Alexander G. Kyriakos
The present paper is the continuation of the paper "Nonlinear field theory I". In the paper it is shown that a fully correspondence between the quantum and the nonlinear electromagnetic forms of the Dirac electron theory exists, so that each element of the Dirac theory acquires the known electrodynamics meaning and reversely
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In the present paper it is shown that the Dirac electron theory is the approximation of the special nonlinear electromagnetic field theory
Nonlinear Dirac equations (NLDE) are derived through a group N^2 of nonlinear (gauge) transformation acting in the corresponding state space. The construction generalises a construction for nonlinear Schrödinger equations. To relate N^2 with physically motivated principles we assume: locality (i.e. it contains no explicit derivative and no derivatives of the wave function), separability (i.e. it acts on product states componentwise) and Poincaré invariance. Furthermore we want that a positional density is invariant under N^2. Such nonlinear transformations yield NLDE which describe physically …
Quantum walk (QW) provides a versatile tool to study fundamental physics and also to make a variety of practical applications. We here start with the recent idea of {\it nonlinear} QW and show that introducing {\it nonlinearity} to QW can lead to a wealth of remarkable possibilities, e.g., simulating nonlinear quantum dynamics thus enhancing the applicability of QW above the existing level for a universal quantum simulator. As an illustration, we show that the dynamics of a nonlinear Dirac particle can be simulated on an optical nonlinear QW platform implemented with a measurement-based-feedfo…
The Dirac equation for the Coulomb problem is restated by incorporating a nonlinear effective interaction into the Dirac Hamiltonian: one keeps the $1/r$ dependence for the Coulomb field, but the coupling constant is modified by a factor depending on the n (principal quantum number) power of the mean value of the Hamiltonian. In this simple context we study the Lamb shift and the hyperfine splitting of the s-levels of hydrogenic atoms. We discuss to what extent the corresponding calculations fit the energy splittings to the appropriate order in the fine structure constant.
The effect of noise on the Dirac phase of electron in the presence of screw dislocation is studied. An uncorrelated noise, which coincides with the nature of thermal fluctuations, is adopted. Results indicate that the Dirac phase is robust against existing noise in the system.
Fundamental solutions for the free Dirac electron and Einstein photon equations in position coordinates are constructed as matrix valued functionals on the space of bump functions. It is shown that these fundamental solutions are related by a unitary transform via the Cauchy distribution in imaginary time. We study the way the classical relativistic mechanics of the free particle comes from the quantum mechanics of the free Dirac electron.
The present paper is the continuity of the previous papers "Non-linear field theory" I and II. Here on the basis of the electromagnetic representation of Dirac's electron theory we consider the geometrical distribution of the electromagnetic fields of the electron-positron. This gives the posibility to obtain the explanation and solution of many fundamental problems of the QED.
We show that, in the relativistic Bohm model of a Dirac-like particle, the electron in the ground state of the hydrogen atom is moving, unlike the prediction for the case of a Schrödinger-like particle, where the electron is stationary. This accounts for the empirically observed dilation of the decay time of the muon in the ground state of muonium.
Topological phases and materials have attracted much attention in recent years. Though many progress has been made, the effect of nonlinearity on such system remains untouched. In this paper, by considering the mean-field approximation in a coupled boson-hexagonal lattice system, we obtain a different type of Dirac cone. Due to its special structure of the cone, the Berry phase (two-dimensional Zak phase) of this new Dirac cone is quantized differently, i.e., it has been modified due to the interactions and the critical line between different topological phases has moved, depending on the type…
We construct nonlinear extensions of Dirac's relativistic electron equation that preserve its other desirable properties such as locality, separability, conservation of probability and Poincaré invariance. We determine the constraints that the nonlinear term must obey and classify the resultant non-polynomial nonlinearities in a double expansion in the degree of nonlinearity and number of derivatives. We give explicit examples of such nonlinear equations, studying their discrete symmetries and other properties. Motivated by some previously suggested applications we then consider nonlinear term…
We propose nonlinear Dirac equations where the conformal degree of the self-interaction terms are equal to that of the Dirac operator and the coupling parameters are dimensionless. As such, the massless equation is conformally invariant and preserves the conformal degree for both the linear and nonlinear components. In 2+1 space-time, we use these features to propose three- and four-parameter nonlinear Dirac equation models that might be useful for applications in 2D systems such as graphene sheets, ribbons and nanotubes.
Classical electromagnetism is linear. However, fields can polarize the vacuum Dirac sea, causing quantum nonlinear electromagnetic phenomena, e.g., scattering and splitting of photons, that occur only in very strong fields found in neutron stars or heavy ion colliders.We show that strong nonlinearity arises in Dirac materials at much lower fields $\sim 1\:\text{T}$, allowing us to explore the nonperturbative, extremely high field limit of quantum electrodynamics in solids. We explain recent experiments in a unified framework and predict a new class of nonlinear magneto-electric effects, includ…
In this work, we study the scattering problem of the general nonlinear finitely many Dirac delta potentials with complex coupling constants (or opacities in the context of optics) using the Green's function method and then find the bound state energies and the wave functions for the particular form of the nonlinearity in the case of positive real coupling constants.
Numerical solution of the nonlinear Dirac equation by a splitting variational quantum algorithm
提出分裂变分量子算法Dirac-sVQA，通过线性狄拉克子步和非线性变分修正分解非线性狄拉克方程，利用自旋-傅里叶传播器和测量基变分更新实现数值模拟。
In this paper we study the nonlinear Dirac (NLD) equation on noncompact metric graphs with localized Kerr nonlinearities, in the case of Kirchhoff-type conditions at the vertices. Precisely, we discuss existence and multiplicity of the bound states (arising as critical points of the NLD action functional) and we prove that, in the $L^2$-subcritical case, they converge to the bound states of the NLS equation in the nonrelativistic limit.
Nambu-Type Generalization of the Dirac Equation
讨论了Dirac方程的非线性推广，推广了非线性哈密顿量的标准范式.在新的框架内制定的所有理论都不存在“比光速快的电报”。引入了与Lie-Poisson动力学相关的无限维李代数的一个新度量。
Physics arising from two-dimensional~(2D) Dirac cones has been a topic of great theoretical and experimental interest to studies of gapless topological phases and to simulations of relativistic systems. Such $2$D Dirac cones are often characterized by a $π$ Berry phase and are destroyed by a perturbative mass term. By considering mean-field nonlinearity in a minimal two-band Chern insulator model, we obtain a novel type of Dirac cones that are robust to local perturbations without symmetry restrictions. Due to a different pseudo-spin texture, the Berry phase of the Dirac cone is no longer quan…
The authors give a detailed information about symmetry (Lie, non-Lie, conditional) of nonlinear PDEs for spinor, vector and scalar fields; using advanced methods of group-theoretical, symmetry analysis construct wide families of classical solutions of the nonlinear Dirac, Yang-Mills, Maxwell-Dirac, Dirac-d'Alembert, d'Alembert-Hamilton equations; expound a new symmetry approach to variable separation in linear and nonlinear PDEs, which allows, in particular, to classify separable Schroedinger equations. The book offers a uniform and relatively simple presentation of a considerable amount of ma…
We study dynamics of Dirac solitons in prototypical networks modeling them by the nonlinear Dirac equation on metric graphs. Soliton solutions of the nonlinear Dirac equation on simple metric graphs are obtained. It is shown that these solutions provide reflectionless vertex transmission of the Dirac solitons under suitable conditions. The constraints for bond nonlinearity coefficients, allowing reflectionless transmission over a Y-junction are derived. The analytical results are confirmed by direct numerical simulations.
Rejuvenation of hydrodynamic transport in solids provides a new window to study collective motion of electrons, where electrons behave like a viscous fluid akin to classical liquids. Experimental observations of such exotic states have not been realized until recent years, and an on-going quest is to amplify the hydrodynamic effect in electron fluids. Here we investigate the hydrodynamic properties of Dirac electron fluid in graphene from a microscopic viewpoint, and elucidate a novel way to enhance electron hydrodynamics. In particular, we present strong evidence that the shear viscosity of D…
We give a rigorous deduction of the eigenvalue problem of the nonlinear Schrödinger equation (NLS) at Dirac Points for potential of honeycomb lattice symmetry. Based on a bootstrap method, we observe the bifurcation of the eigenfunctions into eight distinct modes from the two-dimensional degenerated eigenspace of the regressive linear Schrödinger equation. We give the existence, the way of construction, uniqueness in $H^2$ space and the $C^\infty$ continuity of these eigenfunctions.
We discuss the relation of the Kerr-Newman spinning particle to the Dirac electron and show that the Dirac equation may naturally be incorporated into Kerr-Schild formalism as a master equation controlling the Kerr-Newman geometry. As a result, the Dirac electron acquires an extended space-time structure of the Kerr-Newman geometry - singular ring of the Compton size and twistorial polarization of the gravitational and electromagnetic fields. Behavior of this Dirac -- Kerr-Newman system in the weak and slowly changed electromagnetic fields is determined by the wave function of the Dirac equati…
Nonlinear Terahertz Electroluminescence from Dirac Landau Polaritons
本研究通过太赫兹磁反射率光谱观测到狄拉克Landau极子，并发现其非线性电发光特性，揭示了极子在上分支的强耦合与高占率。
We present an analysis of excited-state solutions for a gravitationally localized system consisting of a filled shell of high-angular-momentum fermions, using the Einstein-Dirac formalism introduced by Finster, Smoller, and Yau [Phys. Rev. D 59, 104020 (1999)]. We show that, even when the particle number is relatively low ($N_f\ge 6$), the increased nonlinearity in the system causes a significant deviation in behavior from the two-fermion case. Excited-state solutions can no longer be uniquely identified by the value of their central redshift, with this multiplicity producing distortions in th…
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