QuantumPhysics.dvi

22 The Dirac Field and the Dirac Equation

Among the relativistic wave equations, we have not yet obtained one suitable for spin 1/2, doing
so is the object of the present chapter. From the preceding discussion, there are 3 cases,
(
1
2

, 0

)

left Weyl spinor

(

0 ,

1

2

)

right Weyl spinor

(

1

2

, 0

)

⊕

(

0 ,

1

2

)

Dirac spinor (22.1)

(The case of a Majorana spinor will turn out to be equivalent to that of a Weyl spinor, and will
not be discussed further here.) It is actually convenient totreat all cases at once by concentrating
on the reducible case of the Dirac spinor. Historically, this is also what Dirac did.

The representation (^12 ,0) is well-known: it can be obtained in terms of the Pauli matrices.
Similarly, the case of the representation (0,^12 ) may be given in terms of the Pauli matrices as well;
(here and below,k= 1, 2 ,3),

DL(Ak) =

σk

2

DL(Bk) = 0

DR(Ak) = 0 DL(Bk) =

σk

2

(22.2)

Of course,DLandDRact on two different two-dimensional representation spaces,and so do the
corresponding Pauli matrices. To make this crystal clear, it is preferable to work on the direct sum
representationD=DL⊕DRof the two spinors, so that the representation matrices are given as
follows,

D(Ak) =

( 1

2 σk^0

0 0

)

D(Bk) =

(

0 0

0 12 σk

)

(22.3)

or in terms of the original rotation and boost generators,

D(Jk) =

( 1

2 σk^0

0 12 σk

)

D(Kk) =

( 1

2 σk^0

0 −^12 σk

)

(22.4)

It is not `a priori so easy to write down a Lorentz-covariant equation because the generators are
not labeled by Lorentz vector indices (such as tensors were), but rather by a novel type of spinor
index. To circumvent these problems, we introduce the Diracmatrices.

22.1 The Dirac-Clifford algebra

One defines the Dirac-Clifford algebra in space-time dimension 4 as an algebra of Dirac matrices
γμ, whereμ= 0, 1 , 2 ,3, which satisfy the following defining equation,

{γμ,γν}=γμγν+γνγμ= 2ημνI (22.5)