Two-dimensional Electrodynamics Research Articles

Geometrodynamical description of two-dimensional electrodynamics

Two-dimensional pure electrodynamics is mapped into two-dimensional gravity in the first-order formalism at classical and quantum levels. Due to the fact that the degrees of freedom of these two theories do not match, we are enforced to introduce extra fields from the beginning. These fields are introduced through a BRST exact boundary term, so they are harmless to the physical content of the theory. The map between electromagnetism and gravity fields generate a non-trivial Jacobian, which brings extra features (but also harmless to the physical content of the gravity theory) at quantum level.

Dynamic nature of the anomalous magnetic moment of an electron in a constant magnetic field in topologically massive two-dimensional electrodynamics

An analytical expression is obtained for the anomalous magnetic moment of an electron in a constant magnetic field in topologically massive two-dimensional electrodynamics. In the limiting case of a relatively weak magnetic field, asymptotic formulas are found, which specify the dependence of the anomalous magnetic moment on the Chern-Simons parameter and dynamic parameter of synchrotron radiation. The conditions are identified for the applicability of the computations of the anomalous magnetic moment of an electron, which have previously been made based on the calculation of the vertex function in two-dimensional electrodynamics with the Chern-Simons term.

Anomalous magnetic moment of an electron in a constant magnetic field in ( 2+1 )-dimensional quantum electrodynamics

The anomalous magnetic moment (AMM) for excited states of an electron in a constant magnetic field has been calculated within the framework of two-dimensional electrodynamics. The analytical results for the interaction energy of the anomalous magnetic moment with the external magnetic field are obtained in two limiting cases of nonrelativistic and relativistic energy values in a comparatively weak magnetic field. It is shown that the interaction energy of the spin with the external field does not contain infrared divergence and tends to zero as magnetic field decreases, while the electron's AMM increases logarithmically.

Two-dimensional plasmons in the random impedance network model of disordered thin film nanocomposites

Random impedance networks are widely used as a model to describe plasmon resonances in disordered metal-dielectric nanocomposites. In order to study thin films, two-dimensional networks are often used despite the fact that such networks correspond to a two-dimensional electrodynamics [J.P. Clerc et al, J. Phys. A 29, 4781 (1996)]. In the present work, we propose a model of two-dimensional systems with three-dimensional Coulomb interaction and show that this model is equivalent to a planar network with long-range capacitive connections between sites. In a case of a metal film, we get a known dispersion $\omega \propto \sqrt{k}$ of plane-wave two-dimensional plasmons. In the framework of the proposed model, we study the evolution of resonances with decreasing of metal filling factor. In the subcritical region with metal filling $p$ lower than the percolation threshold $p_c$, we observe a gap with Lifshitz tails in the spectral density of states (DOS). In the supercritical region $p>p_c$, the DOS demonstrates a crossover between plane-wave two-dimensional plasmons and resonances associated with small clusters.

Limitations in the 2D description of the electromagnetic waves propagation in thin dielectric and magnetic layers

The propagation of electromagnetic waves trapped within dielectric and magnetic layers is considered. The description within the three-dimensional theory is compared to the simplified analysis in two dimensions. Two distinct media configurations of different topology are dealt with: a plane slab and a hollow cylinder. Choosing the appropriate values for the geometrical parameters (layer thickness, radius of the cylinder) and for the electromagnetic properties of the media one can trap exactly one mode corresponding to that obtained within the two-dimensional electromagnetism. However, the symmetry between electric and magnetic fields suggests, that the two versions of the simplified electromagnetism ought to be equally considered. Its usual form is incomplete to describe all modes. It is also found that there exists a domain of optimal values of parameters for which the 2D model works relatively correctly. However, in the case of a cylindrical surface we observe several differences which may be attributed to the curvature of the layer, and which exclude the propagation of evanescent modes. The two-dimensional electrodynamics, whichever form is used, turns out still too poor to describe the so-called ‘hybrid modes’ excited in a real layer. The obtained results can be essential for proper description of the propagating waves within thin layers for which 3D approach is not available due to mathematical complexity and reducing the layer to a lower dimensional structure seems the only possible option.

Anomalous magnetic moment of an electron in a magnetized plasma of topologically massive two-dimensional electrodynamics

Anomalous magnetic moment of an electron in a magnetized plasma of topologically massive two-dimensional electrodynamics

Light-Front Quantization of the Vector Schwinger Model with a Photon Mass Term in Faddeevian Regularization

In this talk, we study the light-front quantization of the vector Schwinger model with photon mass term in Faddeevian Regularization, describing two-dimensional electrodynamics with mass-less fermions but with a mass term for the U(1) gauge field. This theory is gauge-non-invariant (GNI). We construct a gauge-invariant (GI) theory using Stueckelberg mechanism and then recover the physical content of the original GNI theory from the newly constructed GI theory under some special gauge-fixing conditions (GFC’s). We then study LFQ of this new GI theory.

Vector Schwinger Model with a Photon Mass Term with Faddeevian Regularization

In this talk, we consider the vector Schwinger model with a photon mass term with Faddeevian Regularization, describing two-dimensional (2D) electrodynamics with mass-less fermions and study its Hamiltonian and path integral quantization. This theory is seen to be gauge-non-invariant (GNI). We then construct a gauge-invariant (GI) theory corresponding to this GNI theory using the Stueckelberg mechanism and then recover the physical content of the original GNI theory from the newly constructed GI theory under some special gauge-fixing conditions.

VECTOR SCHWINGER MODEL WITH A PHOTON MASS TERM: GAUGE-INVARIANT REFORMULATION, OPERATOR SOLUTIONS AND HAMILTONIAN AND PATH INTEGRAL FORMULATIONS

We consider the vector Schwinger model (VSM) describing two-dimensional electrodynamics with massless fermions, where the left-handed and right-handed fermions are coupled to the electromagnetic field with equal couplings, with a mass term for the U(1) gauge field and then study its operator solutions and the Hamiltonian and path integral formulations. We emphasize here that although the VSM has been studied in the literature rather widely but only without a photon mass term (which was a consequence of demanding the regularization for the VSM to be gauge-invariant (GI)). The VSM with a photon mass term is seen to be a gauge-noninvariant (GNI) theory. Using the standard Stueckelberg formalism we then construct a GI theory corresponding to the proposed GNI model. From this reformulated GI theory, we further recover the physical contents of the proposed GNI theory under a very special gauge choice. The theory proposed and studied here presents a new class of models in the two-dimensional quantum electrodynamics with massless fermions but with a photon mass term.

Effective light-front quantization of scalar field theories and two-dimensional electrodynamics.

We introduce a new method to include condensates in the light-cone Hamiltonian. By using a Gaussian approximation to the ordinary vacuum in a theory close to the light front, we derive an effective Hamiltonian on the light cone, which has new terms reflecting the nontriviality of the vacuum. We demonstrate our method for scalar ${\mathrm{\ensuremath{\varphi}}}^{4}$ theory and the massive Schwinger model.

DOES THE STRONG CP-PROBLEM REALLY EXIST?

It is argued that the principle of absolute gauge invariance (which implies that the functional manifold of the gauge fields is a “circle,” where the integer-valued Chern-Simons numbers present the same point at the “circle”) provides a natural solution of the CP-problem. It is shown in simple mechanical models that the θ-term being a time derivative of some function does not change the quantum dynamics just as it does not change the classical one. In this context the two-dimensional electrodynamics is also considered.

Delta expansion for local gauge theories. I. A one-dimensional model.

The principles of the \ensuremath{\delta} perturbation theory were first proposed in the context of self-interacting scalar quantum field theory. There it was shown how to expand a (${\mathrm{\ensuremath{\varphi}}}^{2}$${)}^{1+\mathrm{\ensuremath{\delta}}}$ theory as a series in powers of \ensuremath{\delta} and how to recover nonperturbative information about a ${\mathrm{\ensuremath{\varphi}}}^{4}$ field theory from the \ensuremath{\delta} expansion at \ensuremath{\delta}=1. The purpose of this series of papers is to extend the notions of \ensuremath{\delta} perturbation theory from boson theories to theories having a local gauge symmetry. In the case of quantum electrodynamics one introduces the parameter \ensuremath{\delta} by generalizing the minimal coupling terms to \ensuremath{\psi}\ifmmode\bar\else\textasciimacron\fi{}(\ensuremath{\partial}-ieA${)}^{\mathrm{\ensuremath{\delta}}}$\ensuremath{\psi} and expanding in powers of \ensuremath{\delta}. This interaction preserves local gauge invariance for all \ensuremath{\delta}. While there are enormous benefits in using the \ensuremath{\delta} expansion (obtaining nonperturbative results), gauge theories present new technical difficulties not encountered in self-interacting boson theories because the expression (\ensuremath{\partial}-ieA${)}^{\mathrm{\ensuremath{\delta}}}$ contains a derivative operator. In the first paper of this series a one-dimensional model whose interaction term has the form \ensuremath{\psi}\ifmmode\bar\else\textasciimacron\fi{}[d/dt-ig\ensuremath{\varphi}(t)${]}^{\mathrm{\ensuremath{\delta}}}$\ensuremath{\psi} is considered. The virtue of this model is that it provides a laboratory in which to study fractional powers of derivative operators without the added complexity of \ensuremath{\gamma} matrices. In the next paper of this series we consider two-dimensional electrodynamics and show how to calculate the anomaly in the \ensuremath{\delta} expansion.

Two-dimensional electrodynamics and the global structure of space

Two-dimensional electrodynamics based on the spacetime S1(X)R is formulated using Fourier series and Dirac's constraint theory. The role of the background electric field F as a dynamical variable is established. The authors show that the background field is unstable against a class of charged pair creation processes involving the global topology of S1 provided mod F mod >1/2e, where e is the unit of electric charge. The results do not require any discussion of the Coulomb Green function or of the role of charges on spatial boundaries.

The lattice gauge theory approach to quantum chromodynamics

The author reviews in a pedagogical fashion some of the recent developments in lattice quantum chromodynamics. This review emphasizes explicit examples and illustrations rather than general proofs and analyses. It begins with a discussion of the heavy-quark potential in continuum quantum chromodynamics. Asymptotic freedom and renormalization-group improved perturbation theory are discussed. A simple dielectric model of confinement is considered as an intuitive guide to the vacuum of non-Abelian gauge theories. Next, the Euclidean form of lattice gauge theory is introduced, and an assortment of calculational methods are reviewed. These include high-temperature expansions, duality, Monte Carlo computer simulations, and weak coupling expansions. A $\ensuremath{\Lambda}$- parameter calculation for asymptotically free-spin models is presented. The Hamiltonian formulation of lattice gauge theory is presented and is illustrated in the context of flux tube dynamics. Roughening transitions, Casimir forces, and the restoration of rotational symmetry are discussed. Mechanisms of confinement in lattice theories are illustrated in the two-dimensional electrodynamics of the planar model and the U(1) gauge theory in four dimensions. Generalized actions for SU(2) gauge theories and the relevance of monopoles and strings to crossover phenomena are considered. A brief discussion of the continuity of fields and topological charge in asymptotically free lattice models is presented. The final major topic of this review concerns lattice fermions. The species doubling problem and its relation to chiral symmetry are illustrated. Staggered Euclidean fermion methods are discussed in detail, with an emphasis on species counting, remnants of chiral symmetry, Block spin variables, and the axial anomaly. Numerical methods for including fermions in computer simulations are considered. Jacobi and Gauss-Siedel inversion methods to obtain the fermion propagator in a background gauge field are reviewed. Langevin, pseudofermion Monte Carlo methods, and random walk methods are discussed, as are proposals for including the fermion determinant in computer simulations. Finally, spontaneous chiral symmetry breaking in lattice models is discussed. The review ends with brief remarks about present research directions.

Variable-shift surface terms and regularization procedures in two-dimensional quantum electrodynamics

The vacuum polarization and photon mass within Schwinger's model for two-dimensional electrodynamics are obtained from a finite surface term associated with the difference between variable-shifted linearly divergent integrals. Consistency is demonstrated between this approach and dimensional regularization. Ambiguity in the application of dimensional reduction is also discussed.

Confinement and properties of vacuum states in two-dimensional electrodynamics of massless quarks

Connections between the properties of vacuum states and the confinement phenomenon are investigated for QED 2.

Evolution operator and quark structure of states in two-dimensional massless electrodynamics

In the two-dimensional electrodynamics of massless fermions (quarks) the time evolution operator is constructed. Its diagonalization permits one to find the physical states of the model in the terms of quarks. The existence of charge and chiral states is discussed. It is shown that such states can be found only with vanishing momentum, i.e., as vacuum states. The quark structure of physical particles is obtained in the infinite momentum frame.

The gauge technique for two-dimensional electrodynamics

The authors show how the Schwinger model may be solved by means of the nonperturbative gauge technique.

Chiral gauge symmetry without anomalies

The Ward identities derived by use of chiral gauge symmetry, when examined in a perturbation expansion using Lagrangian field theory, are found to contain the Adler-Bell-Jackiw-Schwinger anomalies. The cause of this shortcoming is examined, and the result is that when one uses a formulation of field theory in which only fields already renormalized are introduced these anomalies disappear completely. The specific cases examined here are massless two-dimensional electrodynamics with a vector coupling and massless four-dimensional electrodynamics with an axial-vector coupling. In the first example, it is also found that the polarization tensor disappears in second order, and does not exhibit a pole at zero momentum transfer that would cause the vector field to generate a mass spontaneously.

Effects of boundary conditions on massless two-dimensional electrodynamics in a static bag

We investigate the behavior of massless two-dimensional quantum electrodynamics when the fields are confined to a static string. Surface modifications of the equal-time commutation relations are studied, and the generating functional is constructed, consistent with the linear boundary conditions of the theory. It is shown that massive scalars exist independent of the size of the cavity. The boundary condition of the initial fields constrain the boson field to vanish at the end points of the bag. We further demonstrate that the short-distance behavior of the theory is unaffected by the introduction of linear boundary conditions.