Dirac-Hestenes spinor fields in Riemann-Cartan spacetime

In this paper we study Dirac-Hestenes spinor fields (DHSF) on a four-dimensional Riemann-Cartan spacetime (RCST). We prove that these fields must be defined as certain equivalence classes of even sections of the Clifford bundle (over the RCST), thereby being certain particular sections of a new bundle named Spin-Clifford bundle (SCB). The conditions for the existence of the SCB are studied and are shown to be equivalent to the famous Geroch's theorem concerning to the existence of spinor structures in a Lorentzian spacetime. We introduce also the covariant and algebraic Dirac spinor fields and compare these with DHSF, showing that all the three kinds of spinor fields contain the same mathematical and physical information. We clarify also the notion of (Crumeyrolle's) amorphous spinors (Dirac-K\"ahler spinor fields are of this type), showing that they cannot be used to describe fermionic fields. We develop a rigorous theory for the covariant derivatives of Clifford fields (sections of the Clifford bundle (CB)) and of Dirac-Hestenes spinor fields. We show how to generalize the original Dirac-Hestenes equation in Minkowski spacetime for the case of a RCST. Our results are obtained from a variational principle formulated through the multiform derivative approach to Lagrangian field theory in the Clifford bundle.Comment: 45 pages, special macros kapproc.sty and makro822.te

Fractional Schrödinger Operator With Delta Potential Localized On Circle

We consider a system governed by the fractional Schödinger operator with a delta potential supported by a circle in R 2. We find out the function counting the number of bound states, in particular, we give the necessary and sufficient conditions for the absence of bound state in our system. Furthermore, we reproduce the form of eigenfunctions and analyze the asymptotic behavior of eigenvalues for the strong coupling constant case. © 2012 American Institute of Physics.533Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H., (2004) Solvable Models in Quantum Mechanics, , 2nd ed. (with Appendix by P. Exner), (American Mathematical Society, Providence, RI)Bandrowski, B., Karczewska, A., Rozmej, P., Numerical solutions to integral equations equivalent to differential equations with fractional time derivative (2010) Int. J. Appl. Math Comput. Sci., 20 (2), pp. 261-269. , 10.2478/v10006-010-0019-1Bandrowski, B., Rozmej, P., On fractional Schrödinger equation (2010) Comput. Methods Sci. Technol., 16 (2), pp. 191-194. , http://www.man.poznan.pl/cmst/2010/_V_16_2/14_Rozmej_G.pdfBraaksma, B.L.J., Asymptotic expansions and analytic continuations for a class of Barnes-integrals (1962) Compos. Math., 15, pp. 239-341. , http://archive.numdam.org/ARCHIVE/CM/CM_1962-1964__15_/CM_1962-1964__15__239_0/CM_1962-1964__15__239_0.pdfCapelas de Oliveira, E., Silva Costa, F., Vaz, J., The fractional Schödinger operator equation for delta potentials (2010) J. Math. Phys., 51, p. 123517. , 10.1063/1.3525976Capelas de Oliveira, E., Vaz, J., Tunneling in fractional quantum mechanics (2011) J. Phys. A: Math. Theor., 44, p. 185303. , 10.1088/1751-8113/44/18/185303Exner, P., Ichinose, T., Geometrically induced spectrum in curved leaky wires (2001) J. Phys. A, 34, pp. 1439-1450. , 10.1088/0305-4470/34/7/315Exner, P., Kondej, S., Curvature-induced bound states for a δ interaction supported by a curve in (2002) Ann. Henri Poincaré, 3, pp. 967-981. , 10.1007/s00023-002-8644-3Exner, P., Kondej, S., Bound states due to a strong delta interaction supported by a curved surface (2003) J. Phys. A, 36, pp. 443-457. , 10.1088/0305-4470/36/2/311Exner, P., Tater, M., Spectra of soft ring graphs (2004) Waves Random Complex MediaMedia, 14, pp. S47-60. , 10.1088/0959-7174/14/1/010Guo, X., Xu, M., Some physical applications of fractional Schrr̈odinger equation (2006) J. Math. Phys., 47, p. 082104. , 10.1063/1.2235026Gradshteyn, I.S., Ryzhik, I.M., (2007) Table of Integrals, Series, and Products, , 7th ed., (Academic, New York)Jeng, M., Xu, S.-L.-Y., Hawkins, E., Schwarz, J.M., On the nonlocality of the fractional Schrödinger equation (2010) J. Math. Phys., 51, p. 062102. , 10.1063/1.3430552Dong, J., Xu, M., Some solutions to the space fractional Schrödinger equation using momentum representation method (2007) J. Math. Phys., 48, p. 072105. , 10.1063/1.2749172Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., (2006) Theory and Applications of Fractional Differential Equations, , (Elsevier, Amsterdam)Laskin, N., Fractional quantum mechanics and Lévy path integrals (2000) Phys. Lett. A, 268, pp. 298-305. , 10.1016/S0375-9601(00)00201-2Laskin, N., Fractional quantum mechanics (2000) Phys. Rev. E, 62, pp. 3135-3145. , 10.1103/PhysRevE.62.3135Laskin, N., Fractal and quantum mechanics (2000) Chaos, 10, pp. 780-790. , 10.1063/1.1050284Mathai, A.M., Saxena, R.K., Haubold, H.J., (2009) The H-Function, , (Springer, New York)Naber, M., Time fractional Schrödinger equation (2004) J. Math. Phys., 45, pp. 3339-3352. , 10.1063/1.1769611Oberhetting, F., (1974) Tables of Mellin Transforms, , (Springer, New York)Reed, M., Simon, B., (1975) Methods of Modern Mathematical Physics. II. Fourier Analysis, , (Academic, New York)Posilicano, A., A Krein-like formula for singular perturbations of self-adjoint operators and applications (2001) J. Funct. Anal., 183, pp. 109-147. , 10.1006/jfan.2000.3730Stollmann, P., Voigt, J., Perturbation of Dirichlet forms by measures (1996) Potential Anal., 5, pp. 109-138. , 10.1007/BF0039677

The Mass Spectrum of Neutrinos

In a previous paper we showed that Weyl equation possess superluminal solutions and moreover we showed that those solutions that are eigenstates of the parity operator seem to describe a coupled pair of a monopole anti-monopole system. This result suggests to look for a solution of Maxwell equation \partialF^{\infty}=-gJ with a current J as source and such that the Lorentz force on the current is null. We first identify a solution where J={\gamma}^{5}J_{m}is a spacelike field (even if F is not a superluminal solution of the homogeneous Maxwell equation). More surprisingly we find that there exists a solution F of the free Maxwell \partialF=0 that is equivalent to the non homogeneous equation for F^{\infty}. Once this result is proved it suggests by itself to look for more general subluminal and superluminal solutions F of the free Maxwell equation equivalent to a non homogeneous Maxwell equation for a field F_{0} with a current term as source which may be subluminal or superluminal. We exhibit one such subluminal solution, for which the Dirac-Hestenes spinor field {\psi} associated the electromagnetic field F_{0} satisfies a Dirac equation for a bradyonic neutrino under the ansatz that the current is ce^{{\lambda}{\gamma}^{5}}g{\psi}{\gamma}^{0}{\psi}, with g the quantum of magnetic charge and {\lambda} a constant to be determined in such a way that the auto-force be null. Together with Dirac's quantization condition this gives a quantized mass spectrum (Eq.49) for the neutrinos, with the masses of the different flavor neutrinos being of the same order of magnitude (Eq.50) which is in accord with recent experimental findings. As a last surprise we show that the mass spectrum found in the previous case continues to hold if the current is taken spacelike, i.e., ce^{{\lambda}{\gamma}^{5}}g{\psi}_{>}{\gamma}^{3}{\psi}_{>} with {\psi}_{>}, in this case, satisfying a tachyonic Dirac-Hestenes equation.Comment: This version corrects some misprints, has a new remark and a new referenc

Revisiting special relativity: A natural algebraic alternative to Minkowski spacetime

Minkowski famously introduced the concept of a space-time continuum in 1908, merging the three dimensions of space with an imaginary time dimension $i c t$, with the unit imaginary producing the correct spacetime distance $x^2 - c^2 t^2$, and the results of Einstein's then recently developed theory of special relativity, thus providing an explanation for Einstein's theory in terms of the structure of space and time. As an alternative to a planar Minkowski space-time of two space dimensions and one time dimension, we replace the unit imaginary $i = \sqrt{-1}$, with the Clifford bivector $\iota = e_1 e_2$ for the plane that also squares to minus one, but which can be included without the addition of an extra dimension, as it is an integral part of the real Cartesian plane with the orthonormal basis $e_1$ and $e_2$. We find that with this model of planar spacetime, using a two-dimensional Clifford multivector, the spacetime metric and the Lorentz transformations follow immediately as properties of the algebra. This also leads to momentum and energy being represented as components of a multivector and we give a new efficient derivation of Compton's scattering formula, and a simple formulation of Dirac's and Maxwell's equations. Based on the mathematical structure of the multivector, we produce a semi-classical model of massive particles, which can then be viewed as the origin of the Minkowski spacetime structure and thus a deeper explanation for relativistic effects. We also find a new perspective on the nature of time, which is now given a precise mathematical definition as the bivector of the plane.Comment: 29 pages, 2 figure