The Dirac equation provides a description of a spin-1/2 particle, such as electrons and neutrinos, consistent with quantum mechanical and relativistic principles [1]. In 1928, Hermann Weyl introduced a generalization of Riemannian geometry in an attempt to formulate a unified field theory. He proposed a two-component equation to describe massless spin-1/2 particles, which violates parity invariance. For this reason, the motion of the neutrino is described by the Weyl-Dirac (WD) equation [2]. The structure of Weyl's geometry is characterized by the symmetric tensor density g_μν and the pseudovector φ_μ (where it is reasonable to assume that the g_μν represent the gravity field and the φ_μ are the components of the vector potential). In contrast to Poincaré group, the spinorial representation of the Lorentz group can be reduced if a couple of projection operators are defined. The massless condition makes reducible the representation of the Poincaré group since the mass term in the Lagrangian density is not invariant under two separate Lorentz transformations [3]. Dirac equation (as well as WD equation) predicts the so called Klein's paradox (the dispersion of a Dirac electron is nearly transparent when the barrier is of the order of the electron mass). When Dirac equation is written in a curved-Weyl, some consequences arise such as the minimal coupling of the electron and the electromagnetic field and the Klein's paradox holds [4]. When the 2D honeycomb lattice of graphene is studied, its band structure is given by an effective mass description (or k.p approximation) describing the states in the vecinity of the first brillouin zone points (Dirac points). The effective Hamiltonian leads to Weyl-Dirac naturally from Schrödinger equation (Tight-Binding approximation). Therefore, the charge carriers of Graphene (as well as C-Nanotubes) behave like WD electrons near the Dirac points, so coherent transport is possible in graphene, due to the lack of back-scattering of impurities. The predictions of the WD particles have been observed from outstanding experiments in graphene as well as relativistic superconductivity. The purpose of the present lecture is to review how the predictions of the relativistic quantum mechanics in curved spaces can be observed (and measured) in solid state systems.
References
[1] J.J. Sakurai. Advanced Quantum Mechanics. 1967.
[2] W. Greiner. Relativistic Quantum Mechanics. 3rd Ed. Springer, 1987.
[3] M. Kaku, Quantum Field Theory (A modern Introduction). Oxford Un. Press. 1993.
[4] T. Ando, J. Phys. Soc. Jpn. 74, 777 (2005)
[5] H. Weyl, Zeits. f. Physik 56, 330 (1929)
[6] M. Novello. Dirac's Equation In a Weyl Space. Cen. Bra. Pes. Fis. LXIV, N4. 1969
[7] A. Castro, F. Guinea, N. Peres, K. Novoselov, and A. Geim, Rev. Mod. Phys. 81, 109 (2009). 1