model on a finite-size lattice. Electron-electron interaction effects are also discussed.

Screening of Coulomb Impurities in Graphene

We calculate exactly the vacuum polarization charge density in the field of a subcritical Coulomb impurity, Z|e|/r, in graphene. Our analysis is based on the exact electron Green's function, obtained by using the operator method, and leads to results that are exact in the parameter Zalpha, where alpha is the "fine-structure constant" of graphene. Taking into account also electron-electron interactions in the Hartree approximation, we solve the problem self-consistently in the subcritical regime, where the impurity has an effective charge Z(eff), determined by the localized induced charge. We find that an impurity with bare charge Z=1 remains subcritical, Z(eff)alpha<1/2, for any alpha, while impurities with Z=2, 3 and higher can become supercritical at certain values of alpha.

Classical electrodynamics with vacuum polarization: electron self-energy and radiation reaction

The region very close to an electron ( r ⪅ r 0 = e 2/ mc 2 ≈ 2.8 × 10 −13 cm) is, according to quantum electrodynamics, a seething maelstrom of virtual electron-positron pairs flashing in and out of existence. To take account of this well-established physical reality, a phenomenological representation for vacuum polarization is introduced into the framework of classical electrodynamics. Such a model enables a consistent picture of classical point charges with finite electromagnetic self-energy. It is further conjectured that the reaction of a point charge to its own electromagnetic field is tantamount to interaction with its vacuum polarization charge or “aura.” This leads to a modification of the Lorentz-Dirac equation for the force on an accelerating electron, a new differential-difference equation which avoids the pathologies of preacceleration and runaway solutions.

Theory of highly-ionized few-electron systems

Quantum electrodynamical corrections in hydrogen-like atoms are reviewed. In particular the Wichmann-Kroll contribution to the total vacuum polarization charge density is investigated. Limits for precision tests of quantum electrodynamics in strong Coulomb fields are given. A significant parity violation effect in helium-like uranium is pointed out.

Higher-order vacuum polarization charge densities for spherical symmetric external potentials

The vacuum polarization contribution of order α(Zα) n withn≧3 on electron binding energies has been calculated following the prescription of Wichmann and Kroll. The numerical techniques for the evaluation of the vacuum polarization charge density are discussed. Results are presented for atoms withZ=73, 82 and 170, respectively.

Higher order vacuum polarization for finite radius nuclei

The calculation of the higher order, α(Zα) n, n ≧ 3 , vacuum polarization charge density induced by high- Z nuclei of finite extent is discussed here. The Wichmann-Kroll formalism relating the vacuum polarization charge density to the Green function of the Dirac equation is reviewed with attention drawn to modifications necessary for very large- Z systems ( Z > 137) encountered in heavy ion collisions. This paper is concerned with the construction of the radial Green functions for the Dirac equation in the field of finite radius nuclei and on the numerical calculation of the higher order vacuum polarization density from those Green functions. Specific calculations are made for muonic Pb and superheavy electronic atoms. The results from these calculations have been published elsewhere but are further elaborated upon here.

Nuclear-Size Effects on Vacuum Polarization in Muonic Pb

The effect of finite nuclear size on the vacuum-polarization charge density is studied. The results to third order $\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{3}$, and to all orders $\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{n}$, $ng~3$, are presented with special attention focused on the $5{g}_{\frac{9}{2}}\ensuremath{-}4{f}_{\frac{7}{2}}$ transition in muonic Pb. In addition, the accuracy of analytic calculations exploiting the smallness of the electron mass and of the nuclear radius is discussed.

α(Zα)2Vacuum-Polarization Correction in Muonic Atoms

We use the Laplace transform obtained by Wichmann and Kroll for the vacuum-polarization charge density of order ${(Z\ensuremath{\alpha})}^{3}$ induced by a point nucleus to determine the first few terms in the expansion of the charge density for small distances from the nucleus. The result is used to estimate the effect of the polarization charge near the nucleus on the muonic x rays studied experimentally by Dixit et al. The effect is found to be somewhat smaller than an earlier estimate, and the discrepancy between theory and experiment is silightly reduced, but the discrepancy remains large for the high-$Z$ muonic atoms, as much as three or four standard devitations in lead and barium.

Electric Polarizability of the Neutron

In connection with a proposed explanation of the anisotropy observed in the scattering of neutrons from various elements at energies of a few hundred kev, the order of magnitude of the neutron polarizability is estimated by making use of data on photoproduction of pions. A polarizability $\ensuremath{\alpha}$ greater than \ensuremath{\sim}2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}42}$ ${\mathrm{cm}}^{3}$ appears unlikely on this basis. There remains an unexplained factor of \ensuremath{\sim}50 which has to be accounted for either in the polarizability or by providing another explanation of the neutron scattering anisotropy. The possibility of explaining the anisotropy on the basis of ordinary scattering theory does not appear to be excluded. The exact proportionality of the coefficient of $cos\ensuremath{\theta}$ to the neutron momentum does call for an ${r}^{\ensuremath{-}4}$ type of potential, but it is not clear whether the energy dependence of the coefficient is sufficiently well determined by the data and whether the compound nucleus features of the interaction are capable of explaining the observations. Nevertheless, a few less usual effects are estimated. These are the interaction of the neutron moment with the vacuum polarization charge and with the external electric field of the nucleus as well as its interaction with the electric charge density at the nuclear surface. The latter is hard to distinguish from other nuclear effects. The former two effects are small and do not resemble the observed effects.