Chapter 47

The Dirac Equation and

Spin

1

2

Fields

The space of solutions to the Klein-Gordon equation gives an irreducible repre-
sentation of the Poincar ́e group corresponding to a relativistic particle of mass
mand spin zero. Elementary matter particles (quarks and leptons) are spin^12
particles, and we would like to have a relativistic wave equation that describes
them, suitable for building a quantum field theory.
This is provided by a remarkable construction that uses the Clifford algebra
and its action on spinors to find a square root of the Klein-Gordon equation,
the Dirac equation. We will begin with the case of real-valued spinor fields,
for which the quantum field theory describes spin^12 neutral massive relativistic
fermions, known as Majorana fermions. In the massless case it turns out that the
Dirac equation decouples into two separate equations for two-component com-
plex fields, the Weyl equations, and quantization leads to a relativistic theory
of massless helicity±^12 particles which can carry a charge, the Weyl fermions.
Pairs of Weyl fermions of opposite helicity can be coupled together to form a the-
ory of charged, massive, spin^12 particles, the Dirac fermions. In the low energy,
non-relativistic limit, the Dirac fermion theory becomes the Pauli-Schr ̈odinger
theory discussed in chapter 34.

47.1 The Dirac equation in Minkowski space

Recall from section 34.4 that for any real vector spaceRr+swith an inner
product of signature (r,s) we can use the Clifford algebra Cliff(r,s) to define a
first-order differential operator, the Dirac operator∂/. For the Minkowski space
case of signature (3,1), the Clifford algebra Cliff(3,1) is generated by elements
γ 0 ,γ 1 ,γ 2 ,γ 3 satisfying

γ 02 =− 1 , γ^21 =γ 22 =γ^23 = +1, γjγk+γkγj= 0 forj 6 =k