Mass, energy, and the electron

The two-component solutions of the Dirac equation currently in use are not separately a particle equation or an antiparticle equation. We present a unitary transformation that uncouples the four-component, force-free Dirac equation to yield a two-component spinor equation for the force-free motion of a relativistic particle and a corresponding two-component, time-reversed equation for an antiparticle. The particle-antiparticle nature of the two equations is established by applying to the solutions of these two-component equations criteria analogous to those applied for establishing the four-component particle and antiparticle solutions of the four-component Dirac equation. Wave function solutions of our two-component particle equation describe both a right and a left circularly polarized particle. Interesting characteristics of our solutions include spatial distributions that are confined in extent along directions perpendicular to the motion, without the artifice of wave packets, and an intrinsic chirality (handedness) that replaces the usual definition of chirality for particles without mass. Our solutions demonstrate that both the rest mass and the relativistic increase in mass with velocity of the force-free electron are due to an increase in the rate of Zitterbewegung with velocity. We extend this result to a bound electron, in which case the loss of energy due to binding is shown to decrease the rate of Zitterbewegung.     

The Riemann–Roch theorem and zero-energy solutions of the Dirac equation on the Riemann sphere

In this paper, we revisit the connection between the Riemann–Roch theorem and the zero-energy solutions of the two-dimensional Dirac equation in the presence of a delta-function-like magnetic field. Our main result is the resolution of a paradox—the fact that the Riemann–Roch theorem correctly predicts the number of zero-energy solutions of the Dirac equation despite counting what seem to be functions of the wrong type.     

Chirality and correlations in graphene

Graphene is described at low energy by a massless Dirac equation whose eigenstates have definite chirality. We show that the tendency of Coulomb interactions in lightly doped graphene to favor states with larger net chirality leads to suppressed spin and charge susceptibilities. Our conclusions are based on an evaluation of graphene's exchange and random-phase-approximation correlation energies. The suppression is a consequence of the quasiparticle chirality switch which enhances quasiparticle velocities near the Dirac point.     

Stable Directions for Small Nonlinear Dirac Standing Waves

We prove that for a Dirac operator, with no resonance at thresholds nor eigenvalue at thresholds, the propagator satisfies propagation and dispersive estimates. When this linear operator has only two simple eigenvalues sufficiently close to each other, we study an associated class of nonlinear Dirac equations which have stationary solutions. As an application of our decay estimates, we show that these solutions have stable directions which are tangent to the subspaces associated with the continuous spectrum of the Dirac operator. This result is the analogue, in the Dirac case, of a theorem by Tsai and Yau about the Schrödinger equation. To our knowledge, the present work is the first mathematical study of the stability problem for a nonlinear Dirac equation     

Index theorem and universality properties of the low-lying eigenvalues of improved staggered quarks

We study various improved staggered quark Dirac operators on quenched gluon backgrounds in lattice QCD generated using a Symanzik-improved gluon action. We find a clear separation of the spectrum into would-be zero modes and others. The number of would-be zero modes depends on the topological charge as expected from the index theorem, and their chirality expectation value is large ( approximately 0.7). The remaining modes have low chirality and show clear signs of clustering into quartets and approaching the random matrix theory predictions for all topological charge sectors. We conclude that improvement of the fermionic and gauge actions moves the staggered quarks closer to the continuum limit where they respond correctly to QCD topology.     

Spinor analysis of Yang-Mills theory in the Minkowski space

Self-duality equations for Yang-Mills fields and the Dirac equation with an external (anti-) selfdual gauge field are studied in the Minkowski space by spinorial methods. For the Dirac equations, all (four) possible combinations of the fermion chirality and duality of the external fields are considered.     

Maximal symmetry and mass generation of Dirac fermions and gravitational gauge field theory in six-dimensional spacetime

The relativistic Dirac equation in four-dimensional spacetime reveals a coherent relation between the dimensions of spacetime and the degrees of freedom of fermionic spinors. A massless Dirac fermion generates new symmetries corresponding to chirality spin and charge spin as well as conformal scaling transformations. With the introduction of intrinsic W-parity, a massless Dirac fermion can be treated as a Majorana-type or Weyl-type spinor in a six-dimensional spacetime that reflects the intrinsic quantum numbers of chirality spin. A generalized Dirac equation is obtained in the six-dimensional spacetime with a maximal symmetry. Based on the framework of gravitational quantum field theory proposed in Ref. [1] with the postulate of gauge invariance and coordinate independence, we arrive at a maximally symmetric gravitational gauge field theory for the massless Dirac fermion in six-dimensional spacetime. Such a theory is governed by the local spin gauge symmetry SP(1,5) and the global Poincar′e symmetry P(1,5)= SO(1,5) P~(1,5) as well as the charge spin gauge symmetry SU(2). The theory leads to the prediction of doubly electrically charged bosons. A scalar field and conformal scaling gauge field are introduced to maintain both global and local conformal scaling symmetries. A generalized gravitational Dirac equation for the massless Dirac fermion is derived in the six-dimensional spacetime. The equations of motion for gauge fields are obtained with conserved currents in the presence of gravitational effects. The dynamics of the gauge-type gravifield as a Goldstone-like boson is shown to be governed by a conserved energy-momentum tensor, and its symmetric part provides a generalized Einstein equation of gravity. An alternative geometrical symmetry breaking mechanism for the mass generation of Dirac fermions is demonstrated.     

Dirac operator zero-modes on a torus

We study Dirac operator zero-modes on a torus for gauge background with uniform field strengths. Under the basic translations of the torus coordinates the wave functions are subject to twisted periodic conditions. In suitable torus coordinates the zero-mode wave functions can be related to holomorphic functions of the complex torus coordinates. Half of the twisted boundary conditions for the holomorphic part of the zero-mode wave function can be made periodic or anti-periodic. The remaining half is until coordinate dependent but diagonal. We completely solve the twisted boundary conditions and construct the zero-mode wave functions. The chirality and the degeneracy of the zero-modes are uniquely determined by the gauge background and are consistent with the index theorem.     

On the connection between the solutions to the Dirac and Weyl equations and the corresponding electromagnetic four-potentials

In this work we study in detail the connection between the solutions to the Dirac and Weyl equations and the associated electromagnetic four-potentials.First,it is proven that all solutions to the Weyl equation are degenerate,in the sense that they correspond to an infinite number of electromagnetic four-potentials.As far as the solutions to the Dirac equation are concerned,it is shown that they can be classified into two classes.The elements of the first class correspond to one and only one four-potential,and are called non-degenerate Dirac solutions.On the other hand,the elements of the second class correspond to an infinite number of four-potentials,and are called degenerate Dirac solutions.Further,it is proven that at least two of these fourpotentials are gauge-inequivalent,corresponding to different electromagnetic fields.In order to illustrate this particularly important result we have studied the degenerate solutions to the forcefree Dirac equation and shown that they correspond to massless particles.We have also provided explicit examples regarding solutions to the force-free Weyl equation and the Weyl equation for a constant magnetic field.In all cases we have calculated the infinite number of different electromagnetic fields corresponding to these solutions.Finally,we have discussed potential applications of our results in cosmology,materials science and nanoelectronics.     

The Einstein-Dirac Equation in Robertson-Walker Space-Time Does Not Admit Standard Solutions

The Einstein-Dirac equation is considered in the Robertson-Walker space-time. Solutions of the equation are looked for in the class of standard solutions of the Dirac equation. It is shown that the Einstein-Dirac equation does not have standard solutions for both massive and massless Dirac field. Also superpositions of massive standard solutions are not solutions of the Einstein-Dirac equation. The result, that is briefly commented, is coherent and complementary to other existing results.     

Modified Dirac equation with Lorentz invariance violation and its solutions for particles in an external magnetic field

The second-order modified Dirac equation leading to the modified dispersion relation due to the Lorentz invariance violation corrections is suggested. The equation is formulated in the 16-component first-order form. I have obtained the projection matrix extracting solutions of the equation with definite spin projections which can be considered as the density matrix for pure spin states. Exact solutions of the equation are found for particles in the external constant and uniform magnetic field. The synchrotron radiation radius within the novel modified Dirac equation is estimated.     

New exact solutions of the dirac equation. XII

The search for exact solutions of the Dirac equation begun in [1] is continued. We find three new types of external electromagnetic fields where the Dirac equation, Klein-Gordon equation, and classical Lorentz equation can be solved exactly. We find fields for which explicit solutions to the Klein-Gordon equation can be found but for which explicit solutions of the Dirac equation cannot.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 81–86, January, 1985.     

Spin and Pseudospin Symmetries of Dirac Equation and the Yukawa Potential as the Tensor Interaction

Considering of a tensor interaction in Dirac equation removes the degeneracy in spin and pseudospin doublets and consequently leads to results consistent with the experimental data. Here, instead of the commonly used Coulomb or linear terms, we investigate a tensor interaction of Yukawa form. We obtain arbitrary state solutions of Dirac equation under vector, scalar and tensor Yukawa potentials via a physical approximation and the Nikiforov-Uvarov methodology. The solutions are discussed in detail.     

Spin and Pseudospin Symmetries of Dirac Equation and the Yukawa Potential as the Tensor Interaction

Considering of a tensor interaction in Dirac equation removes the degeneracy in spin and pseudospin doublets and consequently leads to results consistent with the experimental data. Here, instead of the commonly used Coulomb or linear terms, we investigate a tensor interaction of Yukawa form. We obtain arbitrary state solutions of Dirac equation under vector, scalar and tensor Yukawa potentials via a physical approximation and the Nikiforov-Uvarov methodology. The solutions are discussed in detail.     

An orbifold relative index theorem

In this paper we prove a relative index theorem for pairs of generalized Dirac operators on orbifolds which are the same at infinity. This generalizes to orbifolds a celebrated theorem of Gromov and Lawson.     

A General Method for Obtaining Degenerate Solutions to the Dirac and Weyl Equations and a Discussion on the Experimental Detection of Degenerate States

In this work, a general method is described for obtaining degenerate solutions of the Dirac equation, corresponding to an infinite number of electromagnetic 4-potentials and fields, which are explicitly calculated. More specifically, using four arbitrary real functions, one can automatically construct a spinor that satisfies the Dirac equation for an infinite number of electromagnetic 4-potentials, defined by those functions. An interesting characteristic of these solutions is that, in the case of Dirac particles with nonzero mass, the degenerate spinors should be localized, both in space and time. The method is also extended to the cases of massless Dirac and Weyl particles, where the localization of the spinors is no longer required. Finally, two experimental methods are proposed for detecting the presence of degenerate states.     

Second Quantization of the Dirac Field: Normal Modes in the Robertson–Walker Space-Time

The quantization of the Dirac field in thecontext of the Robertson–Walker spacetime isreconsidered in some of its constitutive elements. Theparticular solutions of the Dirac equation previouslydetermined are used to construct the normal mode solutionsin the case of flat, closed, and open space-time. Theprocedure is based on a general standard definition ofinner product between solutions of the Dirac equation that is applied by making use of anintegral property of the separated time equation. Theopen-space case requires the recurrence relations offunctions associated to solutions of the Diracequation.     

Noncommutative Integration of the Dirac Equation in a Flat Space and in the de Sitter Space

Noncommutative integration of the Dirac massive and massless equations is performed in a four-dimensional flat space and in the de Sitter space of arbitrary signature. A new class of rigorous solutions of the Dirac equation is constructed in these spaces. The properties of the solutions obtained are examined.