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A novel soliton-like solution in quantum electrodynamics is obtained via a self-consistent field method. By writing the Hamiltonian of quantum electrodynamics in the Coulomb gauge, we separate out a classical component in the density operator of the electron-positron field. Then, by modeling the state vector in analogy with the theory of superconductivity, we minimize the functional for the energy of the system. This results in the equations of the self-consistent field, where the solutions are associated with the collective excitation of the electron-positron field---the soliton-like solution. In addition, the canonical transformation of the variables allowed us to separate out the total momentum of the system and, consequently, to find the relativistic energy dispersion relation for the moving soliton.
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More than 15 years ago, a new approach to quantum mechanics was suggested, in which Hermiticity of the Hamiltonian was to be replaced by invariance under a discrete symmetry, the product of parity and time-reversal symmetry, $mathcal{PT}$. It was shown that if $mathcal{PT}$ is unbroken, energies were, in fact, positive, and unitarity was satisifed. Since quantum mechanics is quantum field theory in 1 dimension, time, it was natural to extend this idea to higher-dimensional field theory, and in fact an apparently viable version of $mathcal{PT}$-invariant quantum electrodynamics was proposed. However, it has proved difficult to establish that the unitarity of the scattering matrix, for example, the Kallen spectral representation for the photon propagator, can be maintained in this theory. This has led to questions of whether, in fact, even quantum mechanical systems are consistent with probability conservation when Greens functions are examined, since the latter have to possess physical requirements of analyticity. The status of $mathcal{PT}$QED will be reviewed in this report, as well as the general issue of unitarity.
The Lee-Wick electrodynamics in the vicinity of a conducting plate is investigated. The propagator for the gauge field is calculated and the interaction between the plate and a point-like electric charge is computed. The boundary condition imposed on the vector field is taken to be the one that vanishes, on the plate, the normal component of the dual field strength to the plate. It is shown that the image method is not valid in Lee-Wick electrodynamics.
We investigate some aspects of the Maxwell-Chern-Simons electrodynamics focusing on physical effects produced by the presence of stationary sources and a perfectly conducting plate (mirror). Specifically, in addition to point charges, we propose two new types of point-like sources called topological source and Dirac point, and we also consider physical effects in various configurations that involve them. We show that the Dirac point is the source of the vortex field configurations. The propagator of the gauge field due to the presence of a conducting plate and the interaction forces between the plate and point-like sources are computed. It is shown that the image method is valid for the point-like charges as well as for Dirac points. For the topological source we show that the image method is not valid and the symmetry of spatial refection on the mirror is broken. In all setups considered, it is shown that the topological source leads to the emergence of torques.
Quantum parity conservation is verified at all orders in perturbation theory for a massless parity-even $U(1)times U(1)$ planar quantum electrodynamics (QED$_3$) model. The presence of two massless fermions requires the Lowenstein-Zimmermann (LZ) subtraction scheme, in the framework of the Bogoliubov-Parasiuk-Hepp-Zimmermann-Lowenstein (BPHZL) renormalization method, in order to subtract the infrared divergences induced by the ultraviolet subtractions at 1- and 2-loops, however thanks to the superrenormalizability of the model the ultraviolet divergences are bounded up to 2-loops. Finally, it is proved that the BPHZL renormalization method preserves parity for the model taken into consideration, contrary to what happens to the ordinary massless parity-even $U(1)$ QED$_3$.
The parity-preserving $U(1)times U(1)$ massless QED$_3$ is proposed as a pristine graphene-like planar quantum electrodynamics model. The spectrum content, the degrees of freedom, spin, masses and charges of the quasiparticles (electron-polaron, hole-polaron, photon and Neel quasiparticles) which emerge from the model are discussed. The four-fold broken degeneracy of the Landau levels, similar as the one experimentally observed in pristine graphene submitted to high applied external magnetic fields, is obtained. Furthermore, the model exhibits zero-energy Landau level indicating a kind of anomalous quantum Hall effect. The electron-polaron--electron-polaron scattering potentials in $s$- and $p$-wave states mediated by photon and Neel quasiparticles are computed and analyzed. Finally, the model foresees that two electron-polarons ($s$-wave state) belonging to inequivalent $mathbf{K}$ and $mathbf{K^prime}$ points in the Brillouin zone might exhibit attractive interaction, while two electron-polarons ($p$-wave state) lying both either in $mathbf{K}$ or in $mathbf{K^prime}$ points experience repulsive interaction.
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