## The Dirac Equation

In this chapter we continue the discussion of relativistic equations for the first quantization of particles. The Klein-Gordon equation introduced in the last chapter describes spin zero particles. In this chapter we discuss the Dirac equation [Di 28], which describes particles with the two internal degrees of freedom characteristic of a spin \ particle. Since both electrons and quarks have spin the Dirac equation has many interesting applications, and some of these will be developed in the next chapter.

5.1 THE EQUATION

vs discussed in Sec. 4.5, the two-component form of the KG equation could be written

While this equation is first order in the time derivative, the KG Hamiltonian (4.38) is second order in the space derivatives and hence does not treat space and time in an equivalent fashion. Furthermore, because the conserved norm for the KG theory was not positive definite, the two-component KG "Hamiltonian" is not Hermitian. Finally, the covariance of the KG equation is only manifest in its original, one-component form. It is natural to ask: "Is there a relativistic equation which is first order in time, treats space and time in a manifestly symmetric fashion, has a positive definite conserved norm (implying that H is Hermitian), and is manifestly covariant?" The investigation of this question leads directly to the Dirac equation.

To answer this question, we look for an equation which is first order in both space and time and which is Hermitian. The equation must have the form i-r-ip = H ip .

dt dt

where a and ยก3 are Hermitian matrices and p = -iV. The relativistic energy momentum relation should emerge naturally, so we require d2

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