Quaternionic Lorentz Group and Dirac Equation

We formulate Lorentz group representations in which ordinary complex numbers are replaced by linear functions of real quaternions and introduce dotted and undotted quaternionic one-dimensional spinors. To extend to parity the space-time transformations, we combine these one-dimensional spinors into bi-dimensional column vectors. From the transformation properties of the two-component spinors, we derive a quaternionic chiral representation for the space-time algebra. Finally, we obtain a quaternionic bi-dimensional version of the Dirac equation.     

On a Lorentz Non-Invariant Generalization of the Dirac Equation

A generalized Dirac equation is presented as a model theory of disturbed Lorentz invariance. The physical properties of this model and experimental consequences are discussed. A program is described how such Lorentz noninvariant equations may be produced by cosmological induction of local causal structure.     

Quasi-exact solvability of Dirac equation with Lorentz scalar potential

We consider exact/quasi-exact solvability of Dirac equation with a Lorentz scalar potential based on factorizability of the equation. Exactly solvable and sl (2)-based quasi-exactly solvable potentials are discussed separately in Cartesian coordinates for a pure Lorentz potential depending only on one spatial dimension, and in spherical coordinates in the presence of a Dirac monopole.     

Klein Paradox of Two-Dimensional Dirac Electrons in Circular Well Potential

We study two-dimensional massive Dirac equation in circular well potential. The energies of bound states are obtained. We demonstrate the Klein paradox of this relativistic wave equation:For large enough potential depth, the bound states disappear from the spectra. Applications to graphene systems are discussed.     

Connecting Solutions of the Lorentz Force Equation do Exist

Recent results on the maximization of the charged-particle action in a globally hyperbolic spacetime are discussed and generalized. We focus on the maximization of over a given causal homotopy class of curves connecting two causally related events x ₀≤x ₁. Action is proved to admit a maximum on , and also one in the adherence of each timelike homotopy class C. Moreover, the maximum σ ₀ on is timelike if contains a timelike curve (and the degree of differentiability of all the elements is at least C ²). In particular, this last result yields a complete Avez-Seifert type solution to the problem of connectedness through trajectories of charged particles in a globally hyperbolic spacetime endowed with an exact electromagnetic field: fixed any charge-to-mass ratio q/m, any two chronologically related events x ₀≪x ₁ can be connected by means of a timelike solution of the Lorentz force equation corresponding to q/m. The accuracy of the approach is stressed by many examples, including an explicit counterexample (valid for all q/m≠0) in the non-exact case. As a relevant previous step, new properties of the causal path space, causal homotopy classes and cut points on lightlike geodesics are studied. An erratum to this article is available at .     

Dynamics of a Dirac particle in the theory with Lorentz invariance violation

Solutions to the Dirac equations have been obtained for particles interacting with vector, axial-vector, and tensor condensates within the framework of the Standard Model Extension. Possible applications of these solutions for describing the neutrino behavior in dense matter and electromagnetic field are considered.     

A relationship between the Dirac and Maxwell equations

It is shown that an analogue of Dirac's equation may be constructed from Milner's extension of Maxwell's equations and a comparison of the algebraic and physical features of the equations made.     

A Link between Quantum Logic and Categorical Quantum Mechanics

Abramsky and Coecke (Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, pp. 415–425, IEEE Comput. Soc., New York, 2004) have recently introduced an approach to finite dimensional quantum mechanics based on strongly compact closed categories with biproducts. In this note it is shown that the projections of any object A in such a category form an orthoalgebra Proj  A. Sufficient conditions are given to ensure this orthoalgebra is an orthomodular poset. A notion of a preparation for such an object is given by Abramsky and Coecke, and it is shown that each preparation induces a finitely additive map from Proj  A to the unit interval of the semiring of scalars for this category. The tensor product for the category is shown to induce an orthoalgebra bimorphism Proj  A×Proj  B→Proj (A ⊗ B) that shares some of the properties required of a tensor product of orthoalgebras. These results are established in a setting more general than that of strongly compact closed categories. Many are valid in dagger biproduct categories, others require also a symmetric monoidal tensor compatible with the dagger and biproducts. Examples are considered for several familiar strongly compact closed categories.     

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.     

On a relation between Maxwell and Dirac theories

It is shown that the Dirac equation can be written in a form similar to Maxwell equations, where the Maxwell tensor is written as a bilinear expression of the Dirac field and the current is a simple function of the external potential and the Dirac field. Similarly, the Maxwell equations can be written as a self-coupled Dirac equation where the potential is a simple function of the Dirac field itself. It is illustrated by examples how the new formalism helps to find solutions of the coupled field equations.     

A Characterization of Dirac Morphisms

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields. The second author benefited from a one-year fellowship of the Conseil Général du Finistère.     

洛伦兹变换的一种新推导

对于洛伦兹变换中两个惯性系S和S′系的约定提出了新思路,使x轴与x′轴反向,从而使S和S′系完全对称,简化了推导过程;并根据约定,从狭义相对论的两条基本假设出发,严格地推导出了洛伦兹变换式.     

Distinguishing between Dirac and Majorana neutrinos in neutral-current reactions

We calculate cross sections for neutral-current reactions initiated by massive Dirac and Majorana neutrinos and analyze such reactions as a possible new method of distinguishing between these types of neutrinos.     

Distinguishing between Dirac and Majorana neutrinos with two-particle interferometry

Two-particle interferometry, a second-order interference effect, is explored as another possible tool to distinguish between massive Dirac and Majorana neutrinos. A simple theoretical framework is discussed in the context of several gedanken experiments. The method can in principle provide both the mass scale and the quantum nature of the neutrino for a certain class of incoherent left-handed source currents.     

QED effective action in a constant electromagnetic field under conditions of possible Lorentz invariance violation in the Dirac equation

This paper is devoted to the calculation of corrections to QED effective action associated with the axial-vector condensate b ^μ, which violate the Lorentz invariance. It was shown that the linear in b ^μ contribution to the 1-loop effective action (Chern-Simons term) is absent in the case of a constant electromagnetic field. The contribution, which is quadratic in b ^μ, was calculated for the cases of both constant magnetic and electric fields. Asymptotic estimates of the quadratic in b ^μ term for strong and weak field strengths were obtained.     

A discussion of causality and the Lorentz group

Zeeman (1964) has shown that the group of automorphisms for the relation of causality on Minkowski space is that generated by the orthochronous Poincaré (Halpern, 1968) group and dilatations. Here we prove that the group of automorphisms that preserve the time-like vectors of Minkowski space normwise is the complete Poincaré group. We prove that the timelike structure within the null cone of a single event does define the whole structure of Minkowski space. Further, it is shown that only inertial observers can use Minkowski space to describe space-time.     

Link between KAM tori and nearby cycles

In the discussion of GUTS in the early universe it is assumed that there exists a locally inertial space-time region large enough to contain a sufficient number of particles justifying the use of flat space statistical mechanics. We show that this assumption is false.