130_notes.dvi

backward in time) into the conventional positive exponent solution ifwe change the charge to +e.
We can interpret solutions 3 and 4 as positrons. We will make this switch more carefully when we
study the charge conjugation operator.

The Dirac equation should be invariant under Lorentz boosts and under rotations, both of which are
just changes in the definition of an inertial coordinate system. Under Lorentz boosts,∂x∂μtransforms
like a 4-vector but theγμmatrices are constant. The Dirac equation is shown to beinvariant under
boostsalong thexidirection if we transform the Dirac spinor according to

ψ′ = Sboostψ

Sboost = cosh

χ

2

+iγiγ 4 sinh

χ

2

with tanhχ=β.

TheDirac equation is invariant under rotationsabout thekaxis if we transform the Dirac
spinor according to

ψ′ = Srotψ

Srot = cos

θ

2

+γiγjsin

θ

2

withijkis a cyclic permutation.

Another symmetry related to the choice of coordinate system is parity. Under aparity inversion
operationthe Dirac equation remains invariant if

ψ′=SPψ=γ 4 ψ

Sinceγ 4 =







1 0 0 0

0 1 0 0

0 0 −1 0

0 0 0 − 1





, the third and fourth components of the spinor change sign while

the first two don’t. Since we could have chosen−γ 4 , all we know is thatcomponents 3 and 4
have the opposite parity of components 1 and 2.

From 4 by 4 matrices, we may derive 16 independent components of covariant objects. Wedefine
the product of all gamma matrices.

γ 5 =γ 1 γ 2 γ 3 γ 4

which obviouslyanticommuteswith all the gamma matrices.

{γμ,γ 5 }= 0

For rotations and boosts,γ 5 commutes withSsince it commutes with the pair of gamma matrices.
For a parity inversion, it anticommutes withSP=γ 4.

The simplest set of covariants we can make from Dirac spinors andγmatrices are tabulated below.