Search

This paper is devoted to a general min-max characterization of the eigenvalues in a gap of the essential spectrum of a self-adjoint unbounded operator. We prove an abstract theorem, then we apply it to the case of Dirac operators with a Coulomb-like potential. The result is optimal for the Coulomb potential. An erratum is appended at the end of the tex., The main goal of this submission is to add an erratum to the old paper published in 2000

Clifford algebras are used for constructing spin groups, and are therefore of particular importance in the theory of quantum mechanics. An algebraist’s perspective on the many subgroups and subalgebras of Clifford algebras may suggest ways in which they might be applied more widely to describe the fundamental properties of matter. I do not claim to build a physical theory on top of the fundamental algebra, and my suggestions for possible physical interpretations are indicative only, and may not work. Nevertheless, both the existence of three generations of fermions and the symmetry-breaking of the weak interaction seem to emerge naturally from an extension of the Dirac algebra from complex numbers to quaternions.

A general formula for inversion in a relativistic Clifford–Dirac algebra has been derived. Identifying the base elements of the algebra as those of space and time, the first order differential equations over all quantities proves to encompass the Maxwell equations, leads to a natural extension incorporating rest mass and spin, and allows an integration with relativistic quantum mechanics. Although the algebra is not a division algebra, it parallels reality well: where division is undefined turns out to correspond to physical limits, such as that of the light cone. The divisor corresponds to invariants of dynamical significance, such as the invariant interval, the general invariant quantities in electromagnetism, and the basis set of quantities in the Dirac equation. It is speculated that the apparent 3-dimensionality of nature arises from a beautiful symmetry between the three-vector algebra and each of four sets of three derived spaces in the full 4-dimensional algebra. It is conjectured that elements of inversion may play a role in the interaction of fields and matter.

A detailed Gitman–Lyakhovich–Tyutin analysis for higher-order topologically massive gravity is performed. The full structure of the constraints, the counting of physical degrees of freedom, and the Dirac algebra among the constraints are reported. Moreover, our analysis presents a new structure into the constraints and we compare our results with those reported in the literature where a standard Ostrogradski framework was developed.

We present SuperTracer, a Mathematica package aimed at facilitating the functional matching procedure for generic UV models. This package automates the most tedious parts of one-loop functional matching computations. Namely, the determination and evaluation of all relevant supertraces, including loop integration and Dirac algebra manipulations. The current version of SuperTracer also contains a limited set of output simplifications. However, a further reduction of the output to a minimal basis using Fierz identities, integration by parts, simplification of Dirac structures, and/or light field redefinitions might still be necessary. The code and example notebooks are publicly available at https://gitlab.com/supertracer/supertracer., Comment: 38 pages, 1 figure. The code and example notebooks are publicly available at https://gitlab.com/supertracer/supertracer

In this paper, the classical and quantum theory of $\mathcal{N}=1$ supergravity in four spacetime dimensions will be studied in the framework of loop quantum gravity. We discuss the canonical analysis of the supergravity Holst action as first introduced by Tsuda. In this way, we also derive a compact expression of the supersymmetry constraint, which plays a crucial role in canonical supergravity theories, akin to the role of the Hamiltonian constraint in nonsupersymmetric generally covariant theories. The resulting theory is then quantized using loop quantum gravity methods. In particular, we propose and discuss a quantization of the supersymmetry constraint and derive explicit expressions of the action of the resulting operator. This is important as it is the first step on the way to analyzing the Dirac algebra generated by supersymmetry and Hamiltonian constraint in the quantum theory and for finding physical states. We also discuss some qualitative properties of such solutions of the SUSY constraint.

Second order degenerate Chiral oscillator Lagrangian is reduced into first order Lagrangian and Hamiltonian analysis of this formalism is performed by means of the DiracBergmann constraint algorithm.

International audience; We emphasize the differences between the chirality concept applied to relativistic fermions and the ususal chirality concept in Euclidean spaces. We introduce the gamma groups and we use them to classify as direct or indirect the symmetry operators encountered in the context of Dirac algebra. Then we show how a recent general mathematical definition of chirality unifies the chirality concepts and resolve conflicting conclusions about symmetry operators, and particularly about the so-called chirality operator. The proofs are based on group theory rather than on Clifford algebras. The results are independent on the representations of Dirac gamma matrices, and stand for higher dimensional ones.

We present FeynCalc 9.3, a new stable version of a powerful and versatile Mathematica package for symbolic quantum field theory (QFT) calculations. Some interesting new features such as highly improved interoperability with other packages, automatic extraction of the ultraviolet divergent parts of 1-loop integrals, support for amplitudes with Majorana fermions and $\gamma$-matrices with explicit Dirac indices are explained in detail. Furthermore, we discuss some common problems and misunderstandings that may arise in the daily usage of the package, providing explanations and workarounds., Comment: 21 pages, no figures, updated to match the journal version. To obtain the program, see https://github.com/FeynCalc/feyncalc

Based on the previous work on the Dirac algebra and su*(4) Lie algebra generators, using Lie transfer we have associated spin to line and Complex reps. Here, we discuss the construction of a Lagrangian in terms of invariant theory using lines or linear Complex reps like Fμν, its dual Fαβ, or even quadratic terms like e.g. FμνFμν, or FaμνFaμν with respect to regular linear Complexe. In this context, we sketch briefly the more general framework of quadratic Complexe and show how special relativistic coordinate transformations can be obtained from (invariances with respect to) line transformations. This comprises the action of the Dirac algebra on 4×2 “spinors”, real as well as complex. We discuss a classical picture to relate photons to linear line Complexe so that special relativity emerges naturally from a special line (or line Complex) invariance, and compare to Minkowski’s fundamental paper on special relativity. Finally, we give a brief outlook on how to generalize this approach to general relativity using advanced projective and (line) Complex geometry related to P5 and the Plucker–Klein quadric as well as transfer principles.

Applications of Clifford analysis to three-dimensional elasticity are addressed in the present paper. The governing equation for the displacement field is formulated in terms of the Dirac operator and Clifford algebra valued functions so that a general solution is obtained analytically in terms of one monogenic function and one multiple-component spatial harmonic function together with its derivative. In order to solve numerically the three-dimensional problems of elasticity for an arbitrary domain with complicated boundary conditions, Clifford algebra valued boundary integral equations (BIEs) for multiple-component spatial harmonic functions at an observation point, either inside the domain, on the boundary, or outside the domain, are constructed. Both smooth and non-smooth boundaries are considered in the construction. Moreover, the singularities of the integrals are evaluated exactly so that in the end singularity-free BIEs for the observation point on the boundary taking values on Clifford numbers can be obtained. A Clifford algebra valued boundary element method (BEM) based on the singularity-free BIEs is then developed for solving three-dimensional problems of elasticity. The accuracy of the Clifford algebra valued BEM is demonstrated numerically.

We study a dissipative quantum system which is described by the Doebner–Goldin equation (DGE) model. For time-independent states, the new three-dimensional analytical solutions of the DGE are obtained by binding the vertical relation of velocity and the gradient of density in the system, when the form of a central potential such as hard core or harmonic oscillator is suggested. Through the gauge-invariant parameters which characterize the physical nature of the dissipation, we find a novel set of gauge-invariant parameters which show that the Galilean invariance is broken in this system. Moreover, a subfamily of the DGE can be obtained after a gauge transformation, which describes a dissipative quantum system with the conserved Galilean invariance. It is interesting that this dissipative quantum system is completely equivalent to a charge-monopole system, in which the Dirac potential is supplied with the nonlinear terms and two cases of the velocity potential. Especially, the two gauge potentials given by Wu and Yang emerge from solving the DGE as two cases in our approach. The results not only present some new physical comprehension of the dissipative quantum system, but also might shed light on the Dirac monopole potential, in the sense that the partition into south and north hemisphere is avoided in our new solutions.

An explicit finite difference scheme is presented for the von Neumann equation for (2+1)D Dirac fermions. It is founded upon a staggered space–time grid which ensures a single-cone energy dispersion and performs the time-derivative in one sweep using a three-step leap-frog procedure. It enables a space–time-resolved numerical treatment of the mixed-state dynamics of Dirac fermions within the effective single-particle density matrix formalism. Energy-momentum dispersion, stability and convergence properties are derived. Elementary numerical tests to demonstrate stability properties use parameters which pertain to topological insulator surface states. A method for the simulation of charge injection from an electric contact is presented and tested numerically. Potential extensions of the scheme to a Dirac–Lindblad equation, real-space–time Green's function formulations, and higher-order finite-difference schemes are discussed.

In this article, we study the existence of solutions for the Dirac system { D ⁢ u = ∂ ⁡ H ∂ ⁡ v ⁢ ( x , u , v ) on ⁢ M , D ⁢ v = ∂ ⁡ H ∂ ⁡ u ⁢ ( x , u , v ) on ⁢ M , \left\{\begin{aligned} \displaystyle Du&\displaystyle=\frac{\partial H}{% \partial v}(x,u,v)\quad\text{on }M,\\ \displaystyle Dv&\displaystyle=\frac{\partial H}{\partial u}(x,u,v)\quad\text{% on }M,\end{aligned}\right. where M is an m-dimensional compact Riemannian spin manifold, u , v ∈ C ∞ ⁢ ( M , Σ ⁢ M ) {u,v\in C^{\infty}(M,\Sigma M)} are spinors, D is the Dirac operator on M, and the fiber preserving map H : Σ ⁢ M ⊕ Σ ⁢ M → ℝ {H:\Sigma M\oplus\Sigma M\rightarrow\mathbb{R}} is a real-valued superquadratic function of class C 1 {C^{1}} with subcritical growth rates. Two existence results of nontrivial solutions are obtained via Galerkin-type approximations and linking arguments.

The Dirac equation is considered in five-dimensional spaces with signatures (2,3), (4,1) and (0,5). The algebraic spinor formalism with the application of fermionic variables is used as the basis of real Clifford algebras and the module over this algebra. It is shown that solutions to the five-dimensional Dirac equation in spaces with signatures (2,3) and (4,1) can be expanded over solutions with zero value of the fifth component of the generalized momentum, and the equation is equivalent to an equation in four-dimensional spacetime.

We study the conformable fractional (CF) Dirac system with separated boundary conditions on an arbitrary time scale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{T}$\end{document}T. Then we extend some basic spectral properties of the classical Dirac system to the CF case. Eventually, some asymptotic estimates for the eigenfunction of the CF Dirac eigenvalue problem are obtained on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{T} $\end{document}T. So, we provide a constructive procedure for the solution of this problem. These results are important steps to consolidate the link between fractional calculus and time scale calculus in spectral theory.

In this note, we get estimates on the eigenvalues of the Dirac operator on the manifold admitting a nontrivial parallel one-form, in terms of the eigenvalues of the Laplace–Beltrami operator and the scalar curvature. These estimates are sharp, in the sense that, for the first eigenvalue, they reduce to the result of Alexandrov et al. (1998).

We start from a noncompact Lie algebra isomorphic to the Dirac algebra and relate this Lie algebra in a brief review to low-energy hadron physics described by the compact group SU(4). This step permits an overall physical identification of the operator actions. Then we discuss the geometrical origin of this noncompact Lie algebra and “reduce” the geometry in order to introduce in each of these steps coordinate definitions which can be related to an algebraic representation in terms of the spontaneous symmetry breakdown along the Lie algebra chain su*(4) → usp(4) → su(2) × u(1). Standard techniques of Lie algebra decomposition(s) as well as the (physical) operator identification give rise to interesting physical aspects and lead to a rank-1 Riemannian space which provides an analytic representation and leads to a 5-dimensional hyperbolic space H 5 with SO(5, 1) isometries. The action of the (compact) symplectic group decomposes this (globally) hyperbolic space into H 2 ⊕ H 3 with SO(2, 1) and SO(3, 1) isometries, respectively, which we relate to electromagnetic (dynamically broken SU(2) isospin) and Lorentz transformations. Last not least, we attribute this symmetry pattern to the algebraic representation of a projective geometry over the division algebra H and subsequent coordinate restrictions.

The problem of a Dirac particle moving in a deformed Hulthen potential is solved in the framework of the path integral formalism. With the help of the Biedenharn transformation, the construction of a closed form for the Green’s function of the second-order Dirac equation is done by using a proper approximation to the centrifugal term and the Green’s function of the linear Dirac equation is calculated. The energy spectrum for the bound states is obtained from the poles of the Green’s function. A Dirac particle in the standard Hulthen potential (q = 1) and a Dirac hydrogen-like ion (q = 1 and a → ∞) are considered as particular cases.

Majorana spinors are constructed in terms of the multivectors of relativistic Cl1,3 algebra. Such spinors are found to be multiplied by primitive idempotents which drastically change spinor properties. Running electronic waves are used to solve the real Dirac–Majorana equation transformed to Cl1,3 algebra. From the analysis of the solution it is concluded that free Majorana particles do not exist, because relativistic Cl1,3 algebra requires the massive Majorana particle to move with light velocity.

Noncommutative integration of the Dirac equation on Lie groups with a right-invariant metric and an invariant electromagnetic field tensor is considered. An electromagnetic field, admitting a noncommutative reduction of the Dirac equation, but not admitting either diagonalization of the squared Dirac equation or separation of variables, is found.

Made available in DSpace on 2018-12-11T16:46:11Z (GMT). No. of bitstreams: 0 Previous issue date: 2017-04-01 Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) We establish dynamical localization for random Dirac operators on the d-dimensional lattice, with d∈ {1 , 2 , 3 } , in the three usual regimes: large disorder, band edge and 1D. These operators are discrete versions of the continuous Dirac operators and consist in the sum of a discrete free Dirac operator with a random potential. The potential is a diagonal matrix formed by different scalar potentials, which are sequences of independent and identically distributed random variables according to an absolutely continuous probability measure with bounded density and of compact support. We prove the exponential decay of fractional moments of the Green function for such models in each of the above regimes, i.e., (j) throughout the spectrum at larger disorder, (jj) for energies near the band edges at arbitrary disorder and (jjj) in dimension one, for all energies in the spectrum and arbitrary disorder. Dynamical localization in theses regimes follows from the fractional moments method. The result in the one-dimensional regime contrast with one that was previously obtained for 1D Dirac model with Bernoulli potential. Departamento de Matemática UFSC Departamento de Matemática UFSCar Departamento de Matemática UFMG UNESP UNESP CNPq: 157873/2015-3 CNPq: 441004/2014-8

The problem of the electron spectrum in a quasi-two-dimensional space (such as interfaces, heterostructures, and surfaces) is set up and analytically solved based on the fundamental Dirac equation. It is demonstrated that using the unitary transformation method and the method of spin invariants to solve the Dirac equation leads to identical results. The eigen bispinors of the Dirac equation are found, and the way in which their diversity arises due to the arbitrariness of the spin-quantization axis direction is demonstrated. The features of electron behavior in a parabolic quantum well are considered.

In the paper, we consider the two-dimensional oscillator within the framework of the noncommutative q-deformed Dirac oscillator. We discuss the eigenvalues of q-deformed Dirac oscillator in (1+2)-dimensions in the presence of a magnetic field in a Noncommutative phase space formalism.

In this work we study the existence of ground-state solutions of Dirac equations with potentials which are allowed to vanish at infinity. The approach is based on minimization of the energy functional over a generalized Nehari set. Some conditions on the potentials are given in order to overcome the lack of compactness., Accepted for publication in Journal of Differential Equations

This paper continues the author’s work [1] [2], where a novel framework of the matter-induced physical geometry was built and an intrinsic nonlinearity of the Dirac equation was discovered. The previous analysis of solitary waveforms’ properties [2] is extended to the four-component Dirac field. It is found that the internal spherical symmetry of the Dirac waveforms is broken to the axial one. The nonlinear Dirac equation is solved and the localized configurations are found analytically. A strict proof that the proper time slowdown is the major mechanism of autolocalization is presented. The previous qualitative conjecture regarding stability or instability of the two types of the waveforms and the origin of cosmological charge asymmetry is supported by detailed analysis. A solution of the problem of mapping between the matter-induced geometry of autolocalized waveforms and the geometry of an ambient Minkowski space is proposed. These results resolve the longstanding puzzle of how the physical Dirac field of real matter becomes a finite-sized particle.

In this paper, we show the existence of solutions for the coupled Dirac system \begin{document}$\left\{ \begin{aligned}Du=\frac{\partial H}{\partial v}(x,u,v)\hspace{4mm} {\rm on}\hspace{2mm}M,\\Dv=\frac{\partial H}{\partial u}(x,u,v)\hspace{4mm} {\rm on}\hspace{2mm}M,\end{aligned} \right.$\end{document} where $M$ is an $n$-dimensional compact Riemannian spin manifold, $D$ is the Dirac operator on $M$, and $H:\Sigma M\oplus \Sigma M\to \mathbb{R}$ is a real valued superquadratic function of class $C^1$ in the fiber direction with subcritical growth rates. Our proof relies on a generalized linking theorem applied to a strongly indefinite functional on a product space of suitable fractional Sobolev spaces. Furthermore, we consider the $\mathbb{Z}_2$-invariant $H$ that includes a nonlinearity of the form \begin{document}$H(x,u,v)=f(x)\frac{|u|^{p+1}}{p+1}+g(x)\frac{|v|^{q+1}}{q+1},$\end{document} where $f(x)$ and $g(x)$ are strictly positive continuous functions on $M$ and $p, q>1$ satisfy \begin{document}$\frac{1}{p+1}+\frac{1}{q+1}>\frac{n-1}{n}.$\end{document} In this case we obtain infinitely many solutions of the coupled Dirac system by using a generalized fountain theorem.

We recall the algebra of endomorphisms on the so-called radial algebra of abstract vector variables which generalizes both polynomial and Clifford algebras, and the defining building blocks of a function theory in this abstract framework. These building blocks are given by the abstract versions of the Dirac operator and the directional derivatives. Together with other fundamental endomorphisms, they lead to an algebraic structure which can be considered as the abstract equivalent of orthogonal Clifford analysis. Following a similar approach, we present the axiomatic definitions of the Hermitian radial algebra, leading to the abstract equivalent of Hermitian Clifford analysis.

In this paper, we give a brute-force proof of the Kastler–Kalau–Walze type theorem for 7-dimensional manifolds with boundary about Dirac operators with torsion.

The algebra of first-order symmetry operators of the Dirac equation on four-dimensional Lie groups with right-invariant metric is investigated. It is shown that the algebra of symmetry operators is in general not a Lie algebra. Noncommutative reduction mediated by spin symmetry operators is investigated. For the Dirac equation on the Lie group SO(2,1) a parametric family of particular solutions obtained by the method of noncommutative integration over a subalgebra containing a spin symmetry operator is constructed.

We proposed a new solvable novel pseudo-Coulomb ring-shaped potential and investigate its pseudospin symmetry by solving the Dirac equation under the condition of an equal mixing of scalar and vector potentials using the factorization techniques. We determine in closed form the energy eigenvalues and eigenfunctions of the bound states of the Dirac equation analytically. We also discuss the non-relativistic limits.

The spectrum of the Dirac operator is investigated on a Riemannian manifold one of whose end is asymptotically flat in some sense. It is proved that there is no non-zero eigenvalue and the essential spectrum is equal to R .

The general topic of this thesis is how to define and compute the index of discretised “lattice” versions of the Dirac operator coupled to a topologically nontrivial gauge field. The famous Atiyah-Singer Index Theorem, applied to the usual continuum Dirac operator, says that its index is equal to the topological charge of the gauge field. This fact plays an important role in theoretical particle physics. However, in the physics context one often solves problems by discretizing the physical theory and simulating it on a computer. This raises the question of whether the Index Theorem (and its physical implications) can be maintained in the discretized lattice setting. This is a subtle mathematical issue since the usual definition of the index of the Dirac operator breaks down in the discretized setting due to its finite-dimensional nature. This thesis reviews how the index can be extracted in another, indirect, way in the discretized setting, and then goes on to consider the index of a new discretization of the Dirac operator introduced recently by D. Adams. This new lattice Dirac operator has the virtue of being more computationally efficient for use in computer simulations of particle physics theory, but before it can be used with confidence a secure theoretical foundation needs to be established. One part of this is to show that its index correctly reproduces the continuum index in the continuum limit where the lattice spacing goes to zero. This is the main goal and new research contribution of this thesis. It is done here by adapting the methods and techniques used to establish this result previously for the standard discretization of the Wilson-Dirac operator, and involves dealing with new issues specific to the new discretization. DOCTOR OF PHILOSOPHY (SPMS)

Part 1 Semigroups and partial differential equations: different domains induce different heat semigroups on C0(omega) Gaussian estimates for second order elliptic divergence operators on Lipschitz and C4 domains approximate solutions to the abstract Cauchy problem smart structures and super stability on the structure of the critical spectrum an operator-valued transference principle and maximal regularity on vector-valued Lp-spaces on anomalous asymptotics of heat kernels on some classes of differential operators generating analytic semigroups a characterization on the growth bound of a semigroup via Fourier multipliers Laplace transform theory for logarithmic regions exact boundary controllability of Maxwell's condition for exponential dichotomy of parabolic evolution equations edge-degenerate boundary value problems on cones a theorem on products of noncommuting sectorial operators the spectral radius, hyperbolic operators and Lyapunov's theorem a new approach to maximal Lp-regularity. Part 2 Nonlinear evolution equations: the instantaneous limits of a reaction-diffusion system a semigroup approach to dispersive waves regularity properties of solutions of fractional evolution equations infinite horizon Riccati operators in nonreflexive spaces a hyperbolic variant of Simon's convergence theorem solution of a quasilinear parabolic-elliptic boundary value problem. Part 3 Physical and life sciences: singular cluster interactions in few-body problems Feynman and Wiener path integrals spectral characterization of mixing evolutions in classical statistical physics on stochastic Schrodinger equation as a Dirac boundary-value problem, and an inductive stochastic limit a maximum principle for fully nonlinear parabolic equations with time degeneracy Dirac algebra and Foldy-Wouthuysen transform. Part 4 Stochastic evolution equation.

I examine the groups which underly classical mechanics, non-relativistic quantum mechanics, special relativity, relativistic quantum mechanics, quantum electrodynamics, quantum flavourdynamics, quantum chromodynamics, and general relativity. This examination includes the rotations SO(2) and SO(3), the Pauli algebra, the Lorentz transformations, the Dirac algebra, and the U(1), SU(2), and SU(3) gauge transformations. I argue that general relativity must be generalized to Einstein-Cartan theory, so that Dirac spinors can be described within the framework of gravitation theory.

In this note, we apply the saddle point reduction to a class of nonlinear Dirac equation with a potential on a compact spin manifold. Two results are obtained, including a multiple theorem.