Dirac Matrices Research Articles

Inverse problems for self-adjoint Dirac systems: explicit solutions and stability of the procedure

A procedure to recover explicitly self-adjoint matrix Dirac systems on semi-axis (with both discrete and continuous components of spectrum) from rational Weyl functions is considered. Its stability is proved. GBDT version of Baecklund-Darboux transformation and various important results on Riccati equations are used for this purpose.

Link between the relativistic canonical quantum mechanics of arbitrary spin and the corresponding field theory

The new relativistic equations of motion for the particles with arbitrary spin and nonzero mass have been introduced. The axiomatic level description of the relativistic canonical quantum mechanics of the arbitrary mass and spin has been given. The 64-dimensional ClR(0,6) algebra in terms of Dirac gamma matrices has been suggested. The link between the relativistic canonical quantum mechanics of the arbitrary spin and the covariant local field theory has been found. Different methods of the Dirac equation derivation have been reviewed. The manifestly covariant field equations for an arbitrary spin that follow from the quantum mechanical equations have been considered. The covariant local field theory equations for spin s = (1,1) particle-antiparticle doublet, spin s = (1,0,1,0) particle antiparticle multiplet, spin s = (3/2,3/2) particle-antiparticle doublet, spin s = (2,2) particle-antiparticle doublet, spin s = (2,0,2,0) particle-antiparticle multiplet and spin s = (2,1,2,1) particle-antiparticle multiplet have been introduced. The Maxwell-like equations for the boson with spin s = 1 and nonzero mass have been introduced as well.

Exact solutions and spacetime singularities in nonlocal gravity

We give here a list of exact classical solutions of a large class of weakly nonlocal theories of gravity, which are unitary and super-renormalizable (or finite) at quantum level. It is explicitly shown that flat and Ricci-flat spacetimes as well as maximally symmetric manifolds are exact solutions of the equation of motion. Therefore, well-known physical spacetimes like Schwarzschild, Kerr, (Anti-) de Sitter serve as solutions for standard matter content. In dimension higher than four we can also have Anti-de Sitter solutions in the presence of positive cosmological constant. We pedagogically show how to obtain these exact solutions. Furthermore, for another version of the theory, written in the Weyl basis, Friedmann-Robertson-Walker (FRW) spacetimes are also exact solutions, when the matter content is given by conformal matter (radiation). We also comment on the presence of singularities and possible resolution of them in finite and conformally invariant theories. "Delocalization" is proposed as a way to solve the black hole singularity problem. In order to solve the problem of cosmological singularities it seems crucial to have a conformally invariant or asymptotically free quantum gravitational theory.

Relativistic nuclear theory—nucleons and mesons: origin, current status, and trends

The nuclear shell model (NSM) is a fundamental model of nuclear theory. At the initial stage, the NSM was developed on the basis of the Schrodinger equation, in particular because it was not clear which Dirac matrices should be associated with various components of the (relativistic) shell model potential. In the early 1970s, the relativistic version of the NSM with meson fields as its main components was developed on the basis of the Dirac equation. The relativistic nuclear shell model (RNSM) includes meson fields with different space–time transformation properties (scalar, four-vector, etc.) that determine the behavior of the (corresponding) meson fields under Lorentz transformations. This fact directly indicates that nuclear theory should be relativistic and based on the Dirac equation.

Neutrino seesaw mechanism with texture zeros

In the context of the Type I seesaw mechanism, we carry out a systematic study of the constraints that result from zeros in both the Dirac and right-handed Majorana neutrino mass matrices. We find that most constraints can be expressed in the standard form with one or two element/cofactor zeros alone, while there are 9 classes of nonstandard constraints. We show that all the constraints are stable under one-loop renormalization group running from the lightest right-handed neutrino mass scale to the electroweak scale. We study the predictions of the nonstandard constraints for the lightest neutrino mass, Dirac CP phase and neutrinoless double beta decay.

Point Particle with Extrinsic Curvature as a Boundary of a Nambu–Goto String: Classical and Quantum Model

It is shown how a string living in a higher dimensional space can be approximated as a point particle with squared extrinsic curvature. We consider a generalized Howe–Tucker action for such a “rigid particle” and consider its classical equations of motion and constraints. We find that the algebra of the Dirac brackets between the dynamical variables associated with velocity and acceleration contains the spin tensor. After quantization, the corresponding operators can be represented by the Dirac matrices, projected onto the hypersurface that is orthogonal to the direction of momentum. A condition for the consistency of such a representation is that the states must satisfy the Dirac equation with a suitable effective mass. The Pauli–Lubanski vector composed with such projected Dirac matrices is equal to the Pauli–Lubanski vector composed with the usual, non projected, Dirac matrices, and its eigenvalues thus correspond to spin one half states.

Compatibility of symmetric quantization with general covariance in the Dirac equation and spin connections

By requiring unambiguous symmetric quantization leading to the Dirac equation in a curved space, we obtain a special representation of the spin connections in terms of the Dirac gamma matrices and their space–time derivatives. We also require that squaring the equation gives the Klein–Gordon equation in a curved space in its canonical from (without spinor components coupling and with no first order derivatives). These requirements result in matrix operator algebra for the Dirac gamma matrices that involves a universal curvature constant. We obtain exact solutions of the Dirac and Klein–Gordon equations in 1+1 space–time for a given static metric.

Package-X: A Mathematica package for the analytic calculation of one-loop integrals

Package-X, a Mathematica package for the analytic computation of one-loop integrals dimensionally regulated near 4 spacetime dimensions is described. Package-X computes arbitrarily high rank tensor integrals with up to three propagators, and gives compact expressions of UV divergent, IR divergent, and finite parts for any kinematic configuration involving real-valued external invariants and internal masses. Output expressions can be readily evaluated numerically and manipulated symbolically with built-in Mathematica functions. Emphasis is on evaluation speed, on readability of results, and especially on user-friendliness. Also included is a routine to compute traces of products of Dirac matrices, and a collection of projectors to facilitate the computation of fermion form factors at one-loop. The package is intended to be used both as a research tool and as an educational tool. Program summaryProgram title: Package-XCatalogue identifier: AEXT_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEXT_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 63912No. of bytes in distributed program, including test data, etc.: 1224562Distribution format: tar.gzProgramming language: Mathematica (Wolfram Language).Computer: Any, which supports Mathematica.Operating system: Windows, Mac OS X, Linux (or any system supporting Mathematica 8.0 or higher).RAM: 10 MB, depending on size of computationClassification: 4.4, 11.1.Nature of problem: Analytic calculation of one-loop integrals in relativistic quantum field theory for arbitrarily high-rank tensor integrals and any kinematic configuration of real-valued external invariants and internal masses.Solution method: Passarino–Veltman reduction formula, Denner–Dittmaier reduction formulae, and two new reduction algorithms described in the manuscript.Restrictions: One-loop integrals are limited to those involving no more than three propagator factors.Unusual features: Includes rudimentary routines for tensor algebraic operations and for performing traces over Dirac gamma matrices.Running time: 5 ms to 10 s for integrals typically occurring in practical computations; longer for higher rank tensor integrals.

Photoexcitations of many-electron atoms and ions by twisted light

SynopsisThe photoexcitation of (multi-electron) atoms and ions by twisted light has been explored within the framework of the density matrix theory and Dirac's relativistic equation. It is shown that the population of the excited atoms and, hence, their fluorescence emission are sensitive with regard to the transverse linear momentum and the (projection of the) total angular momentum of the incident light.

On geometry of the (generalized) G2-manifolds

In this paper, we first understand the classical [Formula: see text]-structure and [Formula: see text]-geometry from the viewpoint of spinor, which is a more familiar framework for physicists. Explicit construction of invariant spinor is given via the Dirac gamma matrices. We introduce a notion of multispinor bundle associated with invariant spinor and differential operator on the section of this bundle. Then we study the vector fields satisfy some additional properties on [Formula: see text]-manifold, more precisely, we prove some no-go theorems corresponding to the vector field preserving the associated 4-form on [Formula: see text]-manifold, and we also consider the nowhere-vanishing vector field which induces an integrable complex structure on the vertical direction of tangent bundle. Next we discuss the relation between the variation of metric and that of effective action on the moduli space of integrable [Formula: see text]-structures. In the last section, we deal with the structure operators on generalized [Formula: see text]-manifold after describing the integrability of generalized [Formula: see text]-structure, some identities of structure operators are derived, which are analogues of Kähler-type and Weitzenböck-type identities under the classical case. And finally, we introduce a flow of which a generalized [Formula: see text]-manifold can be realized as the fixed point.

On [formula omitted] in higher-order QCD calculations and the NNLO evolution of the polarized valence distribution

We discuss the prescription for the Dirac matrix γ5 in dimensional regularization used in most second- and third-order QCD calculations of collider cross sections. We provide an alternative implementation of this approach that avoids the use of an explicit form of γ5 and of its (anti-)commutation relations in the most important case of no more than one γ5 in each fermion trace. This treatment is checked by computing the third-order corrections to the structure functions F2 and g1 in charged-current deep-inelastic scattering with axial-vector couplings to the W-bosons. We derive the so far unknown third-order helicity-difference splitting function ΔPns(2)s that contributes to the next-to-next-to-leading order (NNLO) evolution of the polarized valence quark distribution of the nucleon. This function is negligible at momentum fractions x≳0.3 but relevant at x≪1.

Ultrarelativistic decoupling transformation for generalized Dirac equations

The Foldy--Wouthuysen transformation is known to uncover the nonrelativistic limit of a generalized Dirac Hamiltonian, lending an intuitive physical interpretation to the effective operators within Schr\"{o}dinger--Pauli theory. We here discuss the opposite, ultrarelativistic limit which requires the use of a fundamentally different expansion where the leading kinetic term in the Dirac equation is perturbed by the mass of the particle and other interaction (potential) terms, rather than vice versa. The ultrarelativistic decoupling transformation is applied to free Dirac particles (in the Weyl basis) and to high-energy tachyons, which are faster-than-light particles described by a fully Lorentz-covariant equation. The effective gravitational interactions are found. For tachyons, the dominant gravitational interaction term in the high-energy limit is shown to be attractive, and equal to the leading term for subluminal Dirac particles (tardyons) in the high-energy limit.

Microscopic origin ofUA(1)symmetry violation in the high temperature phase of QCD

We investigate the low-lying eigenmodes of the Dirac matrix with the aim to gain more insight into the temperature dependence of the anomalous $U_A(1)$ symmetry. We use the overlap operator to probe dynamical QCD configurations generated with (2+1)-flavors of highly improved staggered quarks. We find no evidence of a gap opening up in the infrared region of the eigenvalue spectrum even at $1.5\,T_c$, $T_c$ being the chiral crossover temperature. Instead, we observe an accumulation of near-zero eigenmodes. We argue that these near-zero eigenmodes are primarily responsible for the anomalous breaking of the axial symmetry still being effective. At $1.5\,T_c$, these near-zero eigenmodes remain localized and their distribution is consistent with the dilute instanton gas picture. At this temperature, the average size of the instantons is $0.223(8)\,\text{fm}$ and their density is $0.147(7)\,\text{fm}^{-4}$.

Spin-base invariance of fermions in arbitrary dimensions

The concept of spin-base invariance is extended to arbitrary integer dimension $d \geq 2$. Explicit formulas for the spin connection as a function of the Dirac matrices are found. We disclose the hidden spin-base invariance of the vielbein formalism and give a detailed motivation for this symmetry from first principles. The common Lorentz symmetric gauge for the vielbein is constructed for the Dirac matrices, even for metrics which are not linearly connected. Under certain criteria, it constitutes the simplest possible gauge, demonstrating why this gauge is so useful.

Time and quantum theory: A history and a prospectus

The historical part of this paper analyzes in detail how ideas and expectations regarding the role of time in quantum theory arose and evolved in the early years of quantum mechanics (from 1925 to 1927). The general theme is that expectations which seemed reasonable from the point of view of matrix mechanics and Dirac׳s q-number formalism became implausible in light of Dirac–Jordan transformation theory, and were dashed by von Neumann׳s Hilbert space formalism which came to replace it. Nonetheless, I will identify two concerns that remain relevant today, and which blunt the force of Hilgevoord׳s (2005) claim that the demand that time feature as an observable arose as the result of a simple conceptual error. First, I advocate the need for event time observables, which provide a temporal probability distribution for the occurrence of a particular event. Second, I claim that Dirac׳s use of the extended phase space to define time and (minus the) energy as conjugates is not subject to ‘Pauli׳s Theorem,’ the result that rules out time observables in von Neumann׳s formalism. I also claim that the need to define these event time observables leads to a novel motivation for considering Dirac׳s extended state space.

Approximation of upper bound for matrix operators on the Fibonacci weighted sequence spaces

Let be a non-negative matrix. Denote by , the infimum of those satisfying the following inequality:where , and is the Fibonacci numbers sequence and is a decreasing, non-negative sequence of real numbers. In this paper, first, the Fibonacci weighted sequence space is introduced. Then, we focus on the evaluation of , where is the Hausdorff matrix operator or the Nörlund matrix operator or the transpose of the Nörlund matrix operator. For the case of Hausdorff matrices, a Hardy-type formula is established as an upper estimate. Also, a general upper estimate is established for the case of Nörlund matrices and their transpose. In particular, we apply our results to Cesàro matrices, Hölder matrices, Euler matrices and Gamma matrices.

The Universal Expression for the Amplitude Square in Quantum Electrodynamics

The universal expression for the amplitude square |u_f M u_i|^2 for any matrix of interaction M is derived. It has obvious covariant form. It allows the avoidance of calculation of products of the Dirac's matrices traces and allows easy calculation of cross-sections of any different processes with polarized and unpolarized particles.

Octonion generalization of Pauli and Dirac matrices

Starting with octonion algebra and its 4 × 4 matrix representation, we have made an attempt to write the extension of Pauli's matrices in terms of division algebra (octonion). The octonion generalization of Pauli's matrices shows the counterpart of Pauli's spin and isospin matrices. In this paper, we also have obtained the relationship between Clifford algebras and the division algebras, i.e. a relation between octonion basis elements with Dirac (gamma), Weyl and Majorana representations. The division algebra structure leads to nice representations of the corresponding Clifford algebras. We have made an attempt to investigate the octonion formulation of Dirac wave equations, conserved current and weak isospin in simple, compact, consistent and manifestly covariant manner.

An Introduction to Geometric Algebra with some Preliminary Thoughts on the Geometric Meaning of Quantum Mechanics

It is still a great riddle to me why Wolfgang Pauli and P.A.M. Dirac had not fully grasped the meaning of their own mathematical constructions. They invented magnificent, fantastic and very important mathematical features of modern physics, but they only delivered half of the interpretations of their own inventions. Of course, Pauli matrices and Dirac matrices represent operators, which Pauli and Dirac discussed in length. But this is only part of the true meaning behind them, as the non-commutative ideas of Grassmann, Clifford, Hamilton and Cartan allow a second, very far reaching interpretation of Pauli and Dirac matrices.An introduction to this alternative interpretation will be discussed. Some applications of this view on Pauli and Dirac matrices are given, e.g. a geometric algebra picture of the plane wave solution of the Maxwell equation, a geometric algebra picture of special relativity, a toy model of SU(3) symmetry, and some only very preliminary thoughts about a possible geometric meaning of quantum mechanics.

On boundary value problems for perturbed Hermitean matrix Dirac equations in a fractal domain

On boundary value problems for perturbed Hermitean matrix Dirac equations in a fractal domain