A. V. Koshelkin
The exact solutions of the Dirac equation in an external non-abelian SU(N) gauge field which is in the form of a plane wave on the light cone is obtained. The whole set of the solutions for both particles and anti-particle is constructed.
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Dirac formalism of Hamiltonian constraint systems is studied for the noncommutative Abelian Proca field. It is shown that the system of constraints are of second class in agreement with the fact that the Proca field is not guage invariant. Then, the system of second class constraints is quantized by introducing Dirac brackets in the reduced phase space.
In this study we, remembering the experience with topological Dirac variables in the non-Abelian Yang-Mills-Higgs (YMH) model with vacuuum BPS monopole solutions, attempt to construct similar for the Abelian $U(1)$ model. We show that QED, as one understands it commonly, is only the topologically trivial sector ($n=0$) of this Abelian $U(1)$ model. For $n\neq 0$ one gets Dirac monopole modes. In both the cases, $n=0$ and $n\neq 0$, the theory can be quantized via the Hamiltonian reduction in terms of Dirac variables.
We investigate monopole solutions for the Born-Infeld Higgs system. We analyze numerically these solutions and compare them with the standard 't Hooft-Polyakov monopoles. We also discuss the existence of a critical value of beta (the Born-Infeld "absolute field parameter") below which no regular solution exists.
We search a canonical basis of Dirac's observables for the classical Abelian Higgs model with fermions in the case of a trivial U(1) principal bundle. The study of the Gauss law first class constraint shows that the model has two disjoint sectors of solutions associated with two physically different phases. In the electromagnetic phase, the electromagnetic field remains massless: after the determination of the Dirac's observables we get that both the reduced physical Hamiltonian and Lagrangian are nonlocal. In the Higgs phase, the electromagnetic field becomes massive and in terms of Dirac's o…
We introduce a new term into the Dirac equation based on the Lorentz symmetry violation background in order to make a theoretical description of the relativistic quantum dynamics of a spin-half neutral particle, where the wave function of the neutral particle acquires a relativistic Abelian quantum phase given by the interaction between a fixed time-like 4-vector background and crossed electric and magnetic fields, which is analogous to the geometric phase obtained by Wei \textit{et al} [H. Wei, R. Han and X. Wei, Phys. Rev. Lett. \textbf{75}, 2071 (1995)] for a spinless neutral particle with …
Topological superconductors classified as type D admit zero-energy Majorana fermions inside vortex cores, and consequently the exchange statistics of vortices becomes non-Abelian, giving a promising example of non-Abelian anyons. On the other hand, types C and DIII admit zero-energy Dirac fermions inside vortex cores. It has been long believed that an essential condition for the realization of non-Abelian statistics is non-locality of Dirac fermions made of two Majorana fermions trapped inside two well-separated vortices as in the case of type D. Contrary to this conventional wisdom, however, …
Within the framework of augmented version of the superfield approach to Becchi-Rouet-Stora-Tyutin (BRST) formalism, we derive the superspace unitary operator (and its Hermitian conjugate) in the context of four (3 + 1)-dimensional (4D) interacting non-Abelian 1-form gauge theory with Dirac fields. The ordinary 4D non-Abelian theory, defined on the flat 4D Minkowski spacetime manifold, is generalized onto a (4, 2)-dimensional supermanifold which is parameterized by the spacetime bosonic coordinates x^μ(with μ= 0, 1, 2, 3) and a pair of Grassmannian variables (θ, \barθ) which satisfy the standar…
A method is proposed for generalizing the Euclidean Taub-NUT space, regarded as the appropriate background of the Dirac magnetic monopole, to non-Abelian Kaluza-Klein theories involving potentials of generalized monopoles. Usual geometrical methods combined with a recent theory of the induced representations governing the Taub-NUT isometries lead to a general conjecture where the potentials of the generalized monopoles of any dimensions can be defined in the base manifolds of suitable principal fiber bundles. Moreover, in this way one finds that apart from the monopole models which are of a sp…
We search a canonical basis of Dirac's observables for the classical non-Abelian Higgs model with fermions in the case of a trivial SU(2) principal bundle with a complex doublet of Higgs fields and with the fermions in a given representation of SU(2). Since each one of the three Gauss law first class constraints can be solved either in the corresponding longitudinal electric field or in the corresponding Higgs momentum, we get a priori eight disjoint phases of solutions of the model. The only two phases with SU(2) covariance are the SU(2) phase with massless SU(2) fields and the Higgs phase wi…
Complete constraint analysis and choice of gauge conditions consistent with equations of motion is done for non-abelian Chern-Simons field interacting with N-component complex scalar field. Dirac-Schwinger condition is satisfied by the reduced phase-space Hamiltonian density with respect to the Dirac bracket.
Exact gauge structures arise in the evolution of spin-1/2 particles in conformally flat space-times. The corresponding Berry potentials can be Abelian or non-Abelian depending on the mass degeneracy of the system considered. Examples include de Sitter universes and maximal acceleration.
We extend our previous analysis on the exchange statistics of vortices having a single Dirac fermion trapped in each core, to the case where vortices trap two Dirac fermions with U(2) symmetry. Such a system of vortices with non-Abelian Dirac fermions appears in color superconductors at extremely high densities, and in supersymmetric QCD. We show that the exchange of two vortices having doublet Dirac fermions in each core is expressed by non-Abelian representations of a braid group, which is explicitly verified in the matrix representation of the exchange operators when the number of vortices …
We derive and solve the Hamiltonian flow equations for a Dirac particle in an external static potential. The method shows a general procedure for the set up of continuous unitary transformations to reduce the Hamiltonian to a quasidiagonal form.
We develop a method for finding the zero modes of the Dirac operator in the presence of BPS monopoles. We use it to find the zero modes in the case of Abelian BPS monopoles in $\mathbb R^3$.
The quantum fluctuations of the Dirac field in external classical gravitational and electromagnetic fields are studied. A self-consistent equation for torsion is calculated, which is obtained using one-loop fermion diagrams.
One of the key properties of Dirac operators is the possibility of a degeneracy of zero modes. For the Abelian Dirac operator in three dimensions the question whether such multiple zero modes may exist has remained unanswered until now. Here we prove that the feature of zero mode degeneracy indeed occurs for the Abelian Dirac operator in three dimensions, by explicitly constructing a class of Dirac operators together with their multiple zero modes. Further, we discuss some implications of our results, especially a possible relation to the topological feature of Hopf maps.
By explicit evaluation of certain disk S-matrix elements in the presence of background B-flux, we find coupling of two open string tachyons to gauge field, graviton, dilaton or Kalb-Ramond antisymmetric tensor on the world-volume of a single non-BPS D$p$-brane. We then propose an extension of the abelian Dirac-Born-Infeld action which naturally reproduces these couplings in field theory. This action includes non-linearly the dynamics of the tachyon field much like the other bosonic modes of the non-BPS D$p$-brane. On the general grounds of gauge and T-duality transformations and the symmetrize…
We study the self-interaction effects for the Dirac particle moving in an external field created by static charges in (1+1)-dimensions. Assuming that the total electric charge of the system vanishes, we show that the asymptotically linearly rising part of the external potential responsible for nonexistence of bound states in the external field problem without self-interaction is cancelled by the self-potential of the zero mode of the Dirac particle charge density. We derive the Dirac equation which includes the self-potential of the non-zero modes and is nonlinear. We solve the spectrum proble…
Non-Abelian Extensions of the Dirac Oscillator: A Theoretical Approach
本文提出在非阿贝尔规范场中协变的狄拉克振子，通过非最小替换实现振子相互作用，并推导出非阿贝尔场强张量，产生矩阵值自旋-异ospin耦合。
We present a supersymmetric version of Dirac-Born-Infeld (DBI) theory in noncommutative space-time, both for Abelian and non-Abelian gauge groups. We show, using the superfield formalism, that the definition of a certain ordering with respect to the * product leads naturally to a DBI action both in the U(1) as well as in the U(N) case. BPS equations are analysed in this context and properties of the resulting theory are discussed.
The Seiberg-Witten formalism has been realized as an electrodynamics in phase space (associated to the Dirac equation written in phase space) and this fact is explored here with non-abelian gauge group. First, a physically heuristic presentation of the Seiberg-Witten approach is carried out for non-abelian gauge in order to guide the calculation procedures. These results are realized by starting with the Lagrangian density for the free Dirac field in phase space. Then a field strength is derived, where the non-abelian gauge group is the SU(2), corresponding to an isospin (non-abelian) field th…
Non-Abelian Dirac oscillator in a uniform Yang--Mills background: spin--isospin mixing and singlet--triplet splitting
研究平面Dirac振子与均匀Yang-Mills背景耦合，通过Pauli约化公式化，得到自旋-同位旋混合和单态-三重态分裂的解析谱。
The equation describing the stochastic motion of a classical particle in 1+1-dimensional space-time is connected to the Dirac equation with external gauge fields. The effects of assigning different turning probabilities to the forward and the backward moving particles in time are discussed.
We show that the Loss-Yau zero modes of the 3d abelian Dirac operator may be interpreted in a simple manner in terms of a stereographic projection from a 4d Dirac operator with a constant field strength of definite helicity. This is an alternative to the conventional viewpoint involving Hopf maps from S^3 to S^2. Furthermore, our construction generalizes in a straightforward way to any odd dimension. The number of zero modes is related to the Chern-Simons number in a nonlinear manner.
The Dirac action describes the physics of the Nambu-Goldstone scalars found on branes. The Born-Infeld action defines a non-linear theory of electrodynamics. The combined Dirac-Born-Infeld (DBI) action describes the leading interactions supported on a single D-brane in string theory. We define a non-abelian analogue of DBI using the $T \overline{T}$ deformation in two dimensions. The resulting quantum theory is compatible with maximal supersymmetry and such theories are quite rare.
Effects of fermion-vacuum polarization by a singular configuration of an external static vector field are considered in (2 + 1)-dimensional spacetime. Expressions for the induced vacuum charge and magnetic flux are obtained.
We investigate the index of the Neuberger's Dirac operator in abelian gauge theories on finite lattices by numerically analyzing the spectrum of the hermitian Wilson-Dirac operator for a continuous family of gauge fields connecting different topological sectors. By clarifying the characteristic structure of the spectrum leading to the index theorem we show that the index coincides to the topological charge for a wide class of gauge field configurations. We also argue that the index can be found exactly for some special but nontrivial configurations in two dimensions by directly analyzing the s…
A kind of Dirac-Connes operator defined in the framework of Connes' NCG is introduced on discrete abelian groups; it satisfies a Junk-free condition, and bridges the NCG composed by Dimakis, Müller-Hoissen and Sitarz and the NCG of Connes. Then we apply this operator to d-dimensional lattices.
Pair creation of spin- 1/2 particles in Minkowski spacetime is investigated by obtaining exact solu- tions of the Dirac equation in the presence of electromagnetic fields and using them for determining the Bogoliubov coefficients. The resulting particle creation number density depends on the strength of the electric and magnetic fields.
The interaction of a nontrivial topological field configuration with the external fields is considered. The approach is based on the calculation of the zero modes excitation probability. We consider the interaction of the t'Hooft-Polyakov monopole with an external weak uniform magnetic field and the field of another monopole.
We investigate the effect of a constant external non-Abelian field on chiral symmetry breaking in a (2+1)-dimensional Nambu-Jona-Lasinio model and in 3D QCD by solving the gap equation and the Bethe-Salpeter equation, and also by RG analysis. In the (2+1)-dimensional NJL model chiral symmetry breaking occurs for any weak coupling constant but in 3D QCD catalysis of chiral symmetry breaking does not occur.
In this paper we show how to construct a Dirac operator on a lattice in complete analogy with the continuum. In fact we consider a more general problem, that is, the Dirac operator over an abelian finite group (for which a lattice is a particular example). Our results appear to be in direct connexion with the so called fermion doubling problem. In order to find this Dirac operator we need to introduce an algebraic structure (that generalizes the Clifford algebras) where we have quantities that work as square-root of the translation operator. Quantities like these square-roots have been used re…
In this paper we explore explicit exact solutions of the $SU(2)$ Yang-Mills (YM) and Yang-Mill-Higgs (YMH) equations with homogeneous and inhomogeneous external sources. Whereas in the case of YM we have confirmed our analytical findings with the numerical simulations, the numerical corroborations in the YMH case yielded the stochastic character of motion for the ensuing fields.
In the present article we present exact solutions of the Dirac equation for electric neutral particles with anomalous electric and magnetic moments. Using the algebraic method of separation of variables, the Dirac equation is separated in cartesian, cylindrical and spherical coordinates, and exact solutions are obtained in terms of special functions.
Complete constraint analysis and choice of gauge conditions consistent with equations of motion is done for Abelian Chern Simons field interacting minimally with a complex scalar field. The Dirac-Schwinger consistency condition is satisfied by the reduced phase space Hamiltonian density with respect to the the Dirac bracket. It is shown that relativistic invariance under boosts can be obtained only if gauge conditions were chosen consistent with the equations of motion. Moreover all gauge invariant quantities are shown to be free of transformation anomaly.
We investigate magnetic monopole solutions of the non-abelian DBI action describing 2 coincident non-BPS D9-branes in flat space. Just as in the case of kink and vortex solitonic tachyon solutions of the full DBI non-BPS actions, as previously analyzed by Sen, these monopole configurations are singular in the first instance and require regularization. We discuss a suitable non-abelian ansatz and show it solves the equations of motion to leading order in the regularization parameter. Fluctuations are studied and shown to describe a codimension 3 BPS D6-brane. A formula is derived for its tensio…
The motion of electron wave packets in the Dirac semimetals A$_3$Bi (A=Na,K,Rb) is studied in a semiclassical approximation. Because of the two-fold degeneracy of the Dirac points and a momentum-dependent gap term in the low-energy Hamiltonian, the associated Berry curvature is non-Abelian. In the presence of background electromagnetic fields, such a Berry curvature leads to a splitting of trajectories for the wave packets that originate from different Dirac points and chiral sectors. The nature of the splitting strongly depends on the background fields as well as the initial chiral compositio…
We discuss the Abelian and non-Abelian monopoles which could exist in condensed matters. We show how the Dirac monopole can be regularized by the charge screening, and argue that the Dirac monopole of mass of hundred meVs could exist in dielectric condensed matters. Moreover, we generalize this result to non-Abelian condensed matters to show the existence of the non-Abelian monopole configuration in two-gap condensed matters, and present explicit monopole solutions.
A detailed Faddeev-Jackiw quantization of an Abelian topological gravity is performed; we show that this formalism is equivalent and more economical than Dirac's method. In particular, we identify the complete set of constraints of the theory, from which the number of physical degrees of freedom is explicitly computed. We prove that the generalized Faddeev-Jackiw brackets and the Dirac ones coincide to each other. Moreover, we perform the Faddeev-Jackiw analysis of the theory at the chiral point, and the full set of constraints and the generalized Faddeev-Jackiw brackets are constructed. Final…
In this paper we present a model which can produce boundary confining condition on Dirac field interacting with Abelian or non Abelian gauge fields. The constraint is generated by a scalar field. This kind of model can be the foundation for bag models which can produce confinement. The present work represents among other things a generalization to the non Abelian case of our previous result where the Abelian case was studied. In the $U(1)$ case the coupling to the gauge field contains a term of the form $g(ϕ)j_μ(A^μ +\partial^μB)$ where $B$ is an auxiliary field and $j_μ$ is the Dirac current.…
In this paper we analyze two higher-derivative theories, the generalized electrodynamics and the Alekseev-Arbuzov-Baikov's effective Lagrangian from the point of view of Faddeev-Jackiw sympletic approach. It is shown that the full set of constraint is obtained directly from the zero-mode eigenvectors, and that they are in accordance with known results from Dirac's theory, a remnant and recurrent issue in the literature. The method shows to be rather economical in relation to the Dirac's one, obviating thus unnecessary classification and calculations. Afterwards, to conclude we construct the tr…
The scheme of isospin separation is suggested for the equation describing the five-dimensional 'charge-dyon' system in a non-Abelian SU(2) model. As a result, we obtain the Schrodinger equation for 'bare' particle, moving in Coulomb potential plus potential of five-dimensional Dirac monopole.
In this work, the Kaluza-Klein reduction of the Dirac equation on a 6 dimensional spacetime $\mathbb{M}^{1+5} := \mathbb{M}^{1+2} \times \mathbb{S}^3$ is studied. Because of the group structure on $\mathbb{S}^3$, $\mathbb{M}^{1+5}$ can be seen as a principal $SU(2)$ bundle over the model Lorentzian spacetime $\mathbb{M}^{1+2}$. The dimensional reduction induces non-minimal $SU(2)$ couplings to the theory on $\mathbb{M}^{1+2}$. These interaction terms will be investigated by comparing with a minimally $SU(2)$ coupled Dirac equation on $\mathbb{M}^{1+2}$. We hope that these results may help us t…
We study the solutions of the (2+1)-dimensional kappa-deformed Dirac oscillator in the presence of a constant transverse magnetic field. We demonstrate how the deformation parameter affects the energy eigenvalues of the system and the corresponding eigenfunctions. Our findings suggest that this system could be used to detect experimentally the effect of the deformation. We also show that the hidden supersymmetry of the non-deformed system reduces to a hidden pseudo-supersymmetry having the same algebraic structure as a result of the k-deformation.
We consider a spherically symmetric, static system of a Dirac particle interacting with classical gravity and an SU(2) Yang-Mills field. The corresponding Einstein-Dirac-Yang/Mills equations are derived. Using numerical methods, we find different types of soliton-like solutions of these equations and discuss their properties. Some of these solutions are stable even for arbitrarily weak gravitational coupling.
We study the properties of an ultracold Fermi gas loaded in an optical square lattice and subjected to an external and classical non-Abelian gauge field. We show that this system can be exploited as an optical analogue of relativistic quantum electrodynamics, offering a remarkable route to access the exotic properties of massless Dirac fermions with cold atoms experiments. In particular we show that the underlying Minkowski space-time can also be modified, reaching anisotropic regimes where a remarkable anomalous quantum Hall effect and a squeezed Landau vacuum could be observed.
We consider a static, spherically symmetric system of a Dirac particle in a classical gravitational and SU(2) Yang-Mills field. We prove that the only black-hole solutions of the corresponding Einstein-Dirac-Yang/Mills equations are the Bartnik-McKinnon black-hole solutions of the SU(2) Einstein-Yang/Mills equations; thus the spinors must vanish identically. This indicates that the Dirac particles must either disappear into the black-hole or escape to infinity.
A pure Dirac's method for abelian and non-abelian massive theories in three dimensions is performed. Our analysis is developed on the extended phase space reporting the relevant structure of the theories, namely, the extended action, the extended Hamiltonian, the full structure of the constraints and the counting of degrees of freedom. In addition, we compare our results with those found in the literature.
We demonstrate that the nilpotent Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry invariance of the Lagrangian density of a four (3 + 1)-dimensional (4D) non-Abelian 1-form gauge theory with Dirac fields can be captured within the framework of the superfield approach to BRST formalism. The above 4D theory, where there is an explicit coupling between the non-Abelian 1-form gauge field and the Dirac fields, is considered on a (4, 2)-dimensional supermanifold, parameterized by the bosonic 4D space-time variables and a pair of Grassmannian variables. We show that the Grassmannian independe…
The low lying spectrum of the massless overlap Dirac operator coupled to abelian fields in three dimensions with three different measures are shown to exhibit two phases: a strong coupling gapped phase and a weak coupling gapless phase. The vanishing of the gap from the strong coupling side with a Maxwell and a conformal measure is governed by a Gaussian exponent. Contrary to this result, the vanishing of the gap from the strong coupling side with a compact Thirring measure is not consistent with a Gaussian exponent. The low lying spectrum with a non-compact Thirring measure does not exhibit a…
In the continuum, a topological obstruction to the vanishing of the non-abelian anomaly in 2n dimensions is given by the index of a certain Dirac operator in 2n+2 dimensions, or equivalently, the index of a 2-parameter family of Dirac operators in 2n dimensions. In this paper an analogous result is derived for chiral fermions on the lattice in the Overlap formulation. This involves deriving an Index Theorem for a family of lattice Dirac operators satisfying the Ginsparg--Wilson relation. The index density is proportional to Luescher's topological field in 2n+2 dimensions.
We study the Dirac equation for spinor wavefunctions minimally coupled to an external field, from the perspective of an algebraic system of linear equations for the vector potential. By analogy with the method in electromagnetism, which has been well-studied, and leads to classical solutions of the Maxwell-Dirac equations, we set up the formalism for non-abelian gauge symmetry, with the SU(2) group and the case of four-spinor doublets. An extended isospin-charge conjugation operator is defined, enabling the hermiticity constraint on the gauge potential to be imposed in a covariant fashion, and…
This talk presents a study of massless relativistic Dirac fields in three Euclidean dimensions, at finite temperature and density, in the presence of a uniform electromagnetic background. Apart from explaining the behavior of Hall's conductivity for graphene, our results show a direct relationship between the selection of a phase for the Dirac determinant and the generation (or lack thereof) of Berry's phases and Chern-Simons terms.
A covariant version of the non-abelian Dirac-Born-Infeld-Myers action is presented. The non-abelian degrees of freedom are incorporated by adjoining to the (bosonic) worldvolume of the brane a number of anticommuting fermionic directions corresponding to boundary fermions in the string picture. The proposed action treats these variables as classical but can be given a matrix interpretation if a suitable quantisation prescription is adopted. After gauge-fixing and quantisation of the fermions, the action is shown to be in agreement with the Myers action derived from T-duality. It is also shown …
We study a system consisting of a non-Abelian SU(2) Proca field interacting with nonlinear scalar (Higgs) and spinor fields. For such a system, it is shown that particle-like solutions with finite energy do exist. It is demonstrated that the solutions depend on three free parameters of the system, including the central value of the scalar field $ξ_0$. For some fixed values of $ξ_0$, we find energy spectra of the solutions. It is shown that for each of the cases under consideration there is a minimum value of the energy $Δ=Δ(ξ_0)$ (the mass gap $Δ(ξ_0)$ for a fixed value of $ξ_0$). The behavior…
As a color confinement mechanism, a dual Meissner effect due to Abelian monopoles involved in QCD has been discussed so far in various ways. But still there is an important problem unsolved. It is gauge invariance of the schemes or, in other words, the reason why non-Abelian color confinement is explained by means of the Abelian dual Meissner effect. Here it is shown that a random Abelian method based on the violation of non-Abelian Bianchi identity (VNABI) could prove SU(3) invariance and explain why color-singlets alone can survive in the confinement phase of QCD in the framework of the Abel…
We compute the functional determinant for a Dirac operator in the presence of an Abelian gauge field on a bidimensional disk, under global boundary conditions of the type introduced by Atiyah-Patodi-Singer. We also discuss the connection between our result and the index theorem.
We study the spin factor problem both in $3+1$ and $2+1$ dimensions which are essentially different for spin factor construction. Doing all Grassmann integrations in the corresponding path integral representations for Dirac propagator we get representations with spin factor in arbitrary external field. Thus, the propagator appears to be presented by means of bosonic path integral only. In $3+1$ dimensions we present a simple derivation of spin factor avoiding some unnecessary steps in the original brief letter (Gitman, Shvartsman, Phys. Lett. {\bf B318} (1993) 122) which themselves need some a…
We derive a semi-classical effective action and the kinetic equation for massive Dirac fermions in electromagnetic fields. The non-Abelian Berry phase structure emerges from two helicity states of massive fermions with positive energy. The classical spin emerges as a vector in SU(2) helicity space. The continuity equations for the fermion number and the classical spin are derived. The fermion number is conserved while the spin charge is not conserved by anomaly. Previous results about the coefficients of the chiral magnetic effect for the fermion and axial currents in the chiral limit can be r…
Using the natural curvature invariants as building blocks in a superfield construction, we show that the use of a symmetric trace is mandatory if one is to reproduce the square root structure of the non-Abelian Dirac-Born-Infeld Lagrangian in the bosonic sector. We also discuss the BPS relations in connection with our supersymmetry construction.
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