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This document summarizes Juan M Guerra's presentation on the Weyl-Dirac equation in condensed matter physics and graphene. It discusses Dirac's original equation, the Dirac and Weyl Lagrangians, the massless Weyl equation and its two uncoupled components. It also covers Weyl's gauge theory formulation, the Klein paradox of chiral tunneling, applications to graphene and nanotubes where electrons behave as Dirac particles, and ongoing research on electric transport in graphene nanostructures.
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Wd equation
- 1. Escuela de Física-Matemática 2011 Universidad de los andes Weyl-Dirac Equation in CondensedMatterPhysics: Graphene Juan M Guerra Universidad Nacional de Colombia Mayo, 2011.
- 2. Contents Dirac equation Dirac Lagrangian and Transformations Massless Weyl Equation Weyl’sFormulation Klein Paradox(Chiral Tunneling) Graphene and Nanotubes Research References
- 3. Dirac Equation (1928) Covariantform, Invarianceunder a Lorentztransformation* 4-component spinor. Decoupledwhen m=0.
- 4. Dirac Lagrangian and Transformation Mass: Flux of thegrav. Fieldthrough a surfaceenclosingtheparticle if DE retainsitsLorentz-invariantformwhen: Spin term (Gen. Of theLorentzgroup) Finitemassparticle
- 5. Massless Weyl Equation Setting m=0 in DE Weyl Eq. 4-comp. Parity 2-comp. Maximalparityviolation (x2 eq). Invariantunderthe global (chirality) transformation: Twouncoupled Weyl equations (reducedrepresentation). In momentum space => Chirality.
- 6. Weyl’sformulation(1) Weyl gauge theory: Invariance and rescaling. Conformal vs. Weyl gauge symmetry. Weyl-Cartangeometry: Thelength of a vector has no absolutegeometricmeaning. Conf. Transformation in Riemannspace: Themappingisconformalif
- 7. Weyl’sformulation(2) GroupWr in V4 (Riemannspace) isdefinedby: Weyl (conformal) rescalings of themetric and transf. Allotherdynamical variables w:= Weyl dimensionof thefield.
- 8. Klein Paradox (Chiral Tunneling) “…Ifthepotentialisontheorder of theelectronmass, thebarrierisnearlytransparent.” RQM can beconsistentlyformulatedonly in terms of fieldsratherthan individual particles.
- 10. Graphene and Nanotubes (2) Electronicstructure of graphene: Dirac points.
- 11. Graphene and Nanotubes (3) New bridge betweencondensedmatterphysics and QED. Dirac-likeelectrons => honecombstructure. Quasiparticles in graphene Pseudospin degree of freedom (Sublattice) e- and e+ (holes) are interconnected. Coherenttransport in graphene (impuritiesdon’tscatter).
- 12. Research Electric transportthrough C-nanotubes and graphene nanoribbons double quantum dotscoupledby a superconductor. Non-equilibriumGreen’sfunctions.
- 13. References [1] P.A.M. Dirac, The quantum theory of theelectron. 1928. [2] E. Marshak, Conceptual foundations of modernparticlephysics. 1993. [3] M. Blagojevic, Gravitation and Gauge symmetries. 2002. [4] H. Weyl, Gravitation and theelectron. 1929. [5] M. Katsnelson, K. Novoselov, Graphene: New bridge betweencond. Matt. Phys. And QED. 2007. [6] Castro et al. Theelectronicproperties of graphene. 2007.
- 14. Tks!