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The Dirac equation is () = If the Dirac equation is to be covariant, then it should have exactly the same form in all Lorentz frames: ′ ′ (′) ′ (′) = The two spinors and ′ should both describe the same physical field, and so should be related by a transformation that does not change any physical observables (charge, current, mass, etc.
An alternative version of the Dirac equation whose Dirac operator remains the square root of the Laplacian is given by the Dirac–Kähler equation; the price to pay is the loss of Lorentz invariance in curved spacetime. Note that here Latin indices denote the "Lorentzian" vierbein labels while Greek indices denote manifold coordinate indices.
The name of Dirac cone comes from the Dirac equation that can describe relativistic particles in quantum mechanics, proposed by Paul Dirac. Isotropic Dirac cones in graphene were first predicted by P. R. Wallace in 1947 [6] and experimentally observed by the Nobel Prize laureates Andre Geim and Konstantin Novoselov in 2005. [7]
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It was first derived in 1928 by Oskar Klein and Yoshio Nishina, constituting one of the first successful applications of the Dirac equation. [1] The formula describes both the Thomson scattering of low energy photons (e.g. visible light ) and the Compton scattering of high energy photons (e.g. x-rays and gamma-rays ), showing that the total ...
The above matrix operator contracts with one bispinor index of ψ at a time (see matrix multiplication), so some properties of the Dirac equation also apply to the BW equations: the equations are Lorentz covariant, all components of the solutions ψ also satisfy the Klein–Gordon equation, and hence fulfill the relativistic energy–momentum ...
In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos.It appears in the plane-wave solution to the Dirac equation, and is a certain combination of two Weyl spinors, specifically, a bispinor that transforms "spinorially" under the action of the Lorentz group.
This anomalous behavior is due to graphene's massless Dirac electrons. In a magnetic field, these electrons form a Landau level at the Dirac point with an energy that is precisely zero. This is a result of the Atiyah–Singer index theorem index theorem and causes the "+1/2" term in the Hall conductivity for neutral graphene. [4] [47]