A Symmetry Connection Between Hyperbolic and Parabolic Equations

Abstract

We give ansatzes obtained from Lie symmetries of some hyperbolic equations which reduce these equations to the heat or Schrödinger equations. This enables us to construct new solutions of the hyperbolic equations using the Lie and conditional symmetries of the parabolic equations. Moreover, we note that any equation related to such a hyperbolic equation (for example the Dirac equation) also has solutions constructed from the heat and Schrödinger equations.     

Space-time structure of weak and electromagnetic interactions

The generator of electromagnetic gauge transformations in the Dirac equation has a unique geometric interpretation and a unique extension to the generators of the gauge group SU(2) × U(1) for the Weinberg-Salam theory of weak and electromagnetic interactions. It follows that internal symmetries of the weak interactions can be interpreted as space-time symmetries of spinor fields in the Dirac algebra. The possibilities for interpreting strong interaction symmetries in a similar way are highly restricted.     

Conservation Laws and Non-Lie Symmetries for Linear PDEs

Abstract

We introduce a method to construct conservation laws for a large class of linear partial differential equations. In contrast to the classical result of Noether, the conserved currents are generated by any symmetry of the operator, including those of the non-Lie type. An explicit example is made of the Dirac equation were we use our construction to find a class of conservation laws associated with a 64 dimensional Lie algebra of discrete symmetries that includes CPT.     

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.     

Bosonic symmetries of the Dirac equation

We have found on the basis of the symmetry analysis of the standard Dirac equation with nonzero mass the new physically meaningful features of this equation. The new bosonic symmetries of the Dirac equation in both the Foldy-Wouthuysen and the Pauli-Dirac representations are found, among which (together with the 32-dimensional pure matrix algebra of invariance) the new spin s=(1,0) multiplet Poincaré symmetry is proved. In order to carry out the corresponding proofs a 64-dimensional extended real Clifford-Dirac algebra is put into consideration.     

Discrete Symmetries and Generalized Fields of Dyons

We have studied the different symmetric properties of the generalized Maxwell’s–Dirac equation along with their quantum properties. Applying the parity (℘), time reversal ( T\mathcal{T} ), charge conjugation (C\mathcal{C}) and their combined effect like parity time reversal (PT\mathcal{PT}), charge conjugation and parity (CP\mathcal{CP}) and CPT\mathcal{CP}T transformations to various equations of generalized fields of dyons, it is shown that the corresponding dynamical quantities and equations of dyons are invariant under these discrete symmetries.     

Angular symmetry breaking induced by electromagnetic field

It is well known that velocities do not commute in the presence of an electromagnetic field. This property implies that angular algebra symmetries, such as the sO(3) and Lorentz algebra symmetries, are broken. To restore these angular symmetries we show the necessity of adding the Poincaré momentum M to the simple angular momentum L. These restorations performed successively in a flat space and in a curved space lead in each case to the generation of a Dirac magnetic monopole. In the particular case of the Lorentz algebra we consider an application of our theory to gravitoelectromagnetism. In this last case we establish a qualitative relation giving the mass spectrum for dyons.     

The nonlinear Dirac equation in Bose-Einstein condensates: Foundation and symmetries

We show that Bose-Einstein condensates in a honeycomb optical lattice can be described by a nonlinear Dirac equation in the long wavelength, mean field limit. Unlike nonlinear Dirac equations posited by particle theorists, which are designed to preserve the principle of relativity, i.e., Poincaré covariance, the nonlinear Dirac equation for Bose-Einstein condensates breaks this symmetry. We present a rigorous derivation of the nonlinear Dirac equation from first principles. We provide a thorough discussion of all symmetries broken and maintained.     

On Remarkable Reductions of the Nonlinear Dirac Equation

Abstract

The three ansatzes are constructed for the nonlinear Dirac equation.     

Relativistic Spin and Pseudospin Symmetries of Inversely Quadratic Yukawa-like plus Mobius Square Potentials Including a Coulomb-like Tensor Interaction

The Dirac equation for the combined Mobius square and inversely quadratic Yukawa potentials including a Coulomb-like interaction term has been investigated in the presence of spin and pseudospin symmetries with arbitrary spin-orbit quantum number κ .We have obtained the explicit energy eigenvalues and the corresponding eigenfunctions by the framework of Nikiforov-Uvarov method.     

Symmetrieeigenschaften achtkomponentiger Spinorfelder im Gitterraum

Starting from the four component Dirac equation for free particles without mass W.Heisenberg und W.Pauli have shown that the interaction term is uniquely defined, if one requires that all symmetries of free particles are preserved. Here we obtain similar results if we start from the eight component Dirac equation for free particles without mass:

1. The symmetry group of the eight component Dirac equation for free particles without mass has 16 parameters. It is isomorph to the direct product of the SU 4 and a one-parametric group: SU 4× (1).
2. The interaction operator is uniquely defined if one requires to preserve as many symmetries as possible of those given in (1).
3. But some of the symmetries in (1) are necessarily broken, in particular that of SU 3. The symmetry of the interaction operator is given by SO 4× (1)× (1).

These results mean:

1. The Heisenberg theory is uniquely defined, only if one assumes that the free particle part of the equation is well known.
2. The theory can be changed without modifying the fundamental idea ofHeisenberg andPauli to deduce an uniquely defined interaction operator if one starts with a modified free particle part.
3. A special kind of modification of the free particle part leads essentially to the SU 4-symmetry including that of SU 3, which is necessarily broken by the interaction term.
4. The question arises if this break of the SU 3-symmetry has something to do with the real break. This question is not yet touched in this paper.

     

How to Find Discrete Contact Symmetries

Abstract

This paper describes a new algorithm for determining all discrete contact symmetries of any differential equation whose Lie contact symmetries are known. The method is constructive and is easy to use. It is based upon the observation that the adjoint action of any contact symmetry is an automorphism of the Lie algebra of generators of Lie contact symmetries. Consequently, all contact symmetries satisfy various compatibility conditions. These conditions enable the discrete symmetries to be found systematically, with little effort.     

Bound State Solutions of the Dirac Equation for the Eckart Potential with Coulomb-Like Yukawa-Like Tensor Interactions

In this paper, we present the approximate bound state solutions of the Dirac equation within the framework of spin and pseudospin symmetries for Eckart potential for arbitrary κ—state using Nikiforov–Uvarov method. The tensor interactions of Coulomb-like and Yukawa-like form are considered and the effects of these tensors and the degeneracy removing role are discussed in detail. Numerical results and figures to show the effect of the tensor interactions are also reported.     

Similarity Reductions for a Nonlinear Diffusion Equation

Abstract

Similarity reductions and new exact solutions are obtained for a nonlinear diffusion equation. These are obtained by using the classical symmetry group and reducing the partial differential equation to various ordinary differential equations. For the equations so obtained, first integrals are deduced which consequently give rise to explicit solutions. Potential symmetries, which are realized as local symmetries of a related auxiliary system, are obtained. For some special nonlinearities new symmetry reductions and exact solutions are derived by using the nonclassical method.     

The Construction of Dirac Wave Packets for a Fermionic Particle Non-Minimally Coupling with an External Magnetic Field

We shall proceed with the construction of normalizable Dirac wave packets for fermionic particles (neutrinos) with dynamics governed by a “modified” Dirac equation with a non-minimal coupling with an external magnetic field. We are not only interested on the analytic solutions of the “modified” Dirac wave equation but also on the construction of Dirac wave packets which can be used for describing the dynamics of some observable physical quantities which are relevant in the context of the quantum oscillation phenomena. To conclude, we discuss qualitatively the applicability of this formal construction in the treatment of chiral (and flavor) oscillations in the theoretical context of neutrino physics. PACS numbers: 02.30.Cj, 03.65.Pm     

Sudden death of a symmetry

Spin polarization symmetries of Dirac particles in the presence of (1 ±β)V(r) potentials are discussed. The effects of a symmetry breaking term εV(r) are explored and applied to proton and anti-proton scattering on nuclei in the framework of Dirac optical models.     

Relativistic Symmetries in the Hulthén-like Potential and Tensor Interaction

In the presence of spin and pseudospin (p-spin) symmetries, the approximate analytical bound states of the Dirac equation for Hulthén-like potential including a Coulomb-like tensor interaction are obtained with any arbitrary spin–orbit coupling number κ using the Pekeris approximation. The generalized parametric Nikiforov–Uvarov (NU) method is used to obtain the energy eigenvalues and the corresponding wave functions in their closed forms. We show that tensor interaction removes degeneracies between spin and p-spin doublets. Some numerical results are also given.     

Dirac Equation with a New Tensor Interaction under Spin and Pseudospin Symmetries

The approximate analytical solutions of the Dirac equation under spin and pseudospin symmetries are examined using a suitable approximation scheme in the framework of parametric Nikiforov-Uvarov method. Because a tensor interaction in the Dirac equation removes the energy degeneracy in the spin and pseudospin doublets that leads to atomic stability, we study the Dirac equation with a Hellmann-like tensor potential newly proposed in this study.The newly proposed tensor potential removes the degeneracy from both the spin symmetry and pseudospin symmetry completely. The proposed tensor potential seems better than the Coulomb and Yukawa-like tensor potentials.     

The Duffin-Kemmer-Petiau equation in a κ-Minkowski space-time

We study a Duffin-Kemmer-Petiau (DKP) equation built in a κ-Minkowski space-time using Dirac derivatives, and analyse its consequences. We show that the κ-deformation preserves parity and time reversal symmetries while it breaks that of the charge conjugation. Then, we investigate effects of this deformation on the energy spectrum of the spin-one DKP oscillator using a perturbation method. These effects are calculated for any value of the total angular momentum number in the case of natural-parity states and for vanishing angular momentum in the case of unnatural-parity states.     

Dirac Equation with a New Tensor Interaction under Spin and Pseudospin Symmetries

The approximate analytical solutions of the Dirac equation under spin and pseudospin symmetries are examined using a suitable approximation scheme in the framework of parametric Nikiforov-Uvarov method. Because a tensor interaction in the Dirac equation removes the energy degeneracy in the spin and pseudospin doublets that leads to atomic stability, we study the Dirac equation with a Hellmann-like tensor potential newly proposed in this study. The newly proposed tensor potential removes the degeneracy from both the spin symmetry and pseudospin symmetry completely. The proposed tensor potential seems better than the Coulomb and Yukawa-like tensor potentials.