1.1.3 Massless Dirac Particles

For massless Dirac particles, m=0, the Dirac equation(1.18)turns to be

which gives rise to a set of two matrix equations

You can rewrite them as

This means that the Dirac equation for the four-component wavefunction is now reduced to a decoupled two-component one actually. Let us define the helicity operator

You can see that the energy eigenstates are also the eigenstates of helicity. The helicity eigenstates with h=+1 (spin parallel to the momentum p) are called right-handed, and those with h=-1 (spin antiparallel to p) are called left-handed. Thus, the right-handed state, ψR= (φ+χ) corresponds to engenenergy E, while the left-handed state, ψL= (φ-χ) corresponds to -E. In other words, if a massless particle is left-handed, its anti-particle will be right-handed, and vice versa.

When m=0, the dispersion relation in the left panel of Fig.1.1 reduces to the one in the right panel. The linear dispersion relation was related to the neutrino problems in high energy physics once upon times, and is related to graphene problems in condensed matter physics nowadays. Grapheneis an atomic-scale honeycomb lattice made of carbon atoms, which is illustrated in Fig.1.3.

Differing from most traditional three-dimensional materials, graphene is a zero-gap semiconductor due to its special band structure(see Fig.1.4).

Such a band structure for the honeycomb lattice was knownearlier in 1984. Near the six corners of the two-dimensional hexagonal Brillouin zone, you can perceive a linear energy-momentum relation that leads to zero effective mass for electrons and holes. Due to this linear(or“conical”) dispersion relation at low energies, electrons and holes near these six points behave like“relativistic”massless particles described by the Dirac equation (1.33), in which the speed of light c should be replaced by the Fermi velocity vF. The massless quasiparticles in graphene have been observed in experimentrecently.

As the most promising application, graphene is expected to replace silicon chips since that technology is now fast reaching its fundamental limits(bellow 10 nanometers). It is now possible to create graphene wafers for electronic applications by growing a single layer of graphene on top of crystals with a matching substrate. Graphene is also a potential material to use in high-frequency transistors in the terahertz regime, or to build miniature printed circuit boards at the nanoscale.

Exercises

1. Given that α × α, check the following commutation relation fulfills

2. Show the following operator identities

where f=f(r).