Symmetry operators for Dirac's equation on two-dimensional spin manifolds (original) (raw)

Non-Lie and discrete symmetries of the Dirac equation

Journal of Nonlinear Mathematical Physics

New algebras of symmetries of the Dirac equation are presented, which are formed by linear and antilinear first-order differential operators. These symmetries are applied to decouple the Dirac equation for a charged particle interacting with an external field.

General Spin Dirac Equation (II)

In an earlier reading [1], we did demonstrate that one can write down a general spin Dirac equation by modifying the usual Einstein energy-momentum equation via the insertion of the quantity " s " which is identified with the spin of the particle. That is to say, a Dirac equation that describes a particle of spin 1 2 s  S σ where is the normalised Planck constant, are the Pauli matrices and

On a General Spin Dirac Equation

2009

In its bare and natural form, the Dirac Equation describes only spin-1/2 particles. The main purpose of this reading is to make a valid and justified mathematical modification to the Dirac Equation so that it describes any spin particle. We show that this mathematical modification is consistent with the Special Theory of Relativity (STR). We believe that the fact that this modification is consistent with the STR gives the present effort some physical justification that warrants further investigations. From the vantage point of unity, simplicity and beauty, it is natural to wonder why should there exist different equations to describe particles of different spins? For example, the Klein-Gordon equation describes spin-0 particles, while the Dirac Equation describes spin-1/2, and the Rarita-Schwinger Equation describes spin-3/2. Does it mean we have to look for another equation to describe spin-2 particles, and then spin-5/2 particles etc? This does not look beautiful, simple, or at the very least suggest a Unification of the Natural Laws. Beauty of a theory is not a physical principle but, one thing is clear to the searching mind-i.e., a theory that possesses beauty, appeals to the mind, and is (posteriori) bound to have something to do with physical reality if it naturally submits itself to the test of experience. The effort of the present reading is to make the attempt to find this equation.

General Spin Dirac Equation

A complete set of symmetry operators for the Dirac equation

Ukrainian Mathematical Journal, 1991

A complete set of symmetry operators of arbitrary finite order admitted by the Dirac equation is found. The algebraic structure of this set is investigated and subsets of symmetry operators that form bases of Lie algebras and superalgebras are isolated.

Spin operator in the Dirac theory

Physical Review A, 2013

We find all spin operators for a Dirac particle satisfying the following very general conditions: (i) spin does not convert positive (negative) energy states into negative (positive) energy states, (ii) spin is a pseudo-vector, and (iii) eigenvalues of the projection of a spin operator on an arbitrary direction are independent of this direction (isotropy condition). We show that there are four such operators and all of them fulfill the standard su(2) Lie algebra commutation relations. Nevertheless, only one of them has a proper non-relativistic limit and acts in the same way on negative and positive energy states. We show also that this operator is equivalent to the Newton-Wigner spin operator and Foldy-Wouthuysen mean-spin operator. We also discuss another operators proposed in the literature.

Symmetries and supersymmetries of the Dirac operators in curved spacetimes

2004

It is shown that the main geometrical objects involved in all the symmetries or supersymmetries of the Dirac operators in curved manifolds of arbitrary dimensions are the Killing vectors and the Killing-Yano tensors of any ranks. The general theory of external symmetry transformations associated to the usual isometries is presented, pointing out that these leave the standard Dirac equation invariant providing the correct spin parts of the group generators. Furthermore, one analyzes the new type of symmetries generated by the covariantly constant Killing-Yano tensors that realize certain square roots of the metric tensor. Such a Killing-Yano tensor produces simultaneously a Dirac-type operator and the generator of a one-parameter Lie group connecting this operator with the standard Dirac one. In this way the Dirac operators are related among themselves through continuous transformations associated to specific discrete ones. It is shown that the groups of this continuous symmetry can ...

General spin and pseudospin symmetries of the Dirac equation

Physical Review A, 2015

In the 70's Smith and Tassie, and Bell and Ruegg independently found SU(2) symmetries of the Dirac equation with scalar and vector potentials. These symmetries, known as pseudospin and spin symmetries, have been extensively researched and applied to several physical systems. Twenty years after, in 1997, the pseudospin symmetry has been revealed by Ginocchio as a relativistic symmetry of the atomic nuclei when it is described by relativistic mean field hadronic models. The main feature of these symmetries is the suppression of the spin-orbit coupling either in the upper or lower components of the Dirac spinor, thereby turning the respective second-order equations into Schrödinger-like equations, i.e, without a matrix structure. In this paper we propose a generalization of these SU(2) symmetries for potentials in the Dirac equation with several Lorentz structures, which also allow for the suppression of the matrix structure of second-order equation equation of either the upper or lower components of the Dirac spinor. We derive the general properties of those potentials and list some possible candidates, which include the usual spin-pseudospin potentials, and also 2and 1-dimensional potentials. An application for a particular physical system in two dimensions, electrons in graphene, is suggested.

A new way to interpret the Dirac Equation in a non-Riemannian Manifold

The idea of internal mass terms introduced in ref. [1], is shown not to be an appropriate hypothesis when it is placed in connection with the components of the generalized (matrix) vierbeins being proportional to the Riemannian (gravitational) vierbeins. It would result in an undesirable canceling of the Electromagnetic and the Yang-Mills components in the generalized metric. Another hypothesis is introduced where the wave function ψ, is Taylor expanded in terms of a small parameter p.

A concept of Dirac-type tensor equations

2002

Considering a four dimensional parallelisable manifold, we develop a concept of Dirac-type tensor equations with wave functions that belong to left ideals of the set of nonhomogeneous complex valued differential forms.

The symmetries of the Dirac--Pauli equation in two and three dimensions

2004

We calculate all symmetries of the Dirac-Pauli equation in twodimensional and three-dimensional Euclidean space. Further, we use our results for an investigation of the issue of zero mode degeneracy. We construct explicitly a class of multiple zero modes with their gauge potentials. 1

Extra Dirac equations

Il Nuovo Cimento B, 1996

Summary Surprising symmetries in the (j, 0) ⊕ (0,j) Lorentz-group representation space are analysed. The aim is to draw the reader’s attention to the possibility of describing the particle world on the grounds of the Dirac «doubles». Several tune points of the variational principle for this kind of equations are briefly discussed.

On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces

Journal of Geometry and Physics, 1996

The eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces of arbitrary dimension are computed by separating variables in geodesic polar coordinates. These eigenfunctions are then used to derive the heat kernel of the iterated Dirac operator on these spaces. They are then studied as cross sections of homogeneous vector bundles, and a group-theoretic derivation of the spinor spherical functions and heat kernel is given based on Harish-Chandra's formula for the radial part of the Casimir operator.

A Clifford Bundle Approach to the Wave Equation of a Spin 1/2 Fermion in the de Sitter Manifold

Advances in Applied Clifford Algebras, 2015

In this paper we give a Clifford bundle motivated approach to the wave equation of a free spin 1/2 fermion in the de Sitter manifold, a brane with topology M = S0(4, 1)/S0(3, 1) living in the bulk spacetime R 4,1 = (M = R 5 ,g) and equipped with a metric field g := −i * g with i : M →M being the inclusion map. To obtain the analog of Dirac equation in Minkowski spacetime in the structureM we appropriately factorize the two Casimir invariants C1 and C2 of the Lie algebra of the de Sitter group using the constraint given in the linearization of C2 as input to linearize C1. In this way we obtain an equation that we called DHESS1, which in previous studies by other authors was simply postulated..Next we derive a wave equation (called DHESS2) for a free spin 1/2 fermion in the de Sitter manifold using a heuristic argument which is an obvious generalization of a heuristic argument (described in detail in Appendix D) permitting a derivation of the Dirac equation in Minkowski spacetime and which shows that such famous equation express nothing more than the fact that the momentum of a free particle is a constant vector field over timelike integral curves of a given velocity field. It is a remarkable fact that DHESS1 and DHESS2 coincide. One of the main ingredients in our paper is the use of the concept of Dirac-Hestenes spinor fields. Appendices B and C recall this concept and its relation with covariant Dirac spinor fields usually used by physicists.